Quantum dots
7
Quantum dots
Quantum dots, popularly known as "artificial atoms" i, where the confinement potential replaces the potential of the nucleus [I], [30][R7], are fascinating objects. On one hand, these systems are thought to have vast potential for future technological applications, such as possible applications in memory chips [i0], quantum computation [3136], quantum cryptography [37], in roomtemperature quantumdot lasers [38], and so on. But the fundamental physical concepts we have learned from these systems are no less enticing. We shall discuss many of those basic concepts in this review. Some examples of those concepts are: magic numbers in the ground state angular momentum, the spin singlettriplet transition, the socalled generalized Kohn theorem [i], [30][R7], [3941], and its implications, shell structure, singleelectron charging, diamond diagrams, etc. In the last few years we have witnessed a profusion of new results and ideas in quantumconfined zerodimensional electron systems. Experimental advances in fabricating quantum dots and precise measurements of various electronic and optical properties have generated an exciting situation both for the theoreticians and experimentalists. As we shall demonstrate in this review, there have been several interesting developments where the theoretical predictions and experimental surprises have resulted in deeper understanding of these systems. In our review of the properties of quantum dots we shall mostly concern ourselves about the case where an external perpendicular magnetic field i Interestingly, as far as we know, this popular name was introduced in the literature by Maksym and Chakraborty [I], [30][R7]. One other appropriate name "designer atoms" was introduced by Reed [4]. There are, however, significant differences between quantum dots (QDs) and real atoms: QDs are larger than atoms and number of electrons in the dot can be independent of the size of the dot.
8
T. Chakraborty
0.3
el
PS
~o0 I
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o 1
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i  
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i I
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el oO
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ol
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,, /
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=
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,,
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ol
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el
".  .
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e0
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e0 (a)
(b)
(c)
(d)
(e)
F i g u r e 2 Energy levels of two electrons in a long, narrow box (Ly = 10Lx). The energy spacings are measured relative to the ground state energy scaled by Rs i.e., AE/Rs for (a) noninteracting electrons Ly = 20a0, Eg = 1.905Rs, Rs = 105 x Re, Re is the effective Rydberg and ao is the Bohr radius. In the case of interacting electrons the parameters, Ly/ao, Eg/Rs, and Rs/Re are respectively, (b) 20, 1.907, 105, (c) 200, 1.923, 103; (d) 2000, 1.987, 10; and (e) 2 x 104, 2.314, 0.1. The parity and total spin of each state are also indicated [45] [R4].
is present [30] [R7], [42]. However, our discussion of the electron correlation effects would be incomplete if we did not mention the important work done on q u a n t u m dots in the absence of a magnetic field One of the first reports of threedimensional quantum confinement in semiconductor nanostructures suitable for measurement of excitation spectra of q u a n t u m dots was by Reed et a l [43][R1], [44] They studied vertical transport in q u a n t u m dot structures realized by etching narrow columns into heterostructures. W h e n the lateral dimension was made sufficiently small, i e , when full threedimensional confinement was achieved,
Q u a n t u m dots
9
the measured currentvoltage characteristics showed a series of peak structures which were attributed to resonant tunneling through the discrete zerodimensional states in the dot. Electron correlation effects in quantum dots in the absence of a magnetic field were first studied theoretically by Bryant [45][R4]. He considered a twoelectron system in a rectangular box with hardwall potentials and studied the interplay of kinetic and interaction energies as a function of the size of the box. In the infinitebarrier model, the kinetic energy matrix elements scale as 1/L 2 where L is the linear dimension of the quantumwell box. While the interaction matrix elements, scale as 1/L when L is changed without changing the box shape. For small L, the electrons behave as independent, uncorrelated particles because the Coulomb interaction is insignificant compared to singleparticle level spacings. However, when L increases the interactions become important and the level spacings change. Electron correlations help electrons form a Wigner crystal. The signature of this state in a confined system is the degeneracy of the levels. In a long, narrow box the evolution of the states into the levels with degeneracies of the Wigner lattice happens for L _> 0.1 #m (Fig. 2). These results show that there is a continuous evolution of the energy level structure, from the singleparticlelike states in the limit of a small dot, to a level structure in larger dots where electronelectron interactions are dominant. We should point out here that this was the very first report on the importance of electron correlations in mesoscopic systems. Macucci et al. [46] used density functional theory to investigate the electronic structure of quantum dots in the absence of a magnetic field. For the exchangecorrelation terms they used the polynomial representations given in the calculations of Tanatar and Ceperley [47] of a twodimensional electron gas. The numerical studies of Macucci et al. were done mainly for the chemical potential and the differential capacitance of the dot as a function of the dot size and the number of electrons. It was found that there is a gradual transition from the state where the quantum effects are dominant (in very small dots where the quantization energy dominates over the Coulomb energy) to an almost classical capacitorlike behavior of large dots. Tarucha et al. [4850], [51][R33] analyzed the electronic states of a fewelectron vertical quantum dot 2. Their vertical dots were disks with diameter about 10 times the thickness. The lateral potential had a cylindrical symmetry with a soft boundary profile that can be approximated by a harmonic potential. These authors noticed that at zero magnetic field, the Coulomb oscillation is irregular in period reflecting a shell structure associated with a twodimensional harmonic potential. At low fields, they observed antiparallel filling of spindegenerate states. Close to zero magnetic field, they noticed the 2 For details on vertical dots and the work of Tarucha et al., see Sect. 2.4.6.
10
T. Chakraborty
filling of states with parallel spins in accordance with Hund's rule. According to this rule degenerate states in a shell are filled first with parallel spins up to a point where the shell is half filled. Halffilled shells correspond to a maximum spin state t h a t has relatively low energy due to exchange interactions. Observation of the shell filling has prompted several theoretical investigations [5256] to find the underlying physical reasons for that effect. In the absence of an external magnetic field, Fujito et al. [57] calculated the total energy, chemical potential, capacitance, and conductance peak positions for anisotropic parabolic quantum dots. Here the Hamiltonian is written (in effective atomic units) N
%t =
N
i
i=1 1
1 2 ~Vi +  ~ 2 (x~ + y~) + ~zZ~
=
t(ri)
1
~t(ri)+Erij 2
1
with an anisotropic parabolic potential with the cylindrical symmetry of characteristic _1
_!
frequencies ~x and ~z corresponding to oscillator lengths fx = ~x 2 and gz  ~z 2 respectively. The manyelectron ground states were obtained by means of an unrestricted HartreeFock method. The ground state was found to be ferromagnetic up to 12 electrons in large dots (gx  49.5 nm). For small dots (gx = 7.425 nm), oneelectron levels calculated selfconsistently are occupied in accordance with Hund's rule. The differential capacitance c
(N) 


where p ( N )  E ( N )  E ( N  1) is the chemical potential and E(N) the total energy of the ground state of N electrons in a dot, shows a characteristic oscillation with electron numbers. Local maxima or minima of capacitance reflect the shell structure. Shell structure was also seen in the chemical potential and conductance peaks versus the gate voltage. The latter was calculated as follows: a tunneling current flows through the dot between two leads with a small bias voltage when the chemical potential of the dot aligns with that in the leads. The conductance peaks appear at the gate voltages V
(N) 
where ~L is the chemical potential of the leads. More on these topics will be discussed in Sect. 2.4.1. In an interesting paper, Kumar et al. [58][R8] considered a model of a single quantum dot of 300 nm square area. They solved the SchrSdinger equation (using Hartree approximation, i.e., ignoring exchange and correlation effects) and Poisson equations selfconsistently. In the absence of an external magnetic field they found that although the
Quantum
dots
11
0.35
1
1
5 0 0 nm DOT
0.30
0.25 :::k
0.20
0.15
L
0.15
_
1
0.20
. . . . . .
1
0.25
1
0.30
0.35
x (~ m) Figure 3 Lateral potential contours of a quantum dot in a plane below the GaAs/AIGaAs interface. The innermost contour is 15 meV below the Fermi level, which is indicated in the figure [58][R8].
dot geometry was square, the lateral potential had nearly circular symmetry (the angular momentum was approximately a good quantum number). Further, the effective size of the quantum dot (with a diameter of ~ I00 nm) was considerably smaller than the geometrical size of the structure. The calculated quantum dot potential by Kumar et al. is shown in Fig. 3 in a plane 8 nm below the interface of a square, gated GaAs/AiGaAs clot similar to what was experimentally investigated by Hansen et al. [59][R2]. Clearly, the potential contours are nearly circular, particularly at low energies. These authors studied the evolution of energy levels in the presence of an external perpendicular magnetic field. The calculated energy levels for seven electrons per dot showed good qualitative agreement with the results for states in a twodimensional harmonic oscillator potential in a magnetic field, as discussed below.
T. Chakraborty
12
2.1
Oneelectron
systems
The problem of a single ideally twodimensional electron in a circular dot, confined by a parabolic potential grnl*a;02r2 (m* is the electron effective mass) in the presence of an external magnetic field was solved more than half a century ago by Fock [60][R5] (and later by Darwin [61][R6]). It is interesting to note that the same problem (but for zero confinement potential) was studied two years after Fock's work by Landau [62] leading to the term Landau levels. We shall call the energy levels derived below as FockDarwin levels (FDL)[1].
2.1.I Energy spectrum Following the classic work of Fock [60] [R5] the Hamiltonian is written as 7{1( 2m*
e ) 2 ira, p  A + g w02 (x 2 + y2). c
(1By, l~Bx, 0)
If we choose the symmetric gauge vector potential, A = the Schr6dinger equation is written as 2m* h 2 { O~r 02~2 +  r1 0 ~rr
002
1
iz~
e B2r 2
h2 ( di f l df ~ ~ r dr
12 ) r2f
+
Defining the cyclotron frequency c~c ! ~
Sin* c 2
\
i +~m
,~2or2 E)r /
The radial part of the equation is then
2m*
g0 
and e = [e[,
2m*cieBhO~ O0
~. r 1 .2 0 2 . ~ .}
+ Let, ~  ~ f ( r ) e VzTr
(2.1)
ebb ~2m* l cf 8m,c 2 + ~
lm*w2or2E
e B / m * c  h/m*g~,
f=0.
and the magnetic length,
, the radial equation is simplified to
h2
Let us define b 
ft'i' f, (1 +
12f) Jr[.E
2 c) 2 89 4c%/c~
177~*
+ 40202) r 2 +
89
and a new independent variable
] S=0.
(2 .2)
Quantum dots
13
X 
m*wcbr2 _ ar 2 = br2 2h 2~ "
Then the radial equation Eq. (2.2) transforms into
x f , + f~
li f ( E  4z + hwcb
1 1~1 4z + ~ fO.
11 A further simplification of the radial equation follows if we define 3  E/hwcb + ~~,
z f " + f' + / 3   g z  ~
f O.
(2.3)
The singleelectron eigenenergies are then obtained from the solution of Eq. E~l
=
lhwc[b(2n+lll+l)l]
=
(2~ + IZl + 1)(X1 h 2

(2~ + IZI + 1 ) h f t  llhwc,
2
h2
1 _1
(2.3) [20]
hWcl
(2.4)
1 2 where ~2 = (XWc + w~). Here the two quantum numbers are, n = 0, 1,..., the radial quantum number, and 1 = 0, • the azimuthal quantum number. The energy spectrum is displayed in Fig. 4 for hw0 = 4 meV. Some interesting properties of the energy spectrum are immediately obvious: At B = O, Wc = 0 and
E~z  (2~ + IZl + 1)hwo. Similarly, for large B, Wc >> wo, n0,
l_>0,
E
1 ~hw~
nl,
l>_0,
E~
3 ~a2c '
...,
which means that, in the absence of a confinement potential, energies of the positive 1 states are independent of l, but in its presence they increase with I. States with angular momentum 1 < 0 have much higher energies. At low values of the magnetic field such that the magnetic length is larger than or comparable to the size of the confinement potential, there is a hybridization of Landau levels with the levels that arise from spatial confinement. With increasing magnetic field, as the magnetic length becomes much smaller than the radius of the confinement potential, freeelectron behavior (Landautype levels) prevails over spatial confinement. Therefore, one sees a gradual transition from spatial to magnetic quantization that depends on the relative size of the quantum dots as compared to the magnetic length. As an example, if the electrons are confined in a 100~ dot, Landau levels would form in a magnetic field which is above 40 tesla.
14
T. Chakraborty
60 50
,1)
0,3)
(3.5)
/(1,2)
40
A
E LLI
~ ~h~ c
~h~ c
=(1,0) ~(0,,1)
30
,(0,10) ,(0,9)
.(0.8)
20
/(0,7)
.(O,e)
10
~89 c
(~,2[0'4) 0.31 t0 1)
o:o)
0 0
4
8
12
16
20
B(T) F i g u r e 4 Singleelectron energy levels (the FockDarwin levels) in a parabolic dot as a function of the magnetic field. The levels are indicated by their quantum numbers (n, l). The confinement energy is taken to be ha~0 = 4 meV.
The calculated lowenergy excitations of the FockDarwin diagram (Fig. 4) were well reproduced in the magnetotunneling experiments on doublebarrier resonanttunneling structures [6365]. Here one employs a strongly asymmetric doublebarrier heterostructure 3 where the thick barrier is the emitter and the thin one is the collector [64]. Figure 5 sketches the condition for resonant transport via the singleelectron states which is also the basic idea behind the experimental results described below. Here the crucial quantity to understand the experimental results is the addition energy, p ( N ) , the energy required to add one extra electron to a N  1electron dot. To begin, at zero bias (V = 0) the q u a n t u m dot contains no electrons because the ground state energy E(1) of the first electron [E(1) = #(1)] lies above the chemical potential of the emitter ch and collector contacts #E,c. Whenever the addition energy PE exceeds # i ( I )  c~eV where c~ is the voltage to energy conversion coefficient [6365] and p~(1) = E~(1), singleelectron states are available for tunneling from the emitter to the dot and one observes a step A I = 2 F E F c / ( 2 r E + r c ) ~ 2FE, (FE << Fc) in the measured current. Here, FE (Fc) is the tunneling rate through the emitter (collector). The bias position of these steps, 3 F o r m o r e d e t a i l s a b o u t m e a s u r e m e n t s o n v e r t i c a l q u a n t u m d o t s , see Sect. 2.4.6.
Quantum dots
15
~i(1)
2~1(1)
~i (1) ~. LI.E 12
~1(1) ~E
'LtE
~c
__ V:O
V:Vo(1 ) VO(1)
Figure 5 The energy diagram illustrating the transport spectroscopy of singleparticle spectrum of a quantum dot [64]. E(1) = [ # i ( 1 )  #~h] /ec~, provide a direct experimental probe to the singleparticle spectrum including the ground state. The grayscale plot of Fig. 6 shows the differential conductance G = d I / d V as a function of bias voltage and magnetic field (derived from the I  V data) obtained by Schmidt [64]. Some of the lowlying FockDarwin states are clearly discernible from the figure. It is quite remarkable how the energy spectrum of an ideal quantummechanical model system derived by Fock some seventy years ago, is made visible in semiconductor nanostructures.
2.1.2. Singleelectron states To determine the singleelectron wavefunction we return to equation Eq. (2.3) X R## JF R ! Jr
(
fl_
12) R = 0
1 ~x~a
whose solution is [20]
(Zl+~)~]= I 1
1
Rnl(r) aNl+lZl l]!
21Zln.
exp (r2/4a~) r jzl S (  ~ ,
r2)
I~1 + l, 2~H 9
Here we have used the notations
~H

a.

acb= (C~c 2 +4CJo2) 89 = m*wH
1
go
v~
v~'
.
JLA
c
(2.5)
16
T. Chakraborty
F i g u r e 6 Grayscale plot of differential conductance G = dI/dV as a function of magnetic field and bias voltage. The location of the lowest L a n d a u level energy ( 1 hCOc)/eOZ and the second L a n d a u level ( 3haJc)/ec~ are also indicated [64].
Q u a n t u m dots
17
and .7 is the confluent hypergeometric function. If we use the relation
br2) = (n n,l,+}t,)!L~I( ~o br2) ' f ( n, ll[+l,~o we can write the radial function as
RnI(r)
=
=
go (~)
1+IZl
[
so,
2,t,n,
Ill'
"
21~leo2(~+lzl)! ~xp  ~ 0
exp

~o
\ e0
r

(n + lII)iL~I \ 2 g g J
L~l\2e~
'
where L~ I are associated Laguerre polynomials. The singleelectron wavefunction is then O)
27rgo2 21Zl(n + I/l)[
exp
4eo2/\ eo
L~I
(2.6)
We shall make use of this form of the singleelectron wavefunctions on several occasions below.
2.2
Dipole
matrix
elements
In what follows, we assume the wavelength of the incident light to be so large (compared to the size of the quantum dot) that its spatial variation can be neglected within the extent of the electronic wavefunction, i.e., the plane wave describing the radiation field can be approximated as e ik'r ~ 1. In the case of quantum dots this corresponds to far infrared radiation. In this approximation the transition probability from an initial state [p} to a final state Iq) is proportional to the quantity [1], [30][R7], [66], [67][R9], [68][R10], [69, 70], [71][R19]
dipole
approximation
2[ E I(plrlq) "e(~)12" OL
The polarization vectors e (~) are the unit vectors i and j in case of
linear polarization
e(+) = v/~(i 1 i iJ) in case of
circular polarization.
If the incident light is unpolarized we have
or
18
T. Chakraborty Z

I(q[xlp}l 2 + I(qlylp}l 2
=
1_ [[(qlreiOlp)[2 +l(qlre_iolp}[2]. 2
We define the singleparticle matrix elements dA'A
IreiO I /~}

<~'
=
27rSl+l,t,
(2.7)
r2Rn,z,(r)Rnz(r) dr
which hold for circularly symmetric systems (such as parabolic quantum dots) where the quantum number A stands collectively for the principal quantum number n and the angular momentum quantum number 1. Then the singleparticle matrix elements of the operators x and y can be written as 1
XA'A

(a'l~l~)  ~ [d~,x + d ~ , ]
YA',~

(~'lyl~)  ~ [d~,~  d ~ , ] .
1
In occupation representation the dipole operators corresponding to the configuration space operators x and y are
X

E x),),,a~a),,
r
=
ZY~'4a" AA'
It is clear from the above expressions that only transitions with A1 = 1~  1 = +1 are possible. If a photon to be absorbed is polarized in the xdirection, only the operator X is to be used, and for photons polarized in the ydirection, the operator Y is used [71][R19], [72], [73] [R24], [74, 75].
2.3
Basic
properties"
Experiments
Experimental information about the electronic properties of quantum dots is primarily from singleelectron capacitance spectroscopy [76][R21], [77, 78], gated resonant tunneling devices [79], conventional capacitance studies of dot arrays [59][R2], [8085], transport spectroscopy [86][R3], [87][R13], [88][R14], [89], [90][R22], [9194], [95][R23], [96], farinfrared (FIR) magnetospectroscopy [97][Rll], [98], [99][R12], [100102], and very
Quantum dots
19
Figure 7 (a) Schematic diagram of a quantum dot capacitor. (b) Scanningelectron micrograph of quantum dot (300 nm across) arrays [82].
recently, from Raman spectroscopy [103,104]. In conventional capacitance studies, an oscillatory structure in the measured capacitance was attributed to the discrete energy levels of a quantum dot. In the presence of a perpendicular magnetic field, Zeeman bifurcation of the energy levels of a quantum dot was also observed. This splitting is believed to occur due to the interplay between competing spatial and magnetic quantization. Capacitance spectroscopy has been widely used to study the density of states of lowdimensional electron systems [81]. A schematic view of a quantum dot capacitor is shown in Fig. 7 (a). Here the lateral confinement is provided by depletion of highmobility twodimensional electron gas. The 30 nm GaAs cap layer is etched away to deplete the underlying A1GaAs. The electrons are then confined in a quantum dot at the heterojunction. A scanningelectron micrograph of quantum dots is also shown in Fig. 7 (b). The n + GaAs is one electrode of the quantum capacitor and the Ni/Au metal on the top is the other. The differential capacitance (at T = 0) of a confined electron system is determined largely by the thermodynamic density of states:
C = dQ
dn d#
= qJp~dv~ Ds(EF) dV
where dQ is the infinitesimal charge induced by a change in voltage dV, q is the electronic charge, n is number of carriers, # is the chemical potential and Ds = dn/d# is the density of states (DOS) at the Fermi energy EF. A change in the gate voltage changes # and therefore in quantum dot experiments, we can sweep the Fermi energy through the zerodimensional density of states. The measured capacitance (more precisely, the first derivative of the capacitance vs the gate voltage) reveals structures related to the
T. Chakraborty
20
I
rn dots
T=O.7K f=1OkHz
30Onto dots
>
0.40
P
E
0.20
0.00
0.20
0.40
vG (Volts)
Figure 8 Capacitance spectra for three different quantum dots. The size dependence of the spatial quantization is clearly visible in the spectra [82]. zerodimensional quantum levels. Figure 8 shows capacitance results for three quantum dots of different sizes. The wellresolved oscillations in the derivative of the capacitance reflect zerodimensional quantization. The period of the oscillations increases with increasing dot size. Figure 9 shows the gate voltage derivative of the capacitance as a function of the gate voltage at different magnetic fields. The peaks at positive gate voltages show complicated behavior even at very low fields (B < 0.2 tesla). However, as the field is increased to about 1 tesla we can clearly distinguish the peak shifts and splitting of many of the peaks. At higher fields, some peaks increase in size and other weaker peaks move between them. The stronger peaks are the precursors of Landau levels which the smaller peaks join as the field increases and become more degenerate. We plot the peak positions as a function of magnetic field in Fig. 9 (b). The stronger peaks that become Landau levels at high magnetic fields are plotted as solid dots while the weak peaks are shown by open dots. The oscillatory structure in the capacitance seen in Fig. 9 (a) can be attributed to the discrete energy levels of the dot indicating that the quantization has been achieved in all directions. The peak splitting is believed to occur due to the interplay between competing spatial and magnetic quantization 4. 4 Silsbee and Ashoori [84] have presented a different interpretation of these results, where they a t t r i b u t e
Quantum dots
21
~t
T:O'7K ~
250

!
188 ~o 
~^
I o
8
..n~ o ~".....~ ~" o
UO~ ~_. ~,,oo ~"
~
ID

9
9
9
oo
., 9
o
e
""o"~. " o OOo
o ~ 0
9
U)
(....
~.
63~~ ~eo foe
~ ~
9
000
~.,&lg~
>o)
'eeD ! ;""
0
~e . ' ~
0
e,i,.~
o0
00 OoOe " 9 e ~ o o 9 O000e O 0 o e e o oo9 o ee ooO~176 e ee e ee o 0
0L~ _.~~,
_
<1: v >
~"
o
9
0
23
~t
;
9
9

"lD
125 i ~ ' ' ~ " o 9 "" imeeeseol o o o ~ 1 7 6 1o7o6o o
~
oo oeoe
iJ
188
_ I
I
I
......1
~a~NOIONB O O 0 9 9 9 0 0 9 9 9 9 e 0
I ......
0.30 0.20 0.10 0.00 0.10 0.20 0.30
(a) Figure
Vg (V) 9
;meeieeo o 9 e e o c
oO
B=0
250 0.0
(b)
_I 0.4
J J 0.8 1.2
. 1.6
2.0
B (T)
(a) C a p a c i t a n c e s p e c t r a for 300 n m q u a n t u m d o t s at various m a g n e t i c fields.
(b) Gate voltage position of the peaks. Filled and open circles correspond to strong and weak peaks respectively [59][R2]. Another interesting result in capacitance spectroscopy is the observation by Hansen et al. [83] of fractionally quantized states, similar to the fractional quantum Hall effect in a twodimensional electron system [1921]. For dots containing about 30 electrons in a very high magnetic field, the derivative of the capacitance vs the gate voltage shows downward cusps at 51 and 52 filling factors 5 [Fig. 10 (a)]. The temperature dependence of the minima [Fig. 10 (b)] at these two filling factors is also consistent with that of the fractional quantum Hall states.
2.3. I. Singleelectron capacitance spectroscopy The electronic ground state in a parabolic confinement potential described above has indeed been observed in an ingenious experiment reported by Ashoori et al. [76][R21], [77, 78]. The method involved in this experiment is known as singleelectron capacitance spectroscopy, and allows direct measurement of the energy levels of a heelectron dot as a the peak structure of the capacitance spectra as due to successive electron transfer where the peak spacing is determined by the charging energy (see also, Ref. [85]). 5 Defined as tJ = (h/eB)N/A, where A is the area and N the number of electrons in the dot.
T. Chakraborty
22
(a) ~
(b)
l
T=O.7K
.
B=30T
~ v =1/3
=1/3
v=2/3
'
I~\\i
v=2/3
<.% 19T
~
0.3
0.1
~
0.1
!i
0.3
0.5
Vg (Volts)
0.7
.....
0.3
l,
0.1
I
0.1
0.3
0.5
,_
0.7
Vg (Volts)
Figure 10 (a) Derivative of the capacitance vs the gate voltage for 3000 .~ dots. Only the lowest spinsplit Landau level is occupied at the gate voltage shown here. (b) Temperature dependence of the fractional states at B = 30 tesla [83].
function of the magnetic field. The basic configuration of the sample used by Ashoori et al. is shown in Fig. 11 (a). The capacitance was measured between an electrode on top of the QD (the gate) and a conducting layer under the dot that is separated from the dot by a thin tunnel barrier. When the dc gate voltage on the top electrode is varied the Fermi level in the bottom electrode can coincide with the Fermi energy of the dot. That would result in electron tunneling through the thin A1GaAs barrier, as indicated in Fig. 11 (a). Charge modulation in the QD induces a capacitance signal on the gate because of its close proximity to the dot. The capacitance as a function of the gate voltage was found to exhibit a series of uniformly spaced peaks, with separation decreasing with increasing electron number [Fig. 11 (b)]. The peaks are due to the addition of single electrons to the QD. The remarkable aspect of the experiment by Ashoori et al. [76][R21], [77, 78] is that they probed the addition spectrum starting with the very first electron in the dot. The results were plotted by Ashoori et al. [76][R21], [77, 78] as a 2D grayscale plot where the vertical axis is the voltage, the energy scale of electron addition. The capacitance is highest in the bright areas. The bright lines trace the addition energies as a function of the magnetic field B. The ground state energies of systems with up to 35 electrons are presented in the plot of capacitance peaks in Fig. 12. The magnetic field
Q u a n t u m dots
23
b l o c k i n g b a r r i e r i + * § § ++.§ + * 9 + * + § quantumwell =i ~ ~i ........." G a A s tunnel barrie~ ___ ~ " . : IAIGaAs spacer ~e Undoped GaAs ~ . / J . / J J . / . 7 _ ~ J J J / / / ' . / / / / . ~ / ; .
(b)' O9 t"
~
i
~ i ,,,,, 0
o ._1 w
6'3
390
360
330
300
270
240
V(mV) Figure 11 (a) Schematic diagram of the sample. (b) Capacitance data as a function of the gate voltage for a quantum dot. The bottom trace is the signal at the dot in 90 ~ lagging phase [76] [R21].
dependence of the lowest energy state in Fig. 12 (a) is smooth and can be well described by the FockDarwin energy for hw0 = 5.4 meV. Further, the highfield asymptote of this curve also follows the dashed line (plot of ~1 hwc) as expected. The ground state energy curve for the twoelectron system shows a pronounced bump and change of slope around 1.5 tesla. This was interpreted by Ashoori et al. as an indication of spin singlettriplet transition. It is interesting to note that a similar transition for He, predicted at an exceedingly high field of 4 x 105 tesla in the vicinity of white dwarfs and pulsars [105] still remains to be observed. Artificial atoms with weak binding and small electron mass are therefore, in certain aspects, more useful tools than real atoms! The b u m p progresses and shifts monotonically to higher fields as the electron number increases, thereby producing a clear ripple through the data set. We should emphasize at this point that some earlier theoretical work predicted this type of spin transition [106][R15], [107][R16]. Such a spin transition in a twoelectron q u a n t u m dot was also observed experimentally by Su et al. [108] via singleelectron tunneling [for more on these topics, see Sect. 2.4.6]. Ashoori et al. also noticed a distinct loss of intensity as
24
T. Chakraborty
F i g u r e 12 Images of sample capacitance as a function of both magnetic field and energy. The dashed line shows hc~c/2. The number along the traces indicate the electron number [(a) 16, (b) 635] in the dot [76][R21].
the m a g n e t i c field was increased, which resulted from a decreased t u n n e l i n g rate. T h e reason for this is not entirely clear. G r o u n d s t a t e energies of the dot containing ne = 6  35 electrons are shown in Fig. 12 (b). Before we go into the details of these results, let us first take a look at Fig. 13. It is clear from the t h e o r e t i c a l FockDarwin states calculated for the confinement energy of hco0 = 1.12 m e V t h a t the g r o u n d s t a t e energy of the 35th electron should oscillate as the levels cross with increasing m a g n e t i c field. T h e oscillations cease at ~ 2 tesla. If we define the L a n d a u level filling factor t, as the Landaulevel o c c u p a n c y at the dot center, the position of the last level crossing in this figure can be identified as y = 2, i.e., two electrons per flux q u a n t u m .
Quantum
dots
25
I~ I~ 9^ V
' ~
~
~
~162
:
:
'
:
%,"_~':,~:~.':""" .".." :"" .".".: : :':':':'." ." .:.:.:.:.' ".',',',','," "r: ~',:,: "" /:: ." "" " ".' "."":'.".'.'.'.:.:.: " " ' " " " ' " " " . :. . ". . :. . ". . ." : : " :" ..'.'.'.;:.:. " ' " ' " """ "
;
:':..'.'."""''..'.'.''.'':':':':':" :.:.;.~: , , :
:.
"~" [::: .:; "; " ":' . :' . '::: ::::::: :::::::::::. :::' .:: :~: ~: :~. :~: .i: .:: :.:.:.~.:.': , . ::..."'.: ,: :t"""!::::":." ":~':, [:'i,:~:~'..;:';: "~" :~',,: "" . . .;:.: . .. :" .'.':.':':': :..:':.':...';
5
S (T) Figure 13 The FockDarwin s t a t e s of a q u a n t u m d o t w i t h hw0  1.12 m e V ( d o t t e d c u r v e s ) . T h e b o l d s o l i d c u r v e s h o w s t h e m a g n e t i c field e v o l u t i o n o f t h e 3 5  t h e l e c t r o n .
T h u s far, we have discussed only the case of n o n  i n t e r a c t i n g electrons. A simple way to i n c o r p o r a t e the interaction b e t w e e n electrons is to follow the c o n s t a n t  i n t e r a c t i o n (CI) model 6. This a p p r o x i m a t i o n is c o m m o n [87][R13], [110] in studies of the C o u l o m b blockade, where the total energy is w r i t t e n as ne
Z
+
(2.S)
i=1
with U as the interelectron interaction a n d r chemical p o t e n t i a l is t h e n evaluated from #e ~ E ( n e ) 
is the energy of the ith electron. T h e
E(T~e  1) = (he  1) V ~ ~ne
where s~o is the energy of the n~th electron. Going back to Fig. 12 (b), we notice the d e v e l o p m e n t of the ~ = 2 positions (white triangles). For n~ > 10, the ~ = 2 positions for successive electrons agrees well with the CI model with h~0 = 1.1 meV. At large n~, the t u n n e l i n g rates are a t t e n u a t e d a r o u n d y = 2. Ashoori et al. s p e c u l a t e d t h a t , as the electrons in the dot center are in a q u a n t u m Hall state, the incompressibility of t h a t (a h a l l m a r k of the q u a n t u m Hall state) s t a t e causes suppression of the tunneling. 6 For a critical analysis of this approximation, see Ref. [109].
T. Chakraborty
26
Ashoori et al. [111,112] also noted that large dots (lithographic diameter larger than 0.4#m) containing few (~40) electrons reveal some more puzzling features. Instead of being well separated in gate voltage as in the results described thus far, electron additions in these dots are grouped in bunches. With increasing electron number, the bunching pattern develops into a periodic pairing of electron addition peaks in gate voltage, with every fifth peak paired. When the electron number is increased above about 80, the pairing pattern disappears and the usual periodic Coulomb blockade pattern appears. Application of a magnetic field reinstates the pairing phenomenon, where a larger magnetic field Bpairing(rte) is required to create pairing when the dot contains larger number of electrons. Finally, the bunches cease to appear as the magnetic field strength is increased so that the filling factor is at u = 1. These authors surmised that the bunches are caused by electron localization within the QD. The cause of localization giving rise to periodic bunches are also attributed to interactions. It is interesting to note that, for classical particles confined in a parabolic potential [113115], the ground state corresponds to localization of the particles to shells with shell filling somewhat similar to that observed in experiments above. However, along that line of reasoning the pairing mechanism still remains to be explained.
2.3.2.
Optical transitions
In the experimental studies described in this section, the quantum dot structures were created either by etching techniques or fieldeffect confinement (Fig. 14). The samples were prepared from modulationdoped A1GaAs/GaAs heterostructures. For the deepmesaetched quantum dots, an array of photoresist dots (with a period of a = 1000 nm both in x and ydirection) was created by a holographic double exposure. The rectangular 200 nm deep grooves were then etched all the way into the active GaAs layer [Fig. 14 (a)]. Fieldeffect confined quantum dots were prepared by starting from a modulationdoped GaAsheterojunction. Electrons were laterally confined by a gate voltage applied to a NiCrgate. A strong negative gate voltage depletes the carriers leaving isolated electron islands (quantum dots). Details about the fabrication of these dots are available in the literature [2], [97][Rll], [98], [99][R12], [101,116,117]. We have already discussed in Sect. 2.2 the expression for dipole matrix elements. More specifically, transitions from an electronic state ~nt to a state Cn'Z' are governed by the the transition amplitude
dnz,n,Z, = (O~zlre+i~162
(2.9)
with associated oscillator strength f~z,~'z'  (2m*/h)mnz,~,z,
Id~z,~,z, Ie
(2.1o)
Q u a n t u m dots
27
.... n15aAIAs
GaAIAs s p a c e r ODSystern
+ + + + +
~
  '  ~
/////
'"
" J
GaAs
.
_
_
(b)

5ateelectrode Phoforesisf
GaAsHeferosfrucfure QuQntumdof
Figure 14 Sketch of the (a) deepmesaetched quantum dots and (b) fieldeffect confined dots. where a&~z,n,z,  ( E n , l ,  E n l ) / h is the transition frequency. For the singleelectron states in a parabolic quantum dot, Eq. (2.9) can be written in closed form [see, Eq. (2.8)]
dnl,n'l'
(2 )2 
(~nn'(~l ' l + 1 ,
fft* 02 0
1
1
[7t
+ Ill
+
1] ~ 1__

5n,,n+lS1,,l+
1
leading to the selection rules, A1  l '  1 corresponding energies are
2h (1  St,o) rn*wo  •
A E +  hft +
+ 111
as derived above and An  0, 1. The
lh~
c
(2.11)
where the + (  ) sign corresponds to left (right) circular polarization. As the magnetic field increases, A E + approaches the cyclotron energy ha~c, but A E _ decreases. Typical experimental results for these resonance positions are shown in Fig. 15). Liu et al. [70] investigated experimentally the quantized energy levels and the allowed optical transitions of a quantum dot in a magnetic field. They observed the upper branch
T. Chakraborty
28
n 16 0
..13
i
20
~ "
30
.t
,
>
01 o
20 v
i s
i lo
i 15
/
E s
lO ,~
T.4K _.. z%Vg=8K
,,
0
I
I.
I
1.0
2.0
3.0
4.0
B(T) F i g u r e 15 Zeeman splitting of the resonance position of quantum dots. The solid lines are obtained from Eq. (2.11). Independence of the quantization energy on gate voltage and electron number is demonstrated as inset [97][Rll].
of the allowed transitions in a parabolic potential discussed above. These authors also did a theoretical analysis of the energy levels and the dipoleallowed optical transitions and found that the experimental results agree with the calculated transition energies for hcJ0 = 1.5 and 2.8 meV. The FIR spectra observed by Demel et al. [99][R12] for q u a n t u m dot arrays with 210 and 25 electrons per dot are shown in Fig. 16. With increasing magnetic field, the resonance splits into two resonances, as expected. An interesting observation here is the anticrossing behavior of the upper mode at low magnetic fields. The appearance of a resonant anticrossing in the energy levels is believed to be due primarily to a nonlocal interaction in a single dot which is important in low dimensions. Meurer et al. [101] and Meurer [102] pointed out that FIR spectroscopy not only provides the resonant frequency but also the absorption strength, a measure of the electron population per dot. For a parabolic potential at B = 0, all absorption occurs at a single resonant frequency, as demonstrated by the early result in Fig. 15 and more recent results in Fig. 17. However, as the number of electrons in the dot increases, the absorption strength increases proportionately. Figure 18 shows the F I R transmission spectra and the integrated absorption strength vs the gate voltage in a fieldeffect q u a n t u m dot array with a periodicity of 200 nm. Interestingly, the absorption at low electron occupancy
Q u a n t u m dots
29
(a)
2OO
150 ! o
iO0
N ~ 210 R~16Onm
3 5O
(b)
200
150 A
! 10o Nffi25 R  100rim
3 5o
0
2
4
6
8
10
12
14
16
B (T) F i g u r e 16 The magnetic field dispersion of resonant absorption in quantum dot structures with 210 and 25 electrons. The solid lines are obtained from Eq. (2.11) [99][R12].
increases step wise with the gate voltage, i.e., most of the l0 s dots in the array change their charge by one electron at the same gate voltage. Meurer et al. explained this unique behavior as due to the large Coulomb energy t h a t one needs to add a n o t h e r electron to a dot. In Fig. 18 one notices t h a t in order to increase the n u m b e r of electrons from ne = 2 to ne = 3, the voltage must increase by AVg = 30 inV. This value corresponds to a capacitance of C = e/AVg = 5.3 • 10  i s F and a Coulomb charging energy, Ec = e2/2C, of 15 meV. Selfconsistent Hartree calculations for q u a n t u m dot systems with nearly the same dimensions as in this experiment [58][R8] predicted t h a t the voltage intervals required to add an additional electron is AVg = 15 mV, the same as what was estimated by Meurer et al. from their experiments. In the F I R spectra of q u a n t u m dots another type of excitation mode associated with the harmonics 2coc, 3a:c,... has been observed resembling the Bernstein modes [118] seen
T. Chakraborty
30
9
i
!
+ '
+
+. . . ,=. ~ o
12 e  / d o t
25eldot
....
60
60
.,.,.o" ..,.o" , 9
9
i~,9
,. l l ' l ' .O0""
E
40
.00 .+ 9
40
....
....
::~"
O
,
"""o~
....
8~
9 ..... o~ .......
20
20
9
I
0
,
I
'
:
2
!
,
IL
3
4
L
5
0
I
2
4
3
5
B(T)
B (T)
F i g u r e 17 Magnetic field dispersion of the resonant frequency in a quantum dot array with 12 and 25 electrons per dot. The lines are derived from Eq. (2.11) [116]. _+
1.00
'
1
i
"1
,
i
9
J ~
'
0.99
?
,'
i
,
I
'
I
,
l
(b)
//
9 v (v, %
i
X=llSlam
),=ll8la
, x
,j
i i I I
%
0.98   0 . 7 3 5 . . . . . ~ / 0.730 . . . . . . . . ~ "   ' /
I I I!
t._.,
0.97
I
i
0.720 . . . . . . . . . . . ~/ 0.96
Ill
0 7~s . . . . . . . . i . . . . " " S.;S
'
6.00
Vg
'
6.()S
(v)
'
6.10
~ :
0.76
0174,
0.72
0170
'
B (T)
F i g u r e 18 (a) FIR transmission at a fixed laser frequency of 10.4 meV and at various gate voltages for a quantum dot array of period 200 nm. (b) Integrated absorption strength vs the gate voltage for a series of spectra. The stepwise increase indicates the incremental occupation of the dots with one, two and three electrons per dot [101].
earlier in threedimensional [119] and the twodimensional electron systems [120,121]. In the latter case, the explanation for the existence of this mode is t h a t the dynamic spatial modulation of the charge density of a plasmon breaks the isotropy of the system and causes an interaction resulting in an anticrossing when the m a g n e t o p l a s m o n dispersion crosses harmonics nwc (n = 2, 3 , . . . ) of the cyclotron frequency. Experimental evidence of the Bernstein modes in an array of large dots [116,117] (with a period of 2>m, and about 2400 electrons) is shown in Fig. 19. The i m p o r t a n t message of the optical spectroscopy so far is t h a t in F I R spectroscopic measurements on parabolic q u a n t u m dot structures the measured resonant frequency is independent of electron number within the experimental error. These resonances are
Q u a n t u m dots
31
160
,
,
,
,"
,

.._, 120 E o 80 a'40 O0
2
4
7080/[ .......... ~ ._.
60
9r
50
g
30
'
6 B(T) "'
Y
8
+',
~'
1012 '
"
20
1000
1
2
3
4
5
B(T)
F i g u r e 19 Magnetic field dispersion of the various excitations in an array of large quantum dots. The same experimental results are shown in the top and bottom panels and in the inset with different scales. The Burnstein type of anticrossing is shown in the inset [117].
related to singleparticle transition energies in a bare confinement potential [97][Rll], [98], [99][R12], [100102]. As we shall see below, an intriguing theoretical explanation of this important result is available. For q u a n t u m dots with tailored deviations from parabolic confinement (more like a hardwall confinement), Bollweg et al. I122] noticed that the co_ modes display oscillatory behavior [Fig. 20]. Here the experimental resonance frequencies are normalized to a
V//co~ 121
calculated dispersion coc_~_ + ~co~  ~coc which displays no oscillations. The experimental results, in contrast, show two oscillating periods. The m a x i m a and minima are dependent on the filling factor. The maxima of frequency appears at fully occupied Landau levels, while the minima occur at halffilled Landau levels. These authors attribute these oscillations to the formation of compressible and incompressible strips at the edges of the dots. Darnhofer et al. [12a] however presented a different explanation of the observed oscillations in the co_ modes. They performed a selfconsistent calculation of the equilibrium state and the FIR absorption spectrum of electrons in a q u a n t u m dot with nonparabolic
T. Chakraborty
32
180 160 140
, 1.08
E= 80
1.06
~" 60
~,, 1.04
6
40
1.02
4
20
1.00
~
0.98
........
0
0
2
4
6 B (m)
8
10
0.0
b
,o
100
ot ar y
,
~
0.1

8 >
IK:)N~=7.7.101 lcmZ 2 ~I~N_1=7.0 101lcm "2 .
0.2 B1 (T "1)
I
.
~""
0
0.3
F i g u r e 2 0 O b s e r v e d dispersion of the resonant frequencies in a dot array. T h e c a l c u l a t e d dispersion for aJ+ and a~_ (with a~0  86 cm  1 ) are also shown for c o m p a r i s o n . T h e co~ m o d e is the acoustic m o d e of the doublelayer dot array. (b) O b s e r v e d r e s o n a n c e frequencies a~_ normalized to the c a l c u l a t e d dispersion a~ca (left scale) versus B  1 . T h e m o t i o n of individual electrons in the dot for the a~_ m o d e is shown as inset [122].
confinement potential. They used the localdensity approximation to study the ground state properties, ignoring spin degree of freedom of the electrons. At high magnetic fields, they found that the radial extension of the electron density depend on the filling factor. Under dipole excitation the electronic system probes the radiusdependent curvature of the confinement potential, leading to small filling factor dependent oscillation in the co_ mode. The results agree well with the experimental results of Bollweg et al. [122]. Lorke et al. [124][R20] employed FIR spectroscopy to study the coupling of adjacent quantum dots created in GaAsheterojunctions where the coupling of the dots can be tuned by a gate voltage. The resonance peaks as a function of magnetic field strength are shown for the isolated dots in Fig. 21. In the region of coupling, four modes are observed at high magnetic fields (B _> 2.5 tesla). The topmost mode is a magnetoplasmon resonance, and the one adjacent to it is a confined cyclotron resonance. The lower branch is an additional edge mode, at frequencies lower than that observed for an isolated dot, which was identified as a charge moving along a boundary of about twice the length of the perimeter of a single dot. A charge moving along a peanutshaped orbit enclosing two dots, as shown in the inset of Fig. 21 (b), seems to explain the additional edge mode. These results represent a transition from an isolated dot to a electron grid, analogous to the transition from atoms to molecules.
Quantum dots
33
I00
I
80 60
Vg=3.
I
I
.
.
.
.
.
.
I
IV

O 0/.~

4O ~

2O
Ca) ,.~
=
>
0
t20 100
c~
so
~
V =29V 9
50 40 20
.=.
lob)
! 0
 "oo.oo__
i 1
l 2 B
I 3
I 4
._
(T)
Figure 21 Resonance position for (a) isolated and (b) coupled quantum dots. The solid lines are obtained from Eq. (2.11). The inset shows the relevant trajectories for the upper(A) and lower (B) frequency edge modes. The dashed line corresponds to the dispersion for type B edge mode [124][R20].
Before closing this section, we should mention that collective response of twodimensional macroscopic electron disks was studied earlier by Allen et al. [125] by measuring the absorptance of radiation transmitted normal to the surface. In the presence of an external magnetic field a split in the resonance was observed similar to what is observed in fewelectron quantum dots. Finally, Kern et al. [100] prepared very large quantum dots containing 600 electrons by deep mesa etching, and measured an energy spacing of 1 meV. FIR spectroscopy of these systems showed, in addition to the two usual modes, other weak modes and anticrossing behavior. Dahl et al. [126] studied the microwave response in elliptical quantum dots. In this geometry, the plasma resonance shows a large gap at zero magnetic field, where a circular dot shows degenerate dipole modes. Lifting of degeneracy is the major effect of the anisotropic dots. A polarizationdependent higher harmonic of the fundamental
T. Chakraborty
34
Figure 22 The gate geometry with the various gates to create the quantum dot and the constrictions [135].
lowfrequency mode was also observed. Our theoretical understanding of these effects will be discussed below (Sect. 2.11.1).
2.4
Transport spectroscopy
Excellent articles and reviews on transport phenomena in mesoscopic systems exist, e.g., in Refs. [127134, 92]. Here we will briefly discuss some interesting aspects of transport spectroscopy in a mesoscopic system primarily to demonstrate that, just like the optical and capacitance spectroscopies, electrical transport measurements can provide information about the energy spectrum of a 0D system. Quantum dots coupled to two electron reservoirs by weak tunnel junctions allow one to study electron transport through the dots. Tunneling of electrons through the dots is influenced by quantum confinement as well as the charging effects. As described below, addition of an electron in the QD costs a finite charging energy and that allows one to control the number of electrons in the dot which can be changed by one at a time. Detailed reports on the transport results and their qualitative explanation can be found in Refs. [130,135,136,96]. In what follows, we briefly touch upon some of the major experimental works. In Fig. 22, we show one example of QDs created for lateral (parallel to the surface) tunneling. Here the QD is created by depleting the 2DEG, which lies 100 nm below the surface of a GaAs/A1GaAs heterostructure, underneath the gate structure. Electrons are depleted by a large negative voltage (  4 0 0 meV) to gates F, C,
Q u a n t u m dots
35
(b) Singleelectron Tunneling
(a) Coulomb Blockade ,,,
,_,ge(N,l)
Y;["
~1
"J
. . . . . .
'
pe(N) J
ge(N)
7 .... ,f3
eV;s
.........
Figure 23 Schematic picture of (a) Coulomb blockade (CB) [#e < ~1, ~r < ~e(N nI 1)], and (b) Singleelectron tunneling (SET) [#1 > #e(g + 1) > #r] The sourcedrain voltage across the sample VDS = ( # l  #r)/e is very small [eVDs << #e(N + 1)  #~(N)].
1, and 2 forming a QD of diameter ~ 800 nm. Electron transport occurs through the quantum point contact (QPC) constrictions induced by gates 1F and 2F. The charging energy for such dots is estimated to be ~ 0.6 meV, and therefore, the charging energy is the major energy scale at temperatures below 4K.
2.4.1. Charging effects The discreteness of electron charge plays an important role in transport through confined regions that are weakly coupled to the leads. In a QD, charge of a single electron becomes important when the capacitance C of the dot to the surroundings is small. This leads to a charging energy of a single electron, Ec = e2/2C that exceeds the thermal energy kBT. As a result, current through the dot is a discrete transport of single charges rather than a continuous flow of electrons. Charging effects have been studied earlier in classical systems such as granular rims, small metal tunnel junctions, etc [137], but in what follows, we focus entirely on charging effects in QDs. The potential energy landscape of the Q D induced by the gates is shown in Fig. 23. In the two reservoirs at left and at the right of the dot, the states are occupied up to the electrochemical potentials 7 #I and #r. The energy separation between the 0D states in the Q D where N electrons are localized, is E N + I  EN, which is related to the change in the electrochemical potential 2
/~e

# e ( N + 1)  pc(N)  ~ X + l
7 D e f i n e d as t h e s u m of F e r m i e n e r g y E F ( N )

EN
 
EN + ~.
a n d t h e e l e c t r o  s t a t i c p o t e n t i a l eqoN.
(2.12)
36
T. Chakraborty
(a)
N+2
~
(b)
Nd~I N1
i
l
l
' .
,,
w v
d
A e~
rm+qrm+ .....
:" ",

,
:
:2 e
*'C
(c)
AVg
"
Vg
Figure 24 (a) Conductance, (b) number of electrons and (c) the electrostatic energy e~ of the dot as a function of the gate voltage Vg. Depending upon the gate voltages, the energy gap Ape can lead to a blockade in electron tunneling in and out of the dot. This is depicted schematically in Fig. 23 (a). Here electrons can not tunnel out of the dot because #1 and Pr are higher than the highest occupied level pc(N) in the QD. Electrons cannot tunnel into the dot because the resulting electrochemical potential pe(N + 1 ) is at a higher energy level than the electrochemical potentials of the reservoirs. Therefore, at T = 0, electron transport is b l o c k e d  the Coulomb blockade (CB), when p~ (N) < pl, Pr < P~ (N + 1). As shown in Fig. 23 (b), a change in the gate voltage (which in turn, changes the electrostatic potential ~N), pe(N + 1) can be made to line up between p~ and Pr [P~ > p~(N + 1) > Pr]. It is then possible for an electron to tunnel from the left 2DEC reservoir into the dot [pl > p~(N + 1)]. The electrochemical potential in the dot then increases by Ape, and so does the electrostatic potential e~N+l   e ~ N = e 2 / C [Eq. (2.12)]. Since p e ( N + 1) > #r, one electron can now tunnel from the dot to the right reservoir. Then the electrochemical potential drops back to p~(N). A new electron can now tunnel into the dot and the cycle repeats. This process is called the single electron transport (SETR). The conductance of the QD oscillates between zero (CB) and nonzero (SETR) values as one sweeps the gate voltage. These are the Coulomb oscillations [Fig. 24 (a)]. In
Quantum dots
37
VGS VDS dd/D~s . ~ JITopSJ I[3
(a)
Vsl
777"
(b)
3 __
'
'
'
'
I
. . . .
Back
Gate
~X . . . .
\
I )
I
' 1 pm
V
'
/
(
"'
g O 0
0
I
J
I
I
I
5
I
Gate
I
I
I
I
0
I
I
I
I
5
10
Voltage VBS(V)
Figure 25 Crosssection of a quantum dot sample (schematic). (b) Conductance versus the gate voltage. The six gates on top of the 2DEG used to define the quantum dot and the two tunneling barriers is shown as inset [138].
the CB regime, the number of electrons in the dot remains fixed at the minimum of the conductance [Fig. 24 (b)]. At the conductance peak, this number oscillates by one electron and e~ oscillates by e 2 / C [Fig. 24 (c)]. Between two conductance maxima, e~ changes by E N + 1  E N + e 2 / C. Figure 25 (a) depicts the system used by Weis et al. [94], [95][R23], [96,138] to investigate transport spectroscopy in QDs. The dots were made by creating a negatively biased splitgate on top of a standard GaAs/A1GaAs heterostructure. Here the 2DEG lies 86 nm beneath the surface; a negative gate voltage of about Vg =  0 . 7 V depletes the 2DEG under the gates, and as a result, a disk of electrons of diameter 350 nm containing about 10 electrons is formed. The tunneling barriers were tuned by changing the topgate voltages, while the backgate on the reverse side of the GaAs substrate was used to change the electrostatic potential of the QD. Fig. 25 (b) is a typical result for the conductance of the QD as a function of back
T. Chakraborty
38
gate voltage VBs. The observed periodic series of sharp peaks is largely explained by the qualitative picture of Coulomb blockade oscillations presented above. Similar results were also reported earlier by McEuen et al. [87][R13] and Johnson et al. [90][R22].
2.4.2.
The diamond diagrams
Diamond digrams, or charging diagrams provide important information about tunneling spectroscopy of ground and excited states of quantum dots [90] [R22], [91, 94], [95] [R23], [96,138]. The diagrams are generated by plotting the differential conductance dI/dVDs as a function of the backgate voltage VBs for a range (3 mV to +3 mV) of sourcedrain voltages VDS Fig. 26 (a). In this linear grayscale plot, white regions correspond to dIDs/dVDs below 0.1#S and the dark regions correspond to that above 2#S ( S = l / O h m ) . The main structures visible in Fig. 26 (a) are plotted schematically in Fig. 26 (b). In the limit liDS ~ 0, one observes conductance resonance, as described above. Increasing IVDsl, the range of VBS where transport through the QD occurs, is broadened linearly with IVDsl, enclosing in between the diamondshaped CB regions. Along the VDS ~ 0 axis, the electron number N changes to N + 1 when the adjacent diamondshaped zerocurrent region touch, as indicated in Fig. 26 (b). The transport regime  the light gray regions of Fig. 26 (b)  can be identified in the VDS  Vcs plane by extrapolating the boundaries between transport and blockade regimes. Within the SET regimes, additional transport channels through the QD [dark gray regions in Fig. 26 (b)] are also visible at finite VDs. During transport as liDS ~ 0, the QD changes between the ground states of, say, N and N + 1 electron system. At finite VDS, excited states for both N and N + 1 electron systems are also accessible, thereby providing new tunneling channels. This can happen in two ways: either an excited state of the (N + 1)electron system becomes accessible to put the (N + 1)th electron to the Nelectron dot, or the QD is left in an excited state of the Nelectron system, and the (N + 1)th electron leaves the dot. Within the SET regime, one also observes negative differential conductance [white regions in Fig. 26 (a) and dashed lines in Fig. 26 (b)]. Those peaks shift parallel to the boundaries of transport and of the CB regime (upper edge of a diamond). It has been suggested (see Sect. 2.9) that spin selection rules are responsible for these features.
2.4.3. Magnetic field effects In order to extract the magnetic field dependence of Nelectron energy levels in a QD, McEuen et al. [87][R13], [88][R14], [89] investigated transport spectroscopy in the presence of an external magnetic field perpendicular to the plane. Theys started with a 2DEG
Q u a n t u m dots
39
F i g u r e 26 (a) Differential conductance dIDs/dVDs as a function of gate voltage VBS for a range of values of VDS. (b) Sketch of main structures visible in (a). Here light gray areas are regions of singleelectron tunneling, i.e. the regimes where the number of electrons can energetically fluctuate between for instance N and N + 1. Dark gray areas are the regimes where the number of electrons can energetically fluctuate between, for instance, N  1, N and N + 1. Dashed lines indicate the regime of negative differential conductances and the dotted lines correspond to suppressed conductance [138].
in G a A s / A 1 G a A s heterostructure using electrostatic gates to confine and adjust the density of the 2DEG. A negative voltage applied to a lithographicallypatterned split upper gate creates the QD. A positive bias applied to a lower gate adjusts the electron density. The conductance peak positions in Vg, as observed in their experiments [Fig. 27 (a), inset] are a direct measure of the addition energy  the energy to add an electron to the dot" Vg  #e(N)/ec~ (o~ ~ 0.4 is constant). The evolution of the position of a particular conductance peak with increasing field is shown in Fig. 27 (a). From the peak positions, these authors constructed the level spectra [87][R13], [88][R14], [89] which are shown in Fig. 27 for 3 >_ u >_ 2 (b) and for u _< 2 (c). These spectra were obtained by subtracting a constant [AVg  1.175 mV in (b) and AVg  1.35 in (b)] between successive peak position curves in the experiment. McEuen et al. noted t h a t Coulomb interaction dominate the spectra and simple noninteracting models fail to account for the level crossings observed in the spectra. On the other hand, a selfconsistent model t h a t includes both Landaulevel
T. Chakraborty
40
316.5
,:.
il'~ ~
"
'
>
2 ~ 2(20EG) J
,.r'," $& i,o, L
314.5
.
.
.
1.0
.
~ = 4 (2. DEG) .
.
' . . . . . . .
1.5 2.0 B (tesla)
;
2.5
3.0
0.30
0.5

(c)'
0.25
0.4 ;>
0.20
0.3
0.15 >~ 0 . 2 ~5
0.10
0.1
0.05
0
1.5
1.7
1.9 2.1 B (tesla)
0 2.3 2.6
2.7
2.8 B (tesla)
2.9
F i g u r e 27 (a) Position of the conductance peak in backgate voltage Vg as a function of the magnetic field. The measured filling factor of the 2DEG is also indicated. Inset: Conductance versus Vg at B  2.5 tesla. Also shown are the measured addition energies in the regions, (b) 3 _> u > 2 and (c) ~, < 2 [88][R14].
quantization and the proper Coulomb interaction agrees well with the observed spectra. In contrast to the experiment of McEuen et al. where the magnetic field is directed perpendicular to the plane of the dot, Weis et al. [95] [R=a], [96, ~3s] performed experiments with magnetic field directed parallel to the current direction (parallel to the electron plane). The result (Fig. 28) indicates that the q u a n t u m numbers of the ground state energy of the QD changes with an increasing magnetic field. This leads to the observed amplitude modulation of the conductance peaks in Fig. 28.
2.4.4. Electron turnstiles Making use of the Coulomb oscillations described above, Kouwenhoven et al. [9] realized a turnstile operation for electrons in a QD. This is a class of device which can transfer
Q u a n t u m dots
41
O"
15
A
Iv
0
rn
15 5
0
5
10
VB(V) F i g u r e 28 Conductance versus the gate voltage for different values of the magnetic field B between  15 tesla and 15 tesla [138].
individual electrons around a circuit at a welldefined rate. The process of clocking electrons through a QD one by one at a welldefined rate is shown schematically as follows [Fig. 29]" (a) Initially, both barriers are high and hence the probability of an electron tunneling from left contact into the QD is negligibly small. (b) The lefthand barrier is lowered: one electron can now tunnel into the QD. Once it has, the charging energy increases the chemical potential of the QD above that of the lefthand contact, making it energetically unfavorable for another electron to tunnel into the dot (the Coulomb blockade). (c) After the leftbarrier is raised the electron is
trapped in the QD.
(d) W h e n the righthand barrier is lowered, the trapped electron can tunnel into the righthand contact. After the dot is discharged and the righthand barrier is raised again, we are back to step (a) and the system is ready to repeat the process. If the frequency at which the
T. Chakraborty
42 (b)
(a)
~
_N+I
~O ~tr
e~s
.............
(e
(d) _
F i g u r e 29 The four cycles of the turnstile process required to clock a single electron.
electrons are clocked through the QD is a~, then the current flowing is I = ecz, when only one electron passes through the dot during each cycle. Experimental results of Kouwenhoven et al. [9] are shown in Fig. 30, where the I  V curve for an oscillating barrier turnstile excited at a frequency of 10MHz is shown. There are clear current plateaus at values given by In = necz, where n is an integer. Increasing the sourcedrain bias voltage, one can make more than one electron to tunnel in each cycle. Kouwenhoven et al. obtained current plateaus corresponding to controlled transport of up to seven electrons per cycle. In principle, the turnstile can be used as a current standard just as the q u a n t u m Hall effect is used as a resistance standard [139]. One important criterion for such metrological applications is the accuracy of current quantization. One fundamental limitation to the accuracy is a leakage current that arises from macroscopic q u a n t u m t u n n e l i n g this allows an electron to move from source to drain via a virtual tunnel path through forbidden states in the dot. Such a leakage path can, however, be suppressed byreplacing the tunnel barrier with multiple tunnel junctions. On a fundamental level, one could perhaps speculate that this device, under suitable conditions, could be used to transport fractionallycharged Laughlin quasiparticles [140]. Singleelectron t r a n s p o r t in the fractional q u a n t u m Hall regime is still an uncharted territory [141].
Q u a n t u m dots
43
.
.
.
.
.
.
.
lol
.
.
.
.
.
.
.
.
.
.
.
I 1 t I
.<
5
10
...... 4
_
2
.
.
,
.
.
0
.
.
.
L
1,
2
4


V (mV) F i g u r e 30
2.4.5.
Currentvoltage curve for a turnstile at a frequency of 10MHz [9].
Photonassisted
tunneling
Tunneling in the presence of an external microwave field, photonassisted tunneling (PAT), has received increasing attention in recent years [142148]. PAT allows investigation of timedependent tunneling phenomena related to 0D levels. By absorbing or emitting photons from the highfrequency signal during tunneling, electrons can reach the normally inaccessible energy states. In this process, electrons overcome the Coulomb gap and tunnel from the left reservoir in a QD by absorbing discrete photons of energy hu from the microwaves field, see Fig. 31 (a). If the subsequent tunnel process is from the QD to the right reservoir then PAT contributes to the current. The electronturnstile device described above, also produces frequencydependent currents, but the photon energy is much too small at MHz frequencies to be energetically important. Enhancement of tunneling has indeed been observed experimentally when the photon energy corresponds to the energy difference between the incoming electron and an available state of the QD into which it tunnels. The PAT current is observed as a shoulder on the Coulomb oscillation current peaks Fig. 31 (b). PAT can thus be used as an additional spectroscopic tool to investigate the energy levels of a QD. It is expected that this phenomenon can also help the development of highly sensitive microwave detectors [149,150].
T. Chakraborty
44
(a)
(b)
~i
,
Wt~ 0
:: .
,.~ ~
i ,,,,iii ,

~
_
~.~_
1
~.._
,,,
~ I
';', ,,,,

5
0
~
,,
32
31 30 GATE VOLTAGE (mV)
2~
t
F i g u r e 31 (a) Schematic illustration of the PAT process. The solid lines are the occupied levels in the QD, while the dotted lines are unoccupied levels. (b) Observed current versus gate voltage for a QD irradiated by microwave at three frequencies. The solid lines show results for increasing power and the dashed lines are the results for current without microwaves [145].
2.4.6.
Vert ical tunneling
In a vertical QD device the current flows vertically with respect to the heterostructure layers. In these systems vertical confinement of electrons are provided by the various heterostructure layers. The lateral confinement is provided in part by lithographically etching out a pillar in a doublebarrier heterostructure. For the purpose of singleelectron transport studies, these devices have the advantage over the lateral QD devices because here the contribution to the transport current begins already with the first electron in the dot. In contrast to the planar QD devices which have tunable tunnel barriers t h a t are only a few meV high and ~100 nm long, vertical QDs have essentially fixed tunnel barriers t h a t are typically high (a few hundred meV) and thin (typically ~ 1 0 n m ) . In the case of lateral QDs the tunnel barrier increases with decreasing electron numbers in the dot. This means t h a t even in the absence of a Coulomb barrier, one gets a unmeasurably small current [92]. Vertical dot structure is therefore the best way to investigate the properties of a few electron system via electrical measurements. An example of a vertical QD device used for conductance spectroscopy is sketched in Fig. 32. Details of the fabrication process of such structures can be found in [92,151]. The most straightforward measurement one performs with these devices is the simple I  V curve, which exhibits nonohmic features with fine structures related to the energy spectrum of the QD. In t h a t respect, here one does a type of conductance spectroscopy
Quantum dots
45
V
!i
iii
li!!
GaAs:Si 5 nm 10 nm
bb= 6,7,8,9 nm
li!l
AIGaAs GaAs AIGaAs GaAs:Si
Figure 32 Cross section of a doublebarrier tunneling device used for singleelectron spectroscopy. The dashed lines correspond to the depletion layer which confines the electrons to the center of the pillar [64].
because tunneling is enhanced when an available energy level of the QD aligns itself with the Fermi level of one of the contacts. If the barriers are asymmetric and we inject the electrons from the transparent side, electrons accumulate in the QD [63, 65,152155], and the steps in the I  V curves provide a measure of the addition spectrum. When, on the other hand, electrons are injected through the lesstransparent barrier, charge does not accumulate in the dot and structures in the I  V curve correspond to true singleelectron spectrum. S u e t al. [63,152] investigated magnetotunneling in doublebarrier resonant tunneling nanometer device in the singleelectron charging regime. Their I  V results are depicted in Fig. 33 for the case when the collector barrier is less transparent than the emitter barrier and an applied magnetic field parallel to the tunneling direction, i.e., perpendicular
46
T. Chakraborty
"7.5
"i
d. ~ T .f"
180
200
220
24.0
260
v (my) Figure 33 magnetotunneling I  V curves, offeset vertically by 15 pA. Each current step corresponds to an increase of electron in the well by unity [63].
to the barriers. Several steps are observed in the I  V curves which reflect singleelectron charging of the dot. As mentioned earlier, each step in the "staircase" corresponds to an increase of the number of electrons by unity, starting from zero. The change in energy required to add an N  t h electron in the dot is (AE)N  EN  E x  1 , where EN is the total energy of the manyelectron state in the dot. This corresponds to the voltage VN at which the Nth step appears. Therefore, A V N  (VN+I  VN) corresponds to ( A E ) N + I  (AE)N = EN+I  2EN + F,N1. Hence, a step width A V N represents a change in energy needed to add the (N + 1)th electron in the dot. The magnetic field dependence of the voltage extent AV1  c~ ( E 2  2El) and AV2  c~ ( E 3  2E2 + El), where c~ is the voltage to energy conversion coefficient [63] are shown in Fig. 34. Observation of cusps in AV1 and A V2 at the same value of magnetic field was interpreted by S u e t al. as indication of spin singlettriplet transition of the twoelectron state confined in the QD. Using the device structure sketched in Fig. 32 similar results for spin transitions were
Quantum dots
47
20
I
!
i"
i
"!
I
'"1
I ..... ~ ....... 1
i '
T
"';'
ooooOO~
'
oo
15
E
%
10
+#~
<~
++++++++++++
o AV1 t
+ AV2 ~
0
1
2
i
I
4
i
I
6
i
1
1
8
10
I
I
9
12
I
/
14
B (tesla) Figure 34 The voltage extent of the current steps which corresponds to charging of the well by one extra electron versus the magnetic field [63].
reported by Schmidt et al. [64,65]. These authors employed singleelectron tunneling spectroscopy to investigate the manyparticle ground state, in contrast to the singleparticle spectrum of Sect. 2.1.1. observed by Schmidt et al. in a similar type of device. In the present case, however, the thin barrier is the emitter and the thick barrier is the collector [64]. The I  V staircase as a function of the magnetic field oriented parallel to the current direction and a grayscale plot of the differential conductance as a function of bias voltage and the applied magnetic field are shown in Fig. 35. The lowest curve in Fig. 35 (b) corresponds to the ground state energy of a single electron in the QD. The nonmonotonic behavior of the other curves is attributed to transitions in the ground state energy between states of different angular momentum and spin. In this respect, the results correspond to the capacitance measurements of Ashoori et al. [76] [R21]. Finally, vertical quantum dots containing a tunable number of electrons starting from zero were employed by Tarucha et al. [4850], [51][R33] to extract information about the manyparticle ground states via singleelectron transport, as already mentioned in the beginning of this chapter. Their vertical QD is a submicrometer pillar fabricated in an In/A1/GaAs double barrier heterostructure. Here the lateral confinement originates from sidewall depletion that is controlled by a "side gate" surrounding the structure, The number of electrons in the dot was controlled by varying the voltage on this gate.
T. Chakraborty
48
0.8
o.8 "
0.4
0.2 0.0
40
50
60
70
v (mV) 60 .
50
9
.
,
~
.,~
~
40 i
=
30
.... 0
(b) 9
,
2
9
4,
9
6,
''
8,
9
"/ 10
9
1 ,2
9
1'~4
9
1 i6
B (Tesla)
F i g u r e 35 (a) Magnetic field dependence of the I  V staircase. The curves are plotted with a vertical offset (step 0.2 tesla). (b) Magnetic field dependence of the differential conductance in a grayscale plot [64].
The cross section of the q u a n t u m dot device where Coulomb oscillations at zero magnetic field was observed by Tarucha et al. is shown as inset in Fig. 36. We have already indicated in Sect. 2.1.2 that the singleparticle states (FockDarwin states) of a circularly symmetric dot are degenerate. This level degeneracy and consecutive filling of each set of degenerate states causes the shell structure for 2, 6, 12, 2 0 , . . . electrons. Additionally, parallel filling of electrons amongst halffilled degenerate states occur in a shell at electrons numbers ne = 4, 9, 1 6 , . . . due to an exchange effect (Hund's first rule). The addition energy, AEc = #(he + 1 )  #(rte), where the electrochemical potential for
Q u a n t u m dots
49
(~"'/" ~
2O Vd  150 IJV
~
50mK
"~9,~.~~.L~
/(,
<
.... H
II:
oL _
4
"
Dot
I///~DBH 1//4 Side Gate
'
9
161
0
1.5
1.0
0.5
Gate Voltage (V)
Figure 36 Observed Coulomb oscillations in the current vs gate voltage at B  0 tesla in a gated quantum dot device by Tarucha et al. [50], shown as inset.
an heelectron dot corresponds to the position in energy of the current peak in Fig. 36 can now be readily obtained. In fact, the addition energy corresponds to the spacing between the N + 1th and N  t h current peaks. Plotted in Fig. 3T, it has large maxima for N  2,6 and 12 and also relatively large maxima for N = 4,9 and 16 [49, 50]. Interestingly, breaking of the circular symmetry was found to destroy this shell structure and will be discussed in Sect. 2.11. Figure 38 (b) depicts the position of current peaks in the presence of an external magnetic field. Positions of the first three peaks show monotonic behavior with respect to the applied field, while the others show oscillatory behavior with B that increases with the electron number. In general, current peaks shift in pairs with B. The spacing between the peaks is roughly constant for odd N and vary strongly with B when N is even. Most of these features are explained in the framework of singleparticle FockDarwin states and the constantinteraction model [50] discussed in Sect. 2.3.1. For spinless electrons, each state is twofold degenerate. The degeneracies at B = 0 are lifted as B > 0 (Sect. 2.1.1). The corresponding addition energy for even N is strongly magnetic field dependent. On the other hand, addition energy for odd N is determined only by the
T. Chakraborty
50 I '
'
I
I
I
N=2
6

<
00
5
I
I
10
15
.......
I
20
Electron number N
Figure 37 Addition energy vs electron number derived from the Coulomb oscillations in Fig. 36. effect of electronelectron interaction that is responsible for lifting the spin degeneracy. This should naturally lead to a pairing of the current peaks. The good agreement between Fig. 38 (a) and Fig. 38 (b) is an indication that the CI model works reasonably well in the fewelectron regime. However, the model must break down when exchange interaction becomes important, and is seen to be the case for filling of electrons in the second shell. Magnetic field dependence of the third, fourth, fifth and sixth current peaks, i.e., peaks that belong to the second shell, is displayed in Fig. 39. Here one clearly sees pairing of third and fourth peaks and fifth and sixth peaks for B > 0.4 tesla. By following the evolution of the respective pairs with magnetic field, Tarucha et al. identified the quantum numbers (n, l) = (0, 1) with antiparallel spins for the lowlying pairs and ( 0 ,  1) with antiparallel spins for the higher pair. Interestingly, this pairing is rearranged for B < 0.4 tesla. In this field range the third and fifth peaks, and fourth and sixth peaks are paired. Therefore, as B is increased from 0 tesla and exceeds 0.4 tesla, the forth electron undergoes an angular momentum transition from 1 =  1 to 1, while the fifth electron undergoes a reverse transition, i.e., from l = 1 to  1 . Recently, Eto [173] has reported results for the magnetic field dependence of addition spectra using exact diagonalization scheme that closely duplicates the data in Fig. 39.
Q u a n t u m dots
51 4
I'
(a)
50
I"
'
I

(b)
i
> 40
E 3O ,
..... ~ 1
.
' 
L
~c 20
o
.......

4

10 I~ " ~ I ,,,
0
I
,
~

NS 0
I
2 B (Tesla)
3
o
B (Tesla)
Figure 38 (a) Energies of a parabolic dot in the constant interaction model (b) Gate voltage positions of current peaks as a function of the magnetic field [50].
2.5
Manyelectron
systems
Before we discuss the properties of interacting electron systems, let us briefly mention some results for noninteracting manyelectron systems. Geerinckx et al. [69] studied theoretically, the electronic states, energy levels and dipoleallowed optical transitions for six noninteracting electrons in a quantum dot in the presence of a perpendicular magnetic field. They compared the results for hardwall confinement with those for a parabolic confinement potential. In the hardwall case it was found t h a t there are several transitions with different energies allowed. This contrasts sharply with the parabolic case in which only two transitions are allowed. Among the many transitions in the hardwall case, however, only a small number of transitions have sufficient oscillator strength to be observable. As the magnetic field is increased, the resonant frequency approaches the twodimensional result much faster than that for the parabolic confinement potential. Almost sixty years after the pioneering work of Fock [60][R5] on a single electron in a parabolic confinement potential and in an external magnetic field, Maksym and Chakraborty [30][R7] first introduced the electronelectron interaction in that system.
T. Chakraborty
52
1.2
> v
O
c~.13 m
0 >
O
1.4
0
0.5
1.0 1.5 B (Tesla)
2.0
Figure 39 Behavior of the thirdtosixth current peaks in a magnetic field. Spin configurations for electrons in the second shell and the angular momentum quantum numbers are indicated in the figure [50].
In order to calculate the interacting electron states, we assume that the magnetic field is strong enough to keep them spin polarized. This assumption helps us to unambiguously study the interplay between confinement and interaction [30][R7]. (The effect of spin will be discussed later.) In the spin polarized electron system the Hamiltonian is ne
E
ne
1
9 2E 2 1 e2 ~ (p~ + ~A~) ~ + ~~ ~0 ~ + 2J  2m* i=l i=1 "
1
Ir~  rjl
(2.13)
where e is the background dielectric constant. We ignore the neutralizing positive background present in real systems. For an infinite system this cancels the divergence caused by the Coulomb repulsion but for a single dot the matrix elements of the Coulomb interaction are finite. For a periodic array of dots with a large spacing the background
Quantum dots
53
cancellation merely shifts the energy levels for the single dot by a constant. In the following we present explicitly the interaction matrix elements for quantum dots that we have to evaluate numerically to obtain the electronic properties [30][R7], [106] [R15 of interacting parabolic quantum dots.
2.5.1.
Interaction Matrix elements
In order to derive the interaction matrix elements needed for the exact diagonalization scheme, we begin with the singleelectron wave functions derived in Sect. 2.1.2
~ , ( x , 0) = where x =
~/b ~
2~eg (~ + Ill)!
eil~
(2.14)
r and
(~ + IIL)! ~ ( ~ _ ~)!(lll + ~)!~! X 2t~
L~t(x~) = Z (  I ) t~0
The wavefunction can be formulated in a more convenient form, n
@nl(X, O)  Cl (nl) E C2(7Z/, N)eilOe}X2x2~+[l[ t~O where the coefficients are b n! 27rg2 (n ~l/l)!
Cl(nl) 
;
(~ + Ill)!
C2(nl, t~)  (1) ~ ( n  ~)!(Igl + ~>!~!
For the interaction potential we write the Coulomb interaction in the form e2
V([rlr21) =
~lrl  r21
e 2 J" 27r ik.(rlr2)dk. (27r)26 ~~e
(2.15)
The interaction matrix elements are then e 2 47r2g4 [ b Anln2nan4 = 511+12,13+14 g~ b2 L27cg~" Ill21314 6
hi!
!
]2
n4!
•
2~e~ ( ~
+ II~l)!
2~e0~ ( ~
+ Izzl)!
27cg~ ( n 4 f1/41)!
!
]2
T. Chakraborty
54 nl n2 n3 ~4 1 X E E E E [/'~1 3L /'~4 4 2 ( ~c1=0 ~2=0 ~c3=0 ~4=0
(1) ~1+~4
(Ttl
IiI + I/4] k)]'. [~2t
/'63t
1 ~(I/21 + ll31 k)]' .
~l/~l)!(n4I1/41)!
~1!~4!
(~1  ~Cl)!(l/llIt~l)!(Tt4 ~4)!(I/4lI/'~4)!
(_1)~+~3
(~2 Jr IZ=I)!(~3+ IZzl)!
(Tt2  t~2)!(I/21I/~2)!(Tt3 ~3)!(1/31I~3)! /%2!/~3! 1 t~l ~t~4}~(lll I+ IZ~I k) 1 [~1 t ~4 t ~(I/11 + 114I+ k)]' 1 . . [~1 I~4 Ig(l/llIIZ~Ik) ~]'(kJr ~)' s0
E
4 ~(IZ=I + Ilzl + ] [~ + ~z + ~(IZ=l Jr [/31 k) t].,(k 4t),. [~2 nL ~3 1
E t=0
(1) s+t r ( k + s + t + 3 ) 8!t!
1
2 k+s+t+l
'
where F(x) is the G a m m a function. The formula given above works very well for small values of the FockDarwin level index (NFD < 5), defined as NFD = n + ( I l l  1)/2,
(2.16)
but even for moderately high values one runs into numerical problems. The difficulty lies in the severe roundoff errors one accumulates due to cancellations between successive terms of alternating sign. The literature contains some other ways to evaluate the Coulomb integral where we can avoid the numerical problem alluded above. One such approach is due to Stone et al. [156] which is based on an integral representation for 1/r 1 rlr21 and the Taylor expansion of
e 2u2rl"r2
=
.
2
Zoo dueu2(rlr2)2
In the lowest Landau level, the singleparticle states
are r
_
1
r "~ e 
r 2/4 e imO
v/27r2mm! and the Coulomb matrix elements (in units of e2/t~0, where t~  hc/eB) are
r
(2.17)
Tt,~)  / dr I dr2 0 ~+k t (r1)r ~k(r2)r2[(~m(rl)~n(r2) lrI 1 _
Q u a n t u m dots
55 (2m + 2n + 1)!! O<2
• E
(n + p)!(m + k + p)!
p!(k+p)!
p=0
(2k + 4p  1)!! 2k4P(2p + k + n + m + 1)!" (2.18)
This expression has the advantage that all terms in the series in Eq. (2.18) are positive and therefore the roundoff error discussed above is avoided. The convergence is, however, very slow.
Manyelectron spectra
2.5.2.
The eigenstates of the system described by the Hamiltonian Eq. (2.13) are eigenstates of the total angular momentum, which is conserved by the electronelectron interaction. They can be classified by a q u a n t u m number J, which is the sum of singleelectron 1 values. The states are calculated by numerically diagonalizing the manyelectron Hamiltonian. In the following, we provide some technical details regarding the calculation of the eigenstates.
Basis states The problem was formulated using second quantization. Let us recall that the singleparticle states are a~, l O}  ~Pnl(r) where
2
a
n!
~(~ + Ill)!
exp (  i l O  (c~r)2/2) (ar)IL~ j ( ( a r ) 2 ) ,
~ .~,/h] ~
 [(L~o~ + ~c/4) . In order to construct the basis states, the number of electrons ne, total angular moment u m J = ~ 1 and the m a x i m u m of FockDarwin level indices Ntot = ~ NFD in the dot are to be fixed first. As the angular m o m e n t u m increases the number of states belonging to a certain FockDarwin level also increases. For example, for four electrons and J = 6 and total FDL index Ntot = 0 or Ntot = 1, the basis consists of twelve states
a~t%1,lla~?%2,12a~_y~3,13a~n4,1410 ), where q u a n t u m numbers (nl, ll)(1%2,/2)(Tt3,/3)(Tt4,/4) have values for the lowest FDL
(Nto~ =0) (0,0) (0, 1 ) ( 0 , 2 ) ( 0 , 3 )
T. Chakraborty
56 and for the second F D L (Ntot = 1)"
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,I)(0,2)(0,3) (0,I)(0,3)(1,2) (0,I)(0,4)(I,I) (0,I)(0,5)(I,0) (0,2)(0,3)(I,i) (0,2)(0,4)(1,0)
(0,I) (0,I) (0,I) (0,I) (0,I)
(0,0)(0,i)(0,6) (0,0)(0,2)(0,5) (0,0)(0,3)(0,4) (0,I)(0,2)(0,4) (0,2)(0,3)(i,0)
To give an idea of the size of Hamiltonian matrix one needs to deal with: for four electrons restricted to two lowest FockDarwin levels when the total angular m o m e n t u m is J = 22, there are 422 basis states and 24346 offdiagonal nonzero m a t r i x elements. Similarly, if we have five electrons at B = 7.5 tesla in two lowest FockDarwin levels the ground state occurs when J = 30 and we have 1669 basis states and more t h a n 140 000 offdiagonal nonzero m a t r i x elements in the Hamiltonian matrix. The rapid increase of the n u m b e r of matrix elements is a m a j o r problem t h a t severely restricts the n u m b e r of electrons t h a t can be studied in a q u a n t u m dot using the numerical approach. How do FockDarwin levels enter into the computations of the energy eigenstates? Let us begin with the oneparticle energy:
Enl

(2~ + IZl + 1)(h2co~/4 + h2co02)89llhcoc
=
(2NFD + 1 + 1)B  1 C ,
where we denote B  ( h W c2/ 4 + 2 h2co02)89and C _ 1 hcoc. If we assume a basis state of ne noninteracting electrons, the sum of the singleparticle energies is then Etot = 2NtotB + Ltot(B  C) + n e B where Ntot  }]~i~l IVYD is the total FockDarwin level index of the m a n y  b o d y state and J  ntot = }]~=1 ne li is the total orbital angular m o m e n t u m . As described earlier, the FockDarwin levels are degenerate when coc >> coo, i.e., at high magnetic fields. If we now restrict ourselves to Ntot = 0 and choose a certain value for Ltot, then we can construct basis states such t h a t the system is in the lowest FockDarwin level. If we consider the case of Ntot = 1, then the system is in the second level and so on. Introducing the Coulomb interaction between electrons, degeneracy of the FockDarwin levels is lifted and energy bands are formed [30][R7]. We should note that, Ntot is not conserved when interelectron interactions are switched on, but in order to restrict the basis we need to limit the m a x i m u m value of Ntot.
Quantum dots
57
ne=3 B:10T ~~40
M
>.
30
2O
u I
I
I
I
I
I
I
l
1
ne3 B:2T /
~~40
>
M
30
q
2O 1
I
I
I
1
1,
1
l
0
1
2
3
4
5
6
7
,,
I
8
J Figure quantum
40 Energy levels as a function of Y for three electrons in a GaAs dot for magnetic field B  2 tesla and B  i0 tesla [30][R7].
parabolic
Energy spectra The energy levels of a parabolic quantum dot are shown in Fig. 40 for three electrons and in Fig. 41 for four electrons. They were first calculated by Maksym and Chakraborty [30][R7] using the parameters appropriate to GaAs and w0 = 4 meV. The energies are plotted relative to what would be the lowest Landau level, that is, the constant of h (1~w c2 + w02)89per electron is not included. Here the total energies are plotted against Y at magnetic fields representative of low and highfield behavior. As seen in these figures, there are always two sets of broadened levels separated by a gap. In the limit of zero confinement these would be the lowest two Landau levels. The general trend is that the energies increase with J because the singleelectron energies increase with 1. The main difference between low and highfield behavior is the ground state angular momentum. At B = 2 tesla, the ground state appears at the lowest available J, that is, the smallest angular momentum compatible with placing all the electrons in NFD = 0
T. Chakraborty
58
70 >
r
60
pa
50
=4 B = 8 T
_ n


___:
iRBm
40
30 70
m I
I
ne =4
1
I
1
7.
l
1
B2T
60
~
5O
m
~
~~m"~
m
I
4O 30
B
' 0
'2
'
'
'
'
4
6
8
10

'
12
' 14
J Figure 41 Energy levels as a function of J for four electrons in a GaAs parabolic quantum dot for B  2 tesla and B  8 tesla [30][R7].
states. As an example, the three electron system has the ground state at Y = 3. For the noninteracting system, the ground state would have the lowest available J provided B is so high that only NFD = 0 is relevant. The interaction, however, causes the ground state J to increase with B. This effect is caused by the interplay of the singleelectron energies and the interaction energy. In the following, we consider a simple picture where only the NFD =0 states are taken into account. Then the singleelectron contribution to the energy (relative to the lowest
Landau level) is simply h ( 8 8 + a~02)1  ~a~ 1 g. The contribution from the interaction is determined by numerical diagonalization of the Hamiltonian. In Fig. 42, we show these two contributions together with their sum. The singleelectron contribution increases linearly with g because electrons in high angular m o m e n t u m states see a higher confining potential. On the other hand, the interaction contribution decreases because electrons with higher angular momenta move in orbitals of larger radii, thereby reducing their Coulomb energy. The net result is that the total energy as a function of
Q u a n t u m dots
59
ne3 B  I O T no Landau level mixing
... / /o.. ,.I
9  qj/O %%
2O 
.'*\ 9
\./
i""
9
\.," 9 'S
oo
.'"
total
o=o
E
/\
./"

\
UJ
/e 9
0%
>
10 
\
.'" ,/~..,.'_..
~176
/
of
~'~
single eleclron
,,"/ "" "'! ""/" " "I" " 
/
.""
interoction
/
I
0
.........
10
1
I
20
3 Figure 42 Contributions to the total energy as a function of J. Arrows indicate the steps in the interaction energy [30][R7].
J has a minimum. At low B this happens at the lowest available Y because the singleelectron energy increases steeply with Y. At high fields, the increase is much weaker so the minimum occurs at a higher J value.
Magic numbers One very important result here is that the ground state of electrons in a magnetic field occurs only at certain magic values [30][R7] of angular m o m e n t u m (and also spin, to be discussed later) which are dependent on the number of electrons. At these magic J values, which satisfy the relation 1 (no  1) + jn~ J  ~n~ where j is an integer, there are basis states in which electrons are kept apart very effectively. The ground state always occurs at one of these J values and the competition between interaction and confinement determines the optimum J. For ne _< 5 the basis states have all the electrons in a compact cluster in the zeroth Landau level. T h a t is,
60
T. Chakraborty
all the occupied singleelectron orbitals have n = 0 and are adjacent in angular momentum space. For ne > 5, when (he  1)fold symmetry occurs, the basis states have one electron with 1 = 0 and the remaining electrons in a compact cluster. It was shown in Ref. [157][R17], [158,159] that the Coulomb energy of the compact cluster states is reduced by an exchange contributio'n whose magnitude is very large. As a result, the total energy is reduced for these favored values of J. Similar observations were also made by other authors [160], [161][R18]. R ~ n ~t al. [lao] pointed out that the quantummechanical symmetry (Pauli principle and rotational invariance) plays a crucial role in determining the states and hence the magic numbers. These authors also noticed that the magic numbers are insensitive to details of the dynamics of the system. To summarize, the magic numbers occur because the magnetic field compresses the wavefunction of the system and increases the Coulomb energy. At certain critical fields the system can reduce the energy by making a transition to a new ground state which has a larger lateral extent and a higher angular momentum. As the magnetic field is increased we see a series of abrupt changes in total angular m o m e n t u m and system size. The selected values of angular momentum can be explained in terms of the symmetry of the minimum of the combined confinement and interaction potential [109,158,159]. Imamura et al. [162] studied the quantum states of vertically coupled dots in a strong magnetic field. They found that electron correlations in the double dot lead to a series of angular m o m e n t u m magic numbers which are different from those of a single dot. These results correspond to ground states dominated by the interlayer electron correlation. These authors proposed that the magic numbers can be investigated experimentally in vertically coupled dots. The generalized Kohn theorem, to be discussed below, does not hold for two vertically coupled dots with different confining potentials. I m a m u r a et al. surmised that the jump of angular momentum from one magic number to another should show up as discontinuities in the FIR absorption energies of the doubledot versus the magnetic field. Ruan and Cheung [163] studied a system of vertically coupled parabolic QDs, each containing two electrons. The electrons interact via the Coulomb potential. Numerical diagonalization of the Hamiltonian for this coupled system revealed an extra sequence of ground states as a function of increasing magnetic field not expected in uncoupled dots. As discussed above, the study of the interacting electron states in q u a n t u m dots revealed a wealth of very useful information. Not surprisingly then, these systems have attracted a large number of workers and as a consequence a large number of publications [164177] on variations of the work initiated by Maksym and Chakraborty exist in the literature.
Generalized Kohn theorem The magnetooptical results discussed in Sect. 2.3.2 revealed
Quantum dots
61
that the FIR excitation energies are independent of electron number. That finding is, of course, quite surprising. It means that the electronelectron interactions described above do not influence the spectra at all. It turns out that the experimental results demonstrate, albeit in a different situation, a variation of the original Kohn theorem [178]. This theorem states that, in a translationally invariant electron gas, the cyclotron resonance is unaffected by electronelectron interactions. Note that the parabolic confinement potential has the unique property that the Hamiltonian can be written as 1
~  ~~ (P ~QA) 2 ~ 89 ~2]~2 +~'~rel where P  E~'~I Pi and R = y]j rj/ne are the centerofmass (CM) coordinates, Q nee and M = rn*ne. The last term is a function of only the relative coordinates and contains all the effects of the interaction. As a consequence, the wavefunctions are simply ~b(R)~(rij) and the eigenenergies are Ent + Erd. Here we should point out that the CM energy is identical to the singleelectron energy Enl Eq. (2.4) because of the fact that e/m* = Q / M . The dipole operator 7{'= e E E . r j J
e iwt
= ~ E . R e i~t
(2.19)
where E is the applied electric field, is expressed solely in terms of the CM coordinates. It follows that FIR radiation excites the CM but does not affect the relative motion [1], [30][R7], [3941]. The interaction effects can only be probed by either deliberately engineering the dots so that the CM and relative motions are coupled or measuring the thermodynamic properties of the electrons. This important result for quantum dots by Maksym and Chakraborty has been called the generalized Kohn theorem in later publications. There have been a few theoretical studies on the effect of nonparabolicity and consequent coupling of the CM and relative motions. Deviations from the parabolic confinement potential were studied first by Gudmundsson and Gerhardts [179]. They found that a circular symmetric correction, like c< r 4 to the parabolic confinement explains the occurrence of a higher mode observed by Demel et al. [99][R12]. In order to explain the observed anticrossing, they considered the confinement potential with square symmetry, like cv ( x 4 + y4). Pfannkuche and Gerhardts [180] studied numerically the magnetooptical response to the FIR radiation of quantum dots containing two electrons (quantum dot helium). In order to study the possible deviations from the parabolic confinement, they used 7(r) =
1 9 (.042(a~ "4 Jrbx2y 2) ~frt
T. Chakraborty
62 20 14 18 12 16 l0 14
8 12 6 10 4 8 2 6 0 0
1
2
3
B (T)
4
5
0
1
2
3
4
5
B (T)
Figure 43 The lowlying energy values for (a) noninteracting and (b) interacting quantumdot helium as a function of the magnetic field. The dotted curve in (a) is fourfold degenerate. In the interacting system these degeneracies are lifted [181].
where W4, a and b are constants. They concluded that even small deviations from the strictly parabolic case cause rich structure in the FIR spectra. They also observed that the dominant features of the collective excitations are still those of a singleparticle. Anticrossing is also observed in their calculated spectra. The particularly simple system of two quantumconfined electrons (quantumdot helium) has received a lot of a t t e n t i o n (mostly for theoretical studies) because of the relative simplicity of the calculations involved [176,177, 181184]. When the confinement potential is parabolic, the energy spectra (or part of it) for the noninteracting and interacting twoelectron cases are shown in Fig. 43 (a) and Fig. 43 (b) respectively. Obviously, the Coulomb interaction "destroys most of the clear structures immanent in the noninteracting spectrum" [181]. In the noninteracting system, the energy levels tend to bunch up in groups, thereby forming Landau levels in the limit of high magnetic fields. The interaction, on the other hand, causes the energy levels to spread apart from each other. In the noninteracting system, the energy of states in the same FockDarwin level increases with angular momenturn (for a given magnetic field). That is not so in the interacting system. If we compare two states of the same FockDarwin level with adjacent angular momenta, the difference of the energies in the noninteracting system decreases as ~ 1/B. The difference between their Coulomb energy however increases as v ~ (at least at high fields). Therefore, above a certain value of the magnetic field, the state with higher angular m o m e n t u m becomes lower in energy. As in the larger systems [30][R7], this feature influences the ground state of the quantum dot helium. The importance of correlations and the accuracy of the
Q u a n t u m dots
(a)
63
"''"~"",,,,
0.3
l

(b) 0.2
exact
i
''l
,~
~
, .....
exact m
H~
.......
HE
.......
0.15
0.250.2 ....
x
........ , . . . . . . .
01
0.15 0.1
005 0.05 t
0 0
1
2
3
4
5
6
7
0
I
2
3
X
4
5
6
7
8
x
F i g u r e 44 Ground state paircorrelation functions for the exact and HartreeFock (a) L  0 states and (b) L  1 states as a function of dimensionless variable The magnetic field is 1 tesla and a H = 12.79 nm [181].
x = r/aH.
HartreeFock approximations has also been investigated in this system [182]. Calculation of the ground state pair correlation function
g(r)Tra~lZ(~(rri+rj) I where the angular brackets denote the ground state expectation value and ag defined in (2.6) reveals the importance of mixing between singleelectron states of opposite angular m o m e n t u m in the L = 0 state. For q u a n t u m  d o t helium with a parabolic confinement potential, the pair correlation function is simplified because then each state is a product of centerofmass and relative part of the wavefunction. Numerical results for the paircorrelation functions are shown in Fig. 44 for L = 0, 1. In the case of L = 1, the results of HartreeFock (HF) and exact diagonalization agree quite well, but they differ considerably in the case of L = 0. This can be understood as follows: In the HF state, both electrons are in L = 0 states and each electron state has one of its m a x i m a at the origin. As a result, there is a high probability t h a t both electrons are close to the origin in each of the product states t h a t form the HF state, producing a peak at g(0). On the other hand, the exact L = 0 state includes products of single particle states where electrons have opposite angular momenta. Since these states have nonzero values of L they do not have a m a x i m u m at the origin. The electrons are able to avoid each other and hence a peak in is away from the origin.
g(r)
Analytic solutions of QD models Although
the problem of a interacting twodimensional
T. Chakraborty
64
electron gas in a parabolic confinement potential and a perpendicular external magnetic field can be solved in various numerical methods, an exact analytic solution of the problem for a realistic interaction, of course, is far beyond anyone's reach. The singleelectron problem was solved analytically by Fock, as described in Sect. 2.1. Interestingly, for model interelectron interactions or at certain combination of magnetic and confinement potential strength, the interacting manyelectron quantum dot model can be solved analytically. For example, for a model interaction v (ri, rj)  2V0  g1 ?Tt* F 2 I r i  rjl 2
(2.20)
where 170 and F are positive parameters which can be chosen to model different types of dots, Johnson and Payne [165] obtained exact analytic expressions for the energy spectrum as a function of particle number and magnetic field. For the choice of interparticle interaction Eq. (2.20) the Hamiltonian for ne interacting electrons in a parabolic quantum dot 7{ 2m* 1 E[
1 9 w02 E Pi + eAi] 2 + ~m
i
C
i
]rl2 + E
v (ri , ry)  g *# B B E
i
si,~
i
where g* is the effective gfactor, {Si,z} are the spin components along the z axis, #B is the Bohr magneton and p~ = (p~,x,p~,y) and A = (A~,x,A~,v) are the momentum and vector potential associated with ith particle, can be diagonalized exactly by introducing the centerofmass and relative mode ladder operators
A+

1 [4n~m.hf~l
a~

4nem*hPo
{m*f~ (X =t=iY) =1=i ( P x T iPy)}
1
{m'F0 (xij =]=iyij) :7 i (piy,x T ipij,y)}
and similar operators B • b+ associated with opposite angular momentum. Here F0 = It22 neF 2] 89 In addition, the following transformations were used" R  ( X , Y ) = I Y~i ri rij  (xij yij)  r i  rj and the corresponding momentum operators, P (Px, Pv) = ~ p~ and p~j = (p~j,x, p~j,y) = p ~  Pj. For a fully spin polarized system, n e
~
,
the eigenstates of the total Hamiltonian ~ are generated by operating with A +, B +, a +, and b+ on the zeropoint wavefunction whose unnormalized spatial part ff~0 = 10/ is
~0  exp
2h where Z = X  iY and
Zij
Iz jr 2
  n e m * a iZl 2
 Xij

2n~h
i
iyij, with the corresponding energy
Quantum dots
65
E 0   h~'~ ~ ( ~ e  1 ) ~ t F 0 + Tte(Tte  1 ) V 0 
ne(g*m*/4me)hWc. 1
For large magnetic field and strong confinement (~ > n~ F) the ground state is ]L} : Ha+]O} = H ~ o i
i
with the corresponding energy [165]
1 EL = Eo + ~ne(ne  1)h
(F0  1~c) .
Taut [183] obtained analytic solutions for two electrons in a parabolic quantum dot in a magnetic field. The results are, however, available only at certain values of magnetic and confinement potential strength. Analytic expressions for energy levels and magnetization of a twoelectron parabolic dot in a magnetic field were also derived by Dineykhan and Nazmitdinov [184]. As for interelectron interactions, these authors used the Coulomb potentials. For an inversesquare form of the interparticle interaction (~r 2 the twoelectron dot system is exactly solvable for arbitrary values of the magnetic field [166, 167]. The 2Ndimensional quantum problem of N particles (e.g., electrons) with inversesquare interaction, parabolic confinement and an external magnetic field is shown [168] to reduce exactly to a (2N  4)dimensional problem independent of magnetic field and confinement potential strength, for which Johnson and Quiroga [168] evaluated an exact set of relative mode excitations. Though interesting for being exact, the results are of limited use because of their choice of interactions and other constraints. Finally, the problem of N particles of effective mass m* interacting via a logarithmic potential (i.e., satisfying Poisson's equation) in a medium of dielectric constant e confined by a harmonic field was solved exactly by Pino [185] in ThomasFermi approximation in two dimensions. The chemical potential, total energy and differential conductance were calculated and analyzed for various limiting cases in that approximation.
Other important issues of QD m o d e l s  Wagner, Chaplik and Merkt [186] have proposed that, despite the generalized Kohn theorem in a parabolic dot, internal electronic structure can still be analyzed in the FIR spectroscopy, via the quadrupole interaction of FIR radiation with quantum dots. Electron correlation effects and the FIR absorption in a squarewell quantum dot were also studied by various authors [74, 75,187]. Ugajin investigated the absorption spectra and the absorption coefficient of two electrons interacting via the Coulomb interaction and confined in the a squarewell dot
Yc~
Y) 
V0 0
if Ix[ < L / 2 a n d otherwise
[y[ < L / 2 (2.21)
66
T. Chakraborty
for various values of the strength of the Coulomb interaction between the electrons. He found that there are many different types of absorption, some induced by the Coulomb interaction. As the size of the dot was increased the intensity of a few of the absorptions was enhanced, possibly by electron correlations. Although the fewelectron calculations described above, are successful in exploring the electron states, they are limited to only a few electrons because of the size of the Hamiltonian matrix, which, as discussed above, grows rapidly with the number of electrons in the dot. In an interesting paper, Bolton [188] presented a systematic approach to calculate the optimum J astrow wavefunction for a quantum dot containing ne ~ 10 or more electrons. For a parabolic confinement potential, he showed that the neparticle problem (rte >_ 3) can be reduced to a threebody problem, by introducing special derivative operators which act on rij = ri  rj as if they were independent coordinates {rij} i < j. The resulting threebody problem is then solved variationally using the Laughlin [140] function to obtain the optimum pair functions and the ground state energies for upto 10 electrons in a quantum dot. Bolton also reported a quantum MonteCarlo calculation [189] of the ground state properties of 110 electrons in parabolic quantum dots. The generalized Kohn theorem is not valid for quantum dots containing holes (instead of electrons) [190193] because of strong mixing between the valence bands. These systerns therefore exhibit an observable coupling between the CM and internal motion. In the case of a single hole in a quantum dot, the coupling of heavyhole (HH) and lighthole (LH) states results in significant lowering of the aJ+ modes [190]. It also induces transitions with energies comparable to the separation between the lowest HH and the lowest LH subband. For many holes in the quantum dot [191], the F I R response exhibits strong deviations from the usual case that is governed by the generalized Kohn theorein. The energy levels of one and two holes in parabolic quantum dots in the absence of any external magnetic field, have been calculated recently [192]. T h a t work was later extended by these authors to the case of a perpendicular magnetic field. The energy levels in the singlehole case show strong anicrossings as a result of valence band mixing. The Coulomb interaction between two holes results in strong correlation effects. As the applied magnetic field is increased the total angular m o m e n t u m of the ground state increases in order to minimize the Coulomb repulsion. We have established that the ground state of n~electron quantum dots in a magnetic field occurs only at certain magic values of the total angular momentum, and that transitions from one magic value to another should occur as the magnetic field is increased. We also found out that this cannot be probed by infrared spectroscopy because F I R radiation couples to the CM motion and hence is insensitive to the interaction when the confinement potential is parabolic. How should we then observe the effect of interelectrons? Maksym and Chakraborty [30][R7], [106][R15] suggested that thermodynamic quantities such as, electronic heat capacity and magnetization might be sensitive to interelectron
Quantum dots
67
interaction. These quantities are, in principle, measurable, e.9., they have already been measured in a twodimensional electron gas [194197]. Maksym and Chakraborty found that the calculated field dependence of these quantities is oscillatory with discontinuities that occur when the ground state angular momentum changes, a behavior which may be important for observation of the magic angular momenta.
2.5.3. Electronic heat capacity The magnetic field dependence of the electronic heat capacity Cv in a quantum dot was studied by Maksym and Chakraborty [30][R7] from the temperature derivative of the mean energy. The results are shown in Fig. 45 (excluding the Zeeman contribution which is a small, slowly varying background). These calculations do not include the Landau level mixing, which simplifies the problem but do not sacrifice any essential physics. The results for interacting electrons (solid lines) are clearly different than those for noninteracting electrons (dotted lines). In the former case, Cv oscillates as a function of the magnetic field and has minima that are associated with crossovers from one ground state J value to another. The oscillations in C~ are a manybody effect, unlike the lowfield oscillations in C~ for a 2DEG [197]. Their origin is best understood by considering the results for T = 1 K: at this tempere~ture the dominant contribution to C~ comes from two competing ground states. This causes the doublet structure around the crossovers and is understood in terms of the magnetic field dependence of the gap between the corresponding ground states. Far away from the crossover the gap is large and hence C~ is small. Similarly, it is small exactly at a crossover because the gap is then zero. However, on either side of a crossover the gap is nonzero but small. As a result, Cv is nonzero because neither the probability of a thermal excitation nor the heat absorbed in one are vanishingly small. The oscillatory heat capacity should reveal more structures when the ground state spin depends on the magnetic field.
2.5.4. Magnet i:zation Once the manybody eigenvalues and eigenstates are available, as described in the preceding section, the magnetization can be calculated by differentiating the eigenvalues with respect to the magnetic field B. The results of such a calculation for a parabolic quantum dot are shown in Fig. 46. The top panel of each figure gives the magnetization as a function of B, calculated with and without interaction for (a) three electrons and (b) four electrons. The remaining panels show the ground state total angular momentum quantum number (J) and the ground state spin (S). All results are for GaAs quantum dots with hco0 = 4 meV. The calculations were done with the m a x i m u m value of N
T. Chakraborty
68
0.3 ne = 4
10.2 ~
3K
T :
~ > 0.2 E
.
r
0.2 ~ '
"~ ~'0.1 E
.
.
.
.
L:.................
.
_ ..................
.
2
~ .  ~
T=IK
"~,_J_.=_2_2_]J
r;JSJ 2j~_" 2.;o ., .
o
.
.
t
.
Io.1
_
,
,
,
I
J
T =IK
.
.
.

j:6
l
.
J ~2
jg t. . . . . . . . .
j
0
0
4
,
_ne:3
_~i_6 2
3K
~. . . . . 0.1
'
0
ne:4
T =
,
~
"~o. 1
ne : 3
6
8
10 12 14 16 18 20 B(T)
'
2
4
6
8

10 12 14 16 18 20 B(T)
F i g u r e 45 Heat capacity C. as a function of magnetic field for three and four electrons in a q u a n t u m dot. T h e dashed lines in the figure indicates the g r o u n d s t a t e Y [30][R7].
taken to be I; that is, one electron was allowed to have N > 0 and the other electrons had N  0. This truncation is surprisingly accurate, even at low magnetic fields. The absolute value of the magnetization is insensitive to the upper value of the N sum and the only effect of increasing it is that the position of the steps change. This is illustrated in the inset of Fig. 46(a), where the results of allowing the upper limit of the N sum to rise to 2 are shown. Physically, the steps correspond to the changes of the ground state g or both J and S, as can be seen by comparing the three panels of the figure. The magnetization of noninteracting electrons has no step because the lowest two singleelectron levels are unaffected by level crossings as the field is increased. Hence systems of up to four noninteracting electrons in the lowest spin state stay in the same angular momentum state throughout the field range so the magnetization curve is smooth. All the steps in this case are a consequence of the interaction. For five or more noninteracting electrons the magnetization would be affected by negative I levels crossing positive l levels. However, the position of these crossings would be drastically affected by the interaction. In addition, there are relatively few of them when the electron number is small (for five electrons in the lowest spin state there is only one) and they tend to occur at low field. In contrast the steps due to the interaction occur at a regular sequence of J values throughout the field range.
Quantum dots
69
(a)
(b) '
1
#.
0 9

0
"
2.5
2
2 ~~..J..
3
'~

4
10
16
8
12
/
6 4
.
.
.
.
2 0
r./)
.....
J . . . .
0 2 1 0
1 0
5
10
B (T)
15
0
5
10
16
B (T)
Figure 46 Magnetization of a parabolic quantum dot containing (a) three electrons (N = 1), the ground state angular momentum 3 and spin S, and (b) four electrons per dot. The dashdot lines correspond to the noninteracting system results [106][R15].
2.5.5. Spin transitions Spin transitions in a quantum dot with increasing magnetic field were also investigated by Maksym and Chakraborty [106][R15] for three and four electrons in a dot. The effects of spin were found to be important at fields B < 10 T. The system is spin polarized at higher fields. One effect of spin is that it causes extra steps in the magnetization. Each spin state has its own sequence of special J values, and each of those J values correspond to a possible ground state at that spin. Such behavior is a direct consequence of the interelectron interaction. Electron correlations are known to play an important role in ordinary electron systems such as the Hubbard model. The effect of spin we have just described is, however, a unique manifestation of electron correlations in high magnetic fields. A somewhat simplified version of this result was found by Wagner et al. [107][R16], who studied the magnetization and spin transitions in a twoelectron dot. They observed
70
T. Chakraborty
jumps in the ground state angular momenta that are accompanied by jumps in the total spin, which in the twoelectron case is the "spinsinglet  spintriplet" transitions. Ugajin [198] studied the effect of an external electric field, i.e., the Stark effect, on the spinsingletspintriplet transitions of a twoelectron system in a quantum dot. He calculated the energy levels and the absorption coefficient of two electrons confined in a square well dot [74, 75] in the presence of an external electric and magnetic field. Application of an electric field causes level repulsion and makes spin transitions unfavorable. Like the heat capacity and magnetization, there are other physical quantities which show oscillations as a function of increasing magnetic field. One example is the magnetoluminescence energy due to recombination in a quantum dot from an acceptor in the plane of the dot [169]. The magnetic field dependence of the quantity is also oscillatory and undergoes discontinuous jumps as the ground state changes from one angular momentum to another. Pfannkuche et al. [181] performed detailed calculations of the ground state properties of quantumdot helium (energies, paircorrelation functions, densities, etc.) in a magnetic field, using Hartree, HartreeFock, and the exact diagonalization scheme. The results of the Hartree approximation show strong deviations from those of exact diagonalization because of an unphysical selfenergy term in the former approach, which is exactly canceled by the exchange (or Fock) term. The HF results for the triplet state agree well with the exact results, showing the importance of the exchange interaction. The exact results for the singlet state, however, differ significantly from the HF results. This difference is clearly due to correlations, neglected in the HF approximation. Finally, in a typical quantum dot nanostructure, such as that of Ashoori et al. [76], tunneling of electrons between the dot and the surrounding electrodes is controlled by an applied gate voltage. The effect of the gate electrodes on electronelectron interaction within the QD has been investigated by Maksym et al. [199,200]. Given the small size of these nanostructures, one reasonably expects that induced charges on the gate electrodes used to define the nanostructures could have an appreciable screening effects on the interelectron interactions in the dot. For the specific case of a dot in a parallel plate capacitor, the screened interaction at realsitic dotplate separations (~ 800 ~), was found to have a dramatic effect on the energy level spectrum, the electrochemical potential, and on transitions of the total angular momentum and spin of the ground state with magnetic field.
2.5.6. QuantumHall dots One particular aspect of quantum dots has received considerable attention lately, viz., the electronic properties in the very strong magnetic field limit such that only the lowest
Quantum dots
71
Landau level is occupied  quantum Hall dots [201206]. Unfortunately, many of those results are primarily of academic interest and are somewhat removed from real systems as discussed below. Yang et al. [201] studied the addition spectrum (the energy to add one electron to a dot) for quantum dots with 56 electrons, using the exact diagonalization method. As discussed in Sect. 2.1, parameters that determine the ground state in a quantum dot, once the spin degree of freedom is included, are the confinement energy, the magnetic field and the Land6 gfactor. This is because the singleelectron energy depends on two frequencies: ha0 and hac. But in the highfield limit, the singleelectron energy to lowest order in B depends only on y  ~/~c. Yang et al. used 7 as an independent parameter. This means that their results are only valid at extremely high fields and most of the phase diagram is therefore, not very useful for application to real dots [207]. When the magnetic field is strong enough that only the lowest Landau level is filled by the electrons the system enters the fractional quantum Hall effect regime [20, 21] and conductance peaks in the ~ = 1/2p + 1 state are suppressed algebraically in the largene limit. This was predicted by Kinaret et al. [203205], who studied a parabolic QD containing up to eight electrons interacting via a potential of the type: V(r) cx 1/r 2. The tunneling probability, when an electron tunnels into the QD containing ne electrons, is proportional to the square of the singleparticle matrix element
M : (n~ + llc~Ine> for adding an electron in an angular momentum tion~
state c~ between ground state wavefunc
l~e> ~nd I~o + 11.
In the integer quantum Hall effect regime, both these states are Slater determinants made up of angular momentum states in the lowest Landau level (for a circular dot). The overlap matrix element is therefore unity. Due to nontrivial correlations in the fractional quantum Hall state, these authors find
M ~ n e (;1)/4 at u = 1 / 2 p + 1. Palacios et al. [206] performed exact diagonalization of the interacting Hamiltonian in a parabolic dot for up to five electrons. They obtained results for #(ne) with n = 1 (Landau level index) for high fields and n = 2 below 1 tesla. They noticed significant deviations from the results of Yang et al. [201] where the calculations were restricted to n = 0, even at large values of the magnetic field. We mentioned earlier that Ashoori et al. observed a reduction of tunneling rates for large ne around u = 2, where y is the Landau level occupation [76][R21]. Palacios et al. [206] explained that observation as due to electron correlations in the ground state. At low temperatures, the tunneling rates of single electrons are proportional to the spectral function [204,206,208,209]
T. Chakraborty
72
1 _~..~.........................2 L 2 , 2  _ ~
L
~

r
~ . . . .~. .
~
ne = 4
_
.
.
~
~
~
~


~
~
~
~=
....
ll e = .~
.
.
.
~f . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
'.
.
.
.
.
.
'
l
0
Ill
.
.
.
.
"'" ::====: ..........
'1 :: ........
=
.... .............
:
=:==IW ._=
ne = 2
....
'~
.
~
.._
0 
.
~.. .~ = ...............................................
...............................................................
. . . . . . . . . . . . . .
"
~ .
[
_
~
~

2
I
i
1
m
2
,
! 6
t
I~
.I
]rl
8
B (Tesla) Figure
47
T h e s p e c t r a l w e i g h t A ( n e ) for ne  2 
A(ne,~)
E
I ((I)~elc~l(I)~e1)12
5 as a f u n c t i o n o f m a g n e t i c field [206].
(aJ  E~(ne) + Eo(n~  1))
r/,i
(I)~e1 is the ( h e  1)electron manybody ground state wavefunction. In the linear regime, tunneling rates depend on the spectral weight [206,209] i.e., A(ne) = A(ne, #(n~)) and is shown in Fig. 47 for n~ = 2  5 as a function of magnetic field/3. When electrons form a compact droplet with ~, = 2 and minimum ISzl, A(n~) ~ 1. W i t h increasing magnetic field, spin correlations in the ground state reduce A(n~). It is proposed that 1 the spectral weight can even be zero due to spin selection rule I S z ( n e )  S z ( n e  1)I > g. The experimental work of Klein et al. [210] showed the evolution of the chemical potential #no of a dot of 30 electrons, in a magnetic field. These authors reported a divergent spin susceptibility for which a quantitative description was given using the HartreeFock theory. The accuracy of their theoretical analysis is, however, questionable: they used a single Landau level HF approach at fields around 23 tesla which will introduce some error. The important issue of how the calculational errors compare with experimental errors is not clear. The effect of electron correlations in this context was investigated by Ahn et al. [211]. They calculated the ground state energy and chemical potential of a droplet containing a few electrons via an exact treatment of electronelectron interactions with Landau level mixing included. The manyelectron states in the lowest Landau w
h
e
r
e
Quantum dots
73
B=IT ..~176~   ~
...;'..,.,~ E (meV)
i 2
1 3
I 4
I 5
(a)
{b)
L Figure 48 Energy spectrum of a single dot and the dotpair (a) without and (b) with interdot Coulomb coupling. level and in the symmetric gauge was multiplied with a Jastrow correlation factor, so that, "electron correlations are effectively included through the mixing of higher Landau level" [211]. These authors also found bumps in the chemical potential as the magnetic field is varied. This resulted in a ripple in the #  B diagram as observed experimentally. Finally, Pfannkuche and Ulloa [212] studied the overlap matrix element of ne and ( h e  1) electrons states of a parabolic dot, which governs the tunneling rate of an electron passing through the dot. They noticed that, as in FIR spectroscopy [Eq. (2.19)], in transport spectroscopy measurements the excitation spectra are dominated by the CM modes. Working with a 3electron dot they found that electron correlations strongly suppress most other transitions involving excitations of the internal degree of freedom. The CM motion, therefore, dominates the transport resonances.
2.5.7.
Coupled quantum d o t s
In a quantum dot pair coupled in series, the circular symmetry is broken and, as a consequence, radiation couples to the internal motion of the electrons. Such a system was studied by us some time ago [71][R19], [213]. The dotpair is coupled only via the Coulomb interaction (tunneling of the electrons between the dots is not allowed).
T. Chakraborty
74
We consider the Hamiltonian for the Coulombcoupled dotpair in a parabolic confinement to be of the form ~{
=
7{o 
']"{0 Jr ~{ee
~.
E a=1,2
[ 1 ~
_
e
e )2 lm, 2] P a , i  A(ra,i)c + g Wo2(r~,i R~)
"
e2 ~~ee
(
1
E=
,2 i r
e2
+_ye
1 ~,,3 .
lr l i _ r 2 j I
Here the sums over i and j run over the number of electrons in a single dot and R~ is the position of the center of the c~th dot. No attempt is made to include any singleparticle interaction between the dots, i.e., the electrons feel the other dot only through the Coulomb interaction ~ee. We show in Fig. 48 how the Coulomb interaction between the two dots couples the excitations with L = 2 (CM) and L = 5 (CM) of an individual dot. The lower mode of the dipole transition is caused by the transition to L = 4 (CM) level which has no other level to couple to. The coupling of the CM and relative motions causes interesting structures in the dipole transition energies as a function of magnetic field (Fig. 49). As explained above, the lower mode of the transition energies for electrondot pair is always close to the singleparticle mode. On the other hand, the upper mode is seen to exhibit interesting anticrossing behavior due to the coupling described above. Theoretical work on the correlation effects in coupled quantum dots and F I R absorption in those systems has also been reported recently [214,215]. Here a two electrons system was considered in dots which were assumed to be wells with finite barriers (squarewell quantum dots). A square is divided into two in which the spacing plays the role of tunneling barrier. FIR spectra was calculated and the absorption coefficients were found to be strongly influenced by the separation between the dots. In fact, with increasing separation between the dots or increasing the barrier heights, the quantum states of two electrons in coupled quantum dots were found to change from uncorrelated to highly correlated states [215]. F I R absorption reveals the transition from uncorrelated to correlated Wigner lattice states to Mott insulating states as the separation between the dots increases [74]. Kempa et al. [216] studied a twodimensional square array of dots modeled as point dipoles. They noted that if dots that contain a large number of electrons are brought sufficiently close, phase transitions occur in which the dot system polarizes spontaneously. The phase transition is of second order and can lead to both ferroelectric and antiferroelectric arrangements; the latter phase is the most stable.
Quantum dots
75
d :100 nm 000 ~
__
o0o0O~ 9 ....o o o O V J
~

~,,.,,~
o o o o o ~ " ."~ . . . ~
E (meV)

~ ,,' ' ~ ' ~  ~ ~ " ~ '~Y ""~ ~o
."
.[
I 0.0
0.5
o o o
~
0
I
,l,
1.0
1.5
2.0
B(T)
Figure 49 Dipole transition energies and intensities of a threeelectron per dot pair. Dot separation is 100 nm and the confinement energy is hwo = 2.5 meV. Solid lines are oneparticle transition energies. Diameters of the open points are proportional to the intensity of the transition. Palacios and Hawrylak [217] studied correlated fewelectron states in vertical doubledot systems. A critical distance between the dots was calculated where the interdot correlations are found to be important and minimumisospin (or quantum dot index) ground states appear. Similar studies were also reported by Oh et al. [218] who investigated the electronic structure and the optical properties of vertically coupled QDs in magnetic fields. They performed exact diagonalization of the Hamiltonian matrix for one and two electrons in a coupled dot characterized by a parabolic potential in the zyplane. In the growth direction they used the vertical potential consisting of two square wells of equal width (150 .~), a barrier width of 50A, and two buffer layers with thickness of 350 ~. The barrier height of 147 meV was chosen to represent the GaAs system. They found that interdot and interelectron interactions strongly affect the ground state, which can, in principle, be observed via optical spectroscopy. For vertically polarized light, the calculation of resonant energies revealed blueshift and sharp drops with increasing magnetic field that was attributed to the electron_electron interactions.
T. Chakraborty
76
2O
~" > 15 10
5 20
~" 15
lO
5 0
2
4
6
8
10
B (W) F i g u r e 50 Lowest singleelectron energy levels vs magnetic field of a parabolic quantum dot containing a repulsive scatterer (170  10 meV, d = 5 nm) located (a) at the center of the dot and (b) at 5 nm away from the center. In figure (a) some values of the angular momentum quantum number are indicated.
2.6
Electronimpurity systems
While most theoretical studies of q u a n t u m dots involve impurityfree dots, Halonen et al. [73][R24], [219] studied the effects of a repulsive scatterer in a q u a n t u m dot. Repulsive scattering centers in q u a n t u m dots have profound effects on the energy s p e c t r u m of a q u a n t u m dot in a magnetic field. The s y m m e t r y  b r e a k i n g electronimpurity potential introduces additional structures due to level repulsions, lifting of degeneracy etc. in the otherwise well u n d e r s t o o d s p e c t r u m of an impurityfree parabolic q u a n t u m dot discussed above. Halonen et al. analyzed those structures and extracted some simple rules t h a t determine when the level repulsions are supposed to occur. It should be pointed out t h a t t r a n s p o r t properties of q u a n t u m dots with an impurity whose s t r e n g t h can be controlled independently are under active investigations [220] and energy spectra like the ones presented here can be observed in experiments such as singleelectron capacitance We begin with the s t a n d a r d model in which electrons of effective mass m* are confined
Q u a n t u m dots
77
within the z = 0 plane by a parabolic potential and are subjected to a perpendicular magnetic field B. In the presence of a symmetrybreaking impurity potential, the manyelectron Hamiltonian is written as 7{ 1 ne ~e ne e2 1 E (~Az) 2 1 * 2E 2 E Vimp 1 E  277~* i1 Pi Jr  " Jr ~7~'b &O i=1 ri + i=1 (ri) + ~ e ~#j I r ~  r~l (2.22) where ne is the number of electrons in the system and e is the background dielectric constant. We also use the symmetric gauge vector potential, A = g1 (  B y , Bx, 0). The impurities are modeled by a Gaussian potential V imp (r)  V0e (rR)e/de,
where V0 is the potential strength, d is proportional to the width of the impurity potential (the full width at half maximum ~ 1.67d), and R is the position of the impurity. We apply the exact diagonalization method by constructing a basis using singleparticle wavefunctions of a perfect parabolic quantum dot ~nz (r)

Ceil~
where C is the normalization constant, a  v/h/(rn*t2), ft  v/cz02 + a~c2/4, and L~(x) is the associated Laguerre polynomial. The quantum number 1 = 0, +1,t2,... is the orbital angular m o m e n t u m quantum number and n = 0, 1, 2 , . . . is the radial quantum number. In the actual calculations electron spins are taken into account but the Zeeman energy is ignored. The impurity potential V imp (r) introduces an interaction between the singleparticle states ~nt (r) with matrix elements
f
(~'~1/1 (r) V imp (r)Fn212 ( r ) d r =
V0ei(/1/2)O
•
(d/a) 2 (R/a) k e(R/a)2/[l+(d/a) 2]
nl ,
n2 ,
(~1 + Ill I)! (n2 + 121)!
~=  1 ) ~ (n2 + • ~ ( 3! \ n2 /3=0 ~+/3+(IZl I+ I/2Ik)/2
p=O
• s~O = s,
1/2~
c~!
o~0
IZ21)(c~ +/3 ~3
(_1)~ ( n l +
(llll
,, ~1 c~
+ Iz21 k)/2)!
k +p
s
II~l)
[l+(d/a)2] s+p+k+l '
T. Chakraborty
78
'
22
I
'
I
'
I
'
I
'

16
22
>
20
Q)
18
16 0
1
2
3
4
5
B (T)
F i g u r e 51 Lowest energy levels of a twoelectron parabolic quantum dot containing a repulsive scatterer (V0  10 meV and d = 5 nm) located (a) at the center of the dot and (b) at 5 nm away from the center.
where k  [11 12] and the position of the impurity R is represented in polar coordinates
(R,e).
2.6.1. Energy spectrum Some singleelectron energy levels of a parabolic q u a n t u m dot with an impurity at the center are shown in Fig. 50 (a). The material p a r a m e t e r s chosen for the numerical results t h a t follow are a p p r o p r i a t e for GaAs q u a n t u m dots, e = 13, electron effective mass m* = 0.067me, and ha0 = 4 meV. There are several features in the s p e c t r u m which distinguish it from t h a t of an impurityfree parabolic dot (Fig. 4). First of all, the ground state has different angular m o m e n t u m at different magnetic fields. Further, the impurity potential mixes energy levels t h a t have the same angular m o m e n t u m but different principal q u a n t u m number. The degeneracy at B = 0 is partially lifted, and different FockDarwin bands are clearly visible. The level spacings are also different from those in impurityfree parabolic q u a n t u m dots.
Q u a n t u m dots
79
T h e energy levels of a parabolic q u a n t u m dot with an i m p u r i t y which is near b u t not exactly at the center are shown in Fig. 50(b). Here the angular m o m e n t u m is not a g o o d q u a n t u m n u m b e r and all degeneracies at B = 0 tesla are lifted. T h e r e is no a b r u p t change in the g r o u n d s t a t e as the m a g n e t i c field is changed. Anticrossings in the lowest FockD a r w i n b a n d are clearly visible. T h e r e also are several anticrossings in the higher energy levels. C o m p a r i n g these results with those for an impurityfree parabolic dot (Fig. 4) we find t h a t the s t r e n g t h of the energy level repulsion (anticrossing) d e p e n d s strongly on the difference in the q u a n t u m n u m b e r s n a n d 1 of the a p p r o p r i a t e states. T h e level repulsion is s t r o n g e s t for states t h a t have the same value of n a n d t h a t differ as little as possible in I. C o m p a r i n g Fig. 50 (a) a n d Fig. 50 (b) note t h a t level repulsion is s t r o n g for the levels in the lowest F o c k  D a r w i n band. In Fig. 50(a) the 1 = 0 state (i.e., the g r o u n d state at /? = 0 tesla) crosses some of the lowest energy levels of the lowest b a n d as the m a g n e t i c field is increased. B u t as soon as the circular s y m m e t r y is broken by m o v i n g the i m p u r i t y away from the center [Fig. 50(b)] the states in the lowest F o c k  D a r w i n b a n d are mixed. This results in s t r o n g anticrossing such t h a t only hints of the original 1 = 0 energy level can be seen at higher m a g n e t i c fields b e t w e e n energies 10 and 15 meV. T h e track of the original I = 0 energy level becomes m u c h clearer as it crosses the o t h e r levels with higher value of l, i.e., as the m a g n e t i c field is further increased. A strong anticrossing effect is also seen at B = 0 tesla b e t w e e n states with n = 0, 1 =  1 a n d n = 0, 1 = 1, i.e., A n = 0 and A1 = 2. In Fig. 50(a) these two states are d e g e n e r a t e at B = 0 tesla with energy eigenvalue a b o u t 8.6 meV. In Fig. 50(b) this d e g e n e r a c y is clearly lifted due to the b r o k e n circular symmetry. A n o t h e r s t r o n g anticrossing can be seen at B = 1.4 tesla near the energy value of 10.5 meV. This anticrossing c o r r e s p o n d s to the crossing of the energy levels with n = 0, l =  1 a n d n = 0, 1 = 2 of Fig. 50(a). Here A1 = 3 and the level repulsion is clearly weaker t h a n the one b e t w e e n the states with n = 0, 1 =  1 a n d n = 0, 1 = 1 where Al = 2. T h e r e is also an equally s t r o n g anticrossing at higher energy near E = 19 m e V a n d B = 1.4 tesla. This anticrossing results from the level repulsion b e t w e e n states with n = 1, 1 =  1 and n = 1, l = 2, i.e., here also A n = 0 and A1 = 3. A l t h o u g h there seem to be m a n y level crossings in Fig. 50(b) these crossings are actually anticrossings. Because the s t r e n g t h of the level repulsion d e p e n d s strongly on the difference in n and 1 the gap b e t w e e n m a n y of the energy levels is too small to be seen in the Fig. 50(b). T h e fact t h a t there are no crossings of the energy levels m e a n s also t h a t there is no conserved q u a n t i t y other t h a n the energy. In t h a t sense the s y s t e m seems to be chaotic. As the i m p u r i t y is m o v e d further away from the center of the dot, interactions b e t w e e n the states of the impurityfree dot first increase resulting in s t r o n g e r anticrossing effects. B u t w h e n the i m p u r i t y is far e n o u g h its effects are reduced a n d the energy levels begin to resemble the levels of an impurityfree parabolic q u a n t u m dot.
80
T. Chakraborty
The energy levels of a quantum dot containing two interacting electrons and the impurity at the center are shown in Fig. 51 (a). Clearly, the spin singlettriplet transition is moved from about 2.5 tesla (impurityfree case) to about 1.5 tesla due to the presence of the scatterer. Similar results for systems with the impurity is moved away from the center are shown in Fig. 51 (b). As expected, there is no degeneracy at B = 0 tesla. Clear energy level repulsion can also be seen in this case. We have done a detailed analysis of how the energy levels shown in Fig. 51 (a) change as the impurity moves away from the center of the dot. Just like the singleelectron energy levels (Fig. 50) we also find angular momentum selection rules that govern the strength of the level repulsion. If the angular momentum of a pair of crossing energy levels of Fig. 51 (a) differs by two there is a large anticrossing of the corresponding levels in Fig. 51 (b). If the difference is some other even number then the anticrossing is weaker but still not insignificant. However if the difference of the angular momentum q u a n t u m numbers is an odd integer then the level repulsion is almost negligible. Since we are dealing here with two mutually interacting electrons with opposite spin it is evident that the change from the oneelectron case in the rules governing the strength of the level repulsion is due to the Coulomb force. We have done similar calculations for three and fourelectron dots. These calculations support, for electron numbers higher than one, a simple rule that if there is an even (odd) number of electrons, the level repulsion is strong for states that correspond to the states in a corresponding circularly symmetric system (i.e., impurity is at the dot center) whose angular momentums differ by an even (odd) number.
2.6.2. Optical absorption spectrum Figure 52 shows optical absorption of a quantum dot containing one, two, and three electrons and one impurity near but not at the center of the dot [72], [73][R24], [219]. The point here is that any increase of the number of electrons will result in a decrease of the effect of the impurity on the optical absorption. In the oneelectron dot the degeneracy of the absorption modes at B = 0 tesla is clearly lifted. But as the number of electrons is increased the level repulsion decreases substantially. Also the clear structures seen in the one and twoelectron results have almost all disappeared in the threeelectron case; the optical absorption intensity is more or less scattered around the original modes of an impurityfree quantum dot. Figure 53 shows the optical absorption of a twoelectron quantum dot containing one impurity which is near but not exactly at the center of the dot. The absorbed light is now linearly polarized. The main result is that polarization affects absorption only at low magnetic fields. If the electric field of the electromagnetic radiation points along the axis
Quantum dots
81
ne=l
I e
.o O00
. oooOO
00'
OOOOOOoooo0000 _
>" 6
2
_
1:;
n,=2
oOOO ~.~176
..1
~,! ............ I ..........
I
ne=3
oe6
.6;
s at'ee~
_"~176176176176 ~.J 0
J
......1
1
2
........... !_ 3
_
s (T)
Figure 52 Dipoleallowed optical absorption energies and intensities of a parabolic quantum dot containing a repulsive scatterer (V0 = 4 meV and d = 5 nm) located 5 nm away from the center of the dot. Areas of the filled circles are proportional to the calculated absorption intensities. The number of electrons in the dot is indicated inside the figures.
that goes through the center of the dot and the impurity, (say, the z axis), the absorption is strongest for the socalled bulk mode (the upper mode) at low magnetic fields [Fig. 53 (a)]. If the absorbed radiation is polarized perpendicular to that axis, i.e., along y axis, the absorption is strongest for the edge mode (the lower mode) at low fields [Fig. 53 (b)]. As the magnetic field is increased the difference on the absorption intensity between these two polarization directions rapidly disappears. One explanation to this effect could be that the electron density near the impurity decreases as the magnetic field is increased. This is because the electron density moves away from the center towards the edges of the dot as the magnetic field is increased. Finally, in Fig. 54 we show optical absorption of a quantum dot containing five impurities and one, two, and three electrons. Clear similarities with the oneimpurity results (Fig. 52) can be seen here. The primary difference is that the degeneracy at B = 0 is not as strongly lifted, i.e., the system is effectively more circular symmetric than the one impurity case. Here also an increase of the number of electrons reduces the effects of impurities.
T. Chakraborty
82
I
I
I
I
(a)
~
>
v
9 ;
.
6 4
.oo~O *.~176 o
<1
e
O
!
e
9 ~~ e O 0 0 0 0 0 0 _
I
I
i
e
(b)
6
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: oeeeeooo. .
2
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,
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2
3
B (T)
F i g u r e 53 Dipoleallowed optical absorption energies and intensities of a parabolic twoelectron quantum dot containing a repulsive scatterer (V0 = 4 meV and d = 5 nm) located 5 nm away from the center of the dot at x axis. The absorbed electromagnetic radiation is linearly polarized with the electric field pointing (a) along the z axis and (b) along the y axis.
Thus far, we have discussed about systems with V0 being small (210 meV). Let us now consider the case of a repulsive Gaussian potential with V0 = 32 meV placed at the center of the dot. In t h a t case electrons are confined in a wide ring. B o t h the effective radius and width of this ring are taken to be about 20 nm for a single electron. Figure 55 shows electromagnetic absorption energies and intensities of t h a t system with one t h r o u g h three electrons as a function of the magnetic field. Oneelectron results reveal four distinct modes. The strongest of the upper two modes can be i n t e r p r e t e d as a bulk m a g n e t o p l a s m o n mode according to its asymptotic behavior, i.e., its energy approaches hwc as the magnetic field is increased. The origin of the discontinuities near 5 and 8 tesla can be traced back to the fact t h a t the potential forming the ring, in our case, is highly asymmetric: We have a steep Gaussian potential near the center of the dot and the outer edge is formed by a soft parabolic potential. For a s y m m e t r i c potential we expect the bulk m a g n e t o p l a s m o n mode is a s m o o t h function of the magnetic field. The situation is somewhat similar to the experimental work by Dahl et al. [221], who investigated m a g n e t o p l a s m a excitations of a 2DEG confined in a ring [which can alternatively be considered as a disk with a repulsive s c a t t e r e r  also known as an antidot
Q u a n t u m dots
83
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_ _,, . . . ' "
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B (T) F i g u r e 54 Dipoleallowed optical absorption energies and intensities of a parabolic QD containing five repulsive scatterers. The locations of the scatterers, which are denoted as open circles and with parameters V0 = 2 meV and d = 5 nm, are shown as inset. Areas of filled circles are proportional to the calculated absorption intensities. The number of electrons in the dot is also indicated.
(see chapt. 3) at its center (inset of Fig. 56)]. T h e y employed millimeterwave spectroscopy and their observed resonance frequencies as a function of applied m a g n e t i c field are displayed in Fig. 56. Several distinct modes are clearly visible in the data, two of which are seen to split into doublets at finite m a g n e t i c fields. These modes are labeled by con,+, n = 0, 1, 2 , . . . , where n is the n u m b e r of modes in the radial direction and the second, a n g u l a r index can in general assume the values • but was found to be restricted to m = +1 in the present case. Due to circular s y m m e t r y , con+ = con_ a t / ? = 0. In the ring geometry, the upper b r a n c h of the lowest doublet, the coo+ m o d e exhibits a positive dispersion near B = 0, as one would expect for the u p p e r branch of a circular disk. But for B > 0.25 tesla, the coo+ b r a n c h decreases with increasing B, while its oscillator s t r e n g t h decreases rapidly. These are the characteristic signatures of edge m a g n e t o p l a s m o n s ( E M P ) . For larger B, Dahl et al. used an a p p r o x i m a t e formula for dispersion in a disk [222], coemp CX:in d/do + c o n s t a n t , where d is the disk d i a m e t e r and do depends on the sample p a r a m e t e r s . T h e solid line in
T. Chakraborty
84
In e
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B (T) F i g u r e 55 Absorption energies and intensities of a QD including a strong repulsive scatterer (wide ring) with one, two, and three electrons as a function of the magnetic field. The areas of filled circles are proportional to the calculated absorption intensity.
Fig. 56 is for d = 12#m and the dashed line is for d = 50pm. These two values correspond to inner and outer diameters of the rings respectively. Comparing the calculated results with the data, these authors concluded that the coo+ mode corresponds to the E M P localized at the inner boundary of the ring, while the coo mode corresponds to E M P at the outer edge. Coming back to our theoretical results for a wide ring, if we ignore the discontinuities in Fig. 55 discussed above, the two upper modes of the oneelectron spectrum behave clearly the same way as the experimental results of Dahl et al. [221]. However, the two lower modes behave differently (in the oneelectron case) from those experimental results. The lower modes, i.e., the edge magnetoplasmon modes, reveal a periodic structure similar to the case of a parabolic ring [72], [73] [R24]. T h a t is however true only for the oneelectron
Q u a n t u m dots
85
l F i g u r e 56 Frequencies of magnetoplasma resonances for an an array of wide rings. The lines are the dispersions discussed in the text [~21]. I
system. When the number of electrons in the system is increased the periodic structure of the edge modes (the two lowest modes) starts t o disappear. This is, of course, due to the electronelectron interaction. The Coulomb il lteraction is important in wide rings. It should be emphasized that because the spin deg tee of freedom is also included in these calculations, the difference between the one an( twoelectron results is entirely due to the Coulomb force. The lowest mode (which is also the strongest) behaves (even for only three electrons) much the same way as does the lowest mode in ;he experiment [221] (where the system consists of the order of one million electrons). It is interesting to note that this mode is also similar to the observed magnetoplasma resol lance in antidot arrays [223][R37] In the high field regime, the upper mode observed in antidot systems is also qualitatively reproduced in the quantum dots. s For more about this experiment, see Sect. 3.1.1.
86
T. Chakraborty
Figure 57 Capacitance vs voltage for selforganized dots and rings. The ring structures are shown as inset [225].
Hawrylak [224] has studied the effect of a tunable artificial impurity in quantum dots in a magnetic field. The impurity was found to induce electronic spin and charge transitions which modify the chemical potential of the dot and therefore can be observed in resonant tunneling experiments. Our theoretical work presented in Fig. 55 received strong support from recent experiments on quantum rings. With the help of capacitance and FIR spectroscopy, Lorke et al. [225,226] investigated the electronic structure of selforganized quantum rings. In the process of creating selforganized InAs islands (see Sect. 3.3), these authors have been able to change the shape of the islands [227] and create ringlike structures (Fig. 57) of which the inner diameter is ~ 30 nm and the outer diameter ~ 80 nm [225]. Capacitance spectroscopy on these systems revealed structures (Fig. 57) that were distincty different from those of quantum dots. Those structures were interpreted as due to filling of first and second electron level. The FIR response of quantum rings containing two electrons as shown in Fig. 58, exhibit behavior remarkably analogous to those of Fig. 55 (in particular the upper modes), predicted theoretically [73][R24]. Creation of these selfassembled disorderfree quantum rings represents a major development in lowdimensional struc
87
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I
II
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i
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'
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ne=2 
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I
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!
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F i g u r e 58 FIR resonance position of selfas',sembled rings versus the applied magnetic field for twoelectrons per ring [225].
tures where many new and exciting physics driv( ~n by confinement, electroncorrelations, and the influence of an external applied field ca~1 be explored.
2.7
Exciton spectrum /
Theoretical work on the excitons in a q u a n t u m dot in magnetic fields has been reported by Halonen et al. [228] [R25]. Earlier work by Bryant on excitons [229] and biexcitons [230] in q u a n t u m boxes (in the absence of a magnetic fiel( 1) demonstrated the competing effects of q u a n t u m confinement and Coulombinduced elecl :ronhole correlations. More recently, excitons and biexcitons have also been studied in s( miconductor nanocrystallites [231233]. Measurements of the exciton binding energy in t he presence of a magnetic field has been reported in q u a n t u m wells [234236] and quantu m wires [237]. In our work on excitons in a quantum dot im magnetic fields [228][R25] the model Hamiltonian for a twodimensional hydrogenic eXciton" in a parabolic confinement potential and a static external magnetic field is /
 ~e +~J+~h where the electron, hole, and electronhole terms are
T. Chakraborty
88
[
1 ihV~me
e ]2
Ae c
1 2 2 At ~ 71~eCUe T e 2
1 [ih~7h + eAh] 2rnh c e2 1 ~h
I~ 2 2 + ~ 't t t h ~ h l'h
= 6
Ire  rhl"
Here e is the background dielectric constant. Let us now introduce the center of mass (CM) and relative coordinates R = ~ (mere + ? T t h r h ) , r   r e  rh, and the usual notations, M  rne + mh, #  m e m h / M , and 7 = (mh  m~)/M. We also choose the symmetric gauge vector potentials for electrons and holes as A~ = g1B x ( r e  rh) and A h   s B 1 x ( r e  rh). The Hamiltonian can then be rewritten as ~'{  ~I~CM Jr ~'{rel ~ ~{x,
where the various terms are ~CM 
']~rel
h2 _~ 1M 2 M V~M + ~
h2
ihe B
2fiVr~o~+ 2~,~ 1 +3#
ihe
~cB.r
e2B 2 4#2c2
_4_
1
1 (meWe2qmhW~)
R2
.r • Vre, ]
(?TZhad2+ ?Tte(.d~) 7.2
x VOM + ~ ( ~   ~ )
e2 1 r
R.r.
The CM t e r m is the Hamiltonian of a wellknown twodimensionM h a r m o n i c oscillator with energy s p e c t r u m
ECM 
2nCM +/CM + 1) hWCM
1 ('%~+'~h~)
1
~CM 
n C M ~ 0. The other term ~/rel is the Hamiltonian of a twodimensional charged particle in a magnetic field and in parabolic and Coulomb potentials. T h e radial part of the SchrSdinger equation
with
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20
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B (T) F i g u r e 59 The ground state and lowlying excitation energies of a lighthole (rnh = 0.09rn) exciton as a function of the magnetic field (tesla). The confinement potential energy is taken to be 15 meV for both particles.
R" +  1 Rl r

4h2c2 + Mh2
R0
can be solved numerically. All interactions between CM and relative motions are included in the cross t e r m ~ x . The results for the lowest energy levels of a light hole (rnh = 0.09rn) exciton as a function of the magnetic field is shown in Fig. 59 without (a) and with (b) the cross t e r m ~ x added to the Hamiltonian. We plot only those levels t h a t have the total angular m o m e n t u m L = 0. Although the effect of the cross t e r m is much larger for the excited states t h a n for the ground state, one can identify at least some of the lowest levels using the noninteracting energy levels [Fig. 59 (a)]. Due to the confinement, the relative angular m o m e n t u m of the exciton can have different angular m o m e n t u m values to get L = 0. In Fig. 59 (a) the energy levels for l r e ] = 0 (solid lines) and l r e ] = +1 (dashed
T. Chakraborty
90
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Figure 60 Optical absorption energies and intensities of a lighthole exciton as a function of the magnetic field (tesla). Diameters of the filled points are proportional to the calculated intensity of the absorption. lines) are shown. For each energy level of the relative motion there is a spectrum of the CM levels separated from each other by the amount of the confinement potential energy. To illustrate this point more clearly, we have labeled each level by its (exact) quantum numbers: nON ,/CM, nrel, and lrel. Addition of the cross term ~• results in anticrossings in the energy spectrum [Fig. 59 (b)]. Moreover, when the magnetic field is increased some energy levels begin to form the first and second Landau levels. The intensity of optical absorption is calculated from (~(r))  ~
ij
c*cj R~f~,lfe, (0) R L, zL (0) 51~e~,lL5%M,~JCMflSM,l~M,
which gives the probability of finding the electron and hole at the same position. The numerical results are presented in Fig. 60, where a rich antierossing structure of the optically active energy levels is found to be still present. Photoluminescence experiments on quantum dots provide important information about the states of excitons [238], [239] [R26] localized in the quantumconfined lowdimensional systems. Sharp peaks (due to recombination of an exciton from discrete energy levels) in the photoluminescence (PL) spectra are usually attributed to localization of excitons in quantum dots. Zrenner et al. [239][R26], [240,241] reported results on optical spectroscopy on single quantum dots  termed "natural quantum dots". These dots originate from well width fluctuations in a narrow quantum well (Fig. 61). In a narrow well, wellwidth fluctuations of several monolayers (MLs) result in lateral potential variations which can be quite sizable and excitons can be localized in regions with locally enhanced well
Quantum dots
91
Figure 61 Interface roughness in a quantum well on an atomic length scale. Excitons are localized in regions with locally enhanced well w i d t h  the natural quantum dot [241]. width [240,241]. For a GaAs quantum well, a local variation of the well width from 10 ML to 12 ML results in local reduction of the exciton energy by 43 meV [240]. Optical spectroscopy on single quantum dots are possible by spatially resolved PL spectroscopy and resonant charge injection in an electricfieldtunable coupled quantum well structure. Excellent details of the experiments can be found in Refs. [240,241]. A very interesting result by Zrenner et al. [239][R26] was the demonstration of the existence of zerodimensional states in coupled quantum well structures. Because the dots are asymmetric with strong confinement in the z direction and weak confinement in the xyplane, their response to the parallel (BII) and pendicular magnetic field (B• should be very different. In their magnetooptical measurements, those expectations were precisely realized. The positions of the observed emission lines as a function of BII and B• are shown in Fig. 62 (a) and Fig. 62 (b). Clearly, there is negligible influence of BII on the position of the lines, but for B• one observes complicated level shifts, splittings and anticrossings. In fact, the 3 or 4 emission lines on the lowenergy part of the spectrum, show very similar behavior predicted theoretically (Fig. 60) and discussed above [228][R25]. There are a few other interesting investigations of the optical properties of quantum dots reported in the literature [242,243], [244][R27], [245] which provide information about the singleparticle states of electrons and holes. Bayer et al. [242,243] reported results on low intensity magnetoluminescence spectroscopy. They studied electronic transitions corresponding to the ground state and lowlying excited states of GaAs/InGaAs modulated barrier quantum dots. From scanning electron spectroscopy, they concluded that the barrier region is cylindrical. The magnetic field was applied along the growth direction, and therefore the orbital angular momentum around the cylindrical axis was a good quantum number. As the lateral and vertical confinement potentials of these dots, in contrast to a parabolic confinement, are abrupt and rather shallow, the problem is nonseparable in the radial coordinate r and the coordinate in the zdirection. These authors calculated the electronhole states using finitedifference method [242]. The results are shown in Fig. 63 together with the experimentally observed results. Comparison of the theoretical and experimental results clearly show that at zero field the ground state
T. Chakraborty
92 1732 BII
GaAs/AlAs
30A/40A
~" 1730 T = 4.2K
........................ >,, ~1728 i
~
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2
4
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Figure
62
Position of the emission lines as a function of (a) Bil and (b) B •
[239][R26].
is nondegenerate with zero angular momentum, and the first excited electron and hole states are doubly degenerate with angular momenta m + I. At a nonzero magnetic field, the first excited states split according to two orientations of the angular momenta, and a higher state (m = 2) becomes localized in the dots. The magnetic field dependence of the calculated transition energies compare well with the experimental results. Similar results were also obtained by these authors for the deep etched InP/InOaAs/InP quantum dot structures [243]. Rinaldi et al. [244][R27], [245] also reported observation of the Zeeman effect in parabolic InGaAs/GaAs quantum dots in magnetoluminescence experiments. Their photoluminescence spectra show splitting of interband transitions corresponding to quantum dot states with n + Iml < 5 induced by an axial magnetic field. Here, n and m are the principal and azimuthal quantum numbers respectively. The splitting is due to lifting of degeneracy of the excited states (Zeeman effect), similar to what was discussed in the FIR experiments. The authors concluded that the magnetooptical properties of strongly
Q u a n t u m dots
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B (T) F i g u r e 63 Experimental results on magnetophotoluminescence as a function of magnetic field for modulated barrier dots of three different diameters. Theoretical results (solid lines) are discussed in the text.
confined quantum dots reflect the singleparticle states rather than excitonic effects. The effect of magnetic and electric fields on excitons in parabolic quantum dots was investigated by Jaziri and Bennaceur [246]. They calculated the energy and oscillator strength of a Is exciton in a parabolic quantum dot in the presence of parallel electric and magnetic fields. They found that application of an electric field results in a spatial separation of carriers leading to a decrease in the exciton energy and the oscillator strength. Application of a magnetic field leads to additional confinement that in turn leads to additional exciton energy and oscillator strength. For narrow dots (R0 < I00 .~, /~0 ~ V/~/P(~)0, OgO : W e  ~2h), electric and magnetic fields have little effect on the properties of the excitons. Kulakovskii et al. [247] reported magnetophotoluminescence spectroscopy of multiexciton complexes consisting of two and three excitons confined in InGaAs/GaAs quantum dots with lateral dimensions slightly greater than the exciton Bohr radius. They found that the Coulomb correlations in the twoexciton complex enhances the confinement. This additional confinement is strong at zero magnetic field but an increase of magnetic field results in a reduction of the effect. The threeexciton state was found to have an energy greater than three times the single exciton energy. Therefore a threeexciton complex is confined only by the geometric confinement potential of the quantum dot. In such a complex, the excitonexciton repulsion is strongly reduced in a magnetic field [247].
94
T. C h a k r a b o r t y
A detailed account of photoluminescence spectroscopy on multiexciton complexes in InGaAs/GaAs quantum dots in the weakconfinement regime can be found in [248]. Wojs and Hawrylak [2491 studied the coupling of an exciton and an electron, both confined in a quantum dot and subjected to magnetic and electric fields perpendicular to the plane of the dot. They found that the presence of the additional electron in the dot significantly changes the lowenergy absorption spectrum of an exciton. The magnetoexciton dispersion in a quantum dot was also studied by Bockelmann [250] and the interplay of CM and relative motion for an exciton in a quantum disk was studied by Adolph et al. [251]. Heller and Bockelmann [252] performed photoluminescence experiments on single quantum dots in magnetic fields. They studied the ground state and two excited states which split into doublets in the presence of magnetic fields. They found that the ground state spin splitting is smaller than that of the excited levels.
2.8
T i l t e d  f i e l d effects
Meurer, Heitmann, and Ploog [253,254] did FIR spectroscopy of fieldeffect confined quantum dots with diameters ~ 100 nm in GaAsheterojunctions in a tilted magnetic field. For zero tilt angle (0 = 0~ their results are described in Sect. 2.3.2. For the tilt angle, 0 = 18 ~ their results are shown in Fig. 64. The FIR resonances show a splitting of the dispersion caused by resonant interactions with states confined in the growth (i.e., z) direction. The results look similar to the resonant subbandLandau level coupling observed in tiltedfield cyclotron resonance experiments in a twodimensional electron system [20,255]. To understand (at least qualitatively / the experimental results described in Fig. 64, consider a model of threedimensional quantum dots [256]. Here the electrons are confined in a threedimensional potential Vconf  ~m I 9 02r2 ( x 2 + y2 ) + ~1m 9 WzZ 2 2 in the presence of an external magnetic field, where 02r is the frequency of lateral confinement and Wz is the frequency of confinement in the zdirection. Assuming circular symmetry in the xyplane, the resonance frequencies can be obtained by solving the following cubic equation in 022: 026

024 (022 _4_ 02r2 ~ 02z2 ) ~ w2 ( 02202r2 s i n 2 0 +
2 02c2 02z2 COS2 0 4 (Mr4 "4 202r2 02z)

0 2 r4 0 2 z2
"
0.
(2.23) For a choice of Wz which gives a good fit to the data, Eq. (2.23) was numerically solved and the results are also shown in Fig. 64 for comparison. In Fig. 64 (a), for 02r  11 cm 1, Meurer et al. obtained wz  34 c m  1 and in Fig. 64 (b), 02r   2 5 c m  1 and Wz  100 cm 1. The model of a threedimensional dot describes the major features of the data. In particular, it explains why the resonance frequencies at large magnetic fields are determined by the total field, rather than the perpendicular component of the magnetic field.
Q u a n t u m dots
95
F i g u r e 64 Observed magnetic field dispersion of resonance frequencies of quantum dots in a tilted magnetic field. Theoretical results (labeled "Harmonic atom") are also given for comparison.
The case of two interacting electrons in a 3D q u a n t u m dot and a tilted field was investigated by Oh et al. [257]. They noticed considerable difference in the ground state properties in the presence or absence of a tilted field. At 0 = 0 ~ no spin transitions are noticed in the ground state of a spheroidal dot, but a spin transition does appear at 0 = 55 ~ at a magnetic field between 2 and 3 tesla.
2.9
Spin
blockade
in quantum
dots
It has been recently suggested that for a correlated electron system, spin selection rules can also influence the transport properties [258,259]. In contrast to the "charging model" where the excitations are treated within a singleelectron picture, spin selection rules result from fully correlated states. The spin blockade mechanism which is related to the spin polarized states, is supposed to result in negative differential conductances observed experimentally [Sect. 2.3.3]. Basically, the blockade is due to a decreased probability for states with m a x i m u m spin (S = N/2) to decay into states with lower electron number. In contrast to transitions that involve states with S < N/2, transitions in the fully spin polarized case are possible only if the total spin S is reduced. Therefore, the current is reduced at a voltage of the order of excitation energies of the S = N/2 states. Another spin blockade effect would occur in the transitions
(Eo(N), S) +~ (eo(X  ~), S'),
IS S'l > 2,
(2,24)
T. Chakraborty
96
i.e., if the total spins of the ground states corresponding to N and ( N  1) electrons, 1 This should affect the peak heights of the linear conductance. differ by more than 3" These spin effects are suppressed by a high magnetic which renders the ground state fully polarized [258,259]. We have already discussed in Sect. 2.5.5 that spin transitions in quantum dots are entirely due to electron correlations. Imamura, Aoki, and Maksym [260] investigated the spin blockade condition as an effect of total spin dominated by the magic angular momenta. Their system consisted of three and four electrons in parabolic quantum dots in an external magnetic field. The numerical results showed that the spin blockade condition [Eq. (2.24)] is indeed fulfilled. As an example, in their model system with confinement 1 1 3 potential ha~0  6.0 meV, the ground state changes as (Y, S)  (1, 3) ~ (2, 3) ~ (3, 3) for N = 3, and (g, S) = (0, 1) ~ (2, 0) ~ (3, 1) ~ (4, 0) ~ (5, 1) ~ (6,2) for N = 4. The spin blockade is found in the region 4.96 < B < 5.18 tesla. For a double dot system (vertically Coulomb coupled), the spin blockade condition is fulfilled for a wider range of magnetic field than for a single dot.
2.10
Q u a n t u m dot molecules
In addition to the work on the electronic properties of coupled dots described above, there is also a large body of work on the transport properties of two dots in close proximity. Investigations of discrete electronic states of coupled quantum dots placed a tunneling distance apart  the quantumdotmolecule states, began with the work of Reed and his collaborators [44]. Their starting system was a double quantum well triple barrier structure designed to have two quantum dots connected in series between quantum wire contacts. Their observed currentvoltage characteristics indicated significantly sharper peak in the coupleddot spectra compared to the singledot spectra. Tunneling through coupled QDs was expected to be strongly influenced by the quantization of energy levels in individual dots [44,261265]. As the tunneling between the two dots is primarily elastic, the energy states of one dot need to align with the energy states of the other dot for interdot tunneling and hence transport through the entire system to take place [44, 261,262,266]. Using a tunable double QD system with welldeveloped 0D states in each dot, van der Vaart et al. [263] exploited the Coulomb blockade of tunneling to control the number of electrons in the dots. They observed sharp resonances in the current when there is matching of energy of the two 0D States in two different dots. This result demonstrated that transport through a double dot is reasonably enhanced when the energies of the two 0D states match. Waugh et al. [267,268] investigated lowtemperature tunneling at B = 0 through double and triple quantum dots with adjustable interdot coupling. They noticed that interdot
Quantum dots
97
A verticallycoupled doubledot device. The dashed lines define the depletion layer at the pillar surface, that confine the electrons to the interior of the pillar [64].
Figure 65
tunneling leads to a variety of phenomena not observed for single dots. One greatly discussed phenomena is that, as interdot tunnel conductance is increased, Coulomb blockade conductance peaks split into two peaks for double dots and three for triple dots. For weak tunneling, the observed peak splitting approaches zero and in the strong coupling regime where the two dots essentially merge, the splitting saturates. These results are consistent with the theoretical predictions for double dot systems [265,269]. Peak splitting in coupled QDs has been studied experimentally by Livermore et al. [270] who presented a unified picture of the evolution of the coupled dot system from weak to strong tunneling regime. In the former case, capacitive coupling is dominant, while in the other case interdot tunneling dominates. Blick et al. [271,272] investigated the charging diagrams of a doubledot system (DDS), connected in series and coupled by a tunneling barrier. Such a system was described be these authors as an artificial molecule where electrons are shared between the two sites. The charging diagram was generated by measuring the conductance through the DDS while the electrostatic potentials of two independent gates was varied. These authors
T. Chakraborty
98
40
30
~> E O
20
> I
C
>
10
0
4
8
12
16
0
4
8
12
16
B (T) F i g u r e 66 The currentstep position of (a) low and (b) highbias I  V staircases as a function of magnetic field. The step positions are plotted with respect to the first step. The solid lines are the energy levels (with respect to the lowest energy) for a harmonic confinement potential [275].
observed a coherent tunneling mode or molecularlike state in the DDS which leads to a finite conductivity even if the gate voltages do not exactly match the positions where sequential singleelectron transport through the DDS is allowed. This coherent mode manifests itself as a tunnel splitting in addition to the Coulomb interaction in the charging diagram of the DDS [271,272]. Blick et al. also investigated the Rabi oscillations between two discrete states in these artificial molecules induced by externally applied highfrequency radiation [273,274]. In Refs. [64,275], Schmidt et al. investigated singleelectron t r a n s p o r t through a vertically coupled double dot system (Fig. 65) and thereby explored the singleparticle regime of a stronglycoupled DDS. This system was created by imposing a submicron lateral confinement on a triplebarrier heterostructure. Their I  V curve exhibited steps similar to SET in single dots. This is in contrast to the sequential SET through two QDs in
Quantum dots
99
series that leads to sharp peaks in the I  V curve [261263]. Schmidt et al. attributed these current steps to SET through discrete singleparticle states extended over the two identical dots due to coherent interdot coupling. The DDS can then be pictured as ionized
artificial hydrogen molecule. In order to compare the position of the current steps in the I  V curve observed experimentally with the singleparticle energies of the DDS, Schmidt et al. plotted the biasvoltage differences as a function of magnetic field (Fig. 66). The observed results are then compared with the computed energy levels (E~,l  E0,0) for parabolic confinement (see Sect. 2.1.1). The agreement between theoretical and experimental results is generally good except near the degeneracy points of the theoretical curves, where intricate anticrossing behavior is observed. This signifies deviations from the perfectly rotationally symmetric state assumed in the theoretical results. Austing et al. [276] investigated the addition spectra of vertically coupled double quanturn dots. They obtained the Coulomb diamond diagrams from strongly coupled (dots separated by a 25A barrier) and weakly coupled (dots separated by a 75A barrier) and extracted the shell filling just like in single dots. Not all shell fillings were observed for the DDS. The addition spectra of double quantum dots was calculated by Tamura [277], who considered a 3D electron system with parabolic confinement in the xyplane and a squarewell potential in the zdirection. The energies were calculated via the unrestricted selfconsistent HartreeFock approach. It was suggested that the absence of shellfilling according to Hund's rule, observed in experiments by Austing et al. [276], is due to the effect of dot thickness. A detailed theoretical study of the electron correlation effects on the electronic states and conductance in a vertically coupled DDS has been reported by Asano [278]. For the parabolically confined system in the xyplane and squarewell potential in the growth direction, the total spin momentum of the ground states was calculated as a function of the total number of electrons and the distance between the dots by using the numerical diagonalization method. Two different regimes were considered. In the strong coupling regime (i.e., the distance between the dots is small) a correspondence was found between the ground states in the DDS and those in real diatomic molecules bound by electrons in the 2p orbital, i.e., B2, C2, N2, 02 and F2. This happens because the structures in the singleparticle levels in the two systems are qualitatively similar. In the weak coupling regime (i.e., the distance between the dots is large), the effect of electron correlations on the spin structure of the ground states is quite significant. Electronic states with a small spin momentum are stable for even number of electrons while states with a large spin momentum are stable for odd number of electrons. The reason put forward by the author is that for even numbers of electrons, two electrons localized in different dots form a singlet pair due to electron correlations and interdot hopping. The ground state is then well described by a combination of such singlet pairs. For an odd
100
T. Chakraborty
number of electrons, the ground state is similar to Nagaoka's ferromagnetic state, that is the spins of all electrons align parallel in order to decrease the kinetic energy of the electrons. Interestingly, the physical picture of the ground state in the weakly coupled DDS was found to be analogous to that in the Hubbard model near half filling. The conductance of the DDS was also calculated by Asano, who found Coulomb oscillations in the DDS. For large separation of the dots, the amplitudes of several conductance peaks are suppressed due to electrons correlations (spin blockade). Finally, as in singledot systems, PAT (see Sect. 2.4.5) has also been studied in the DDS [51][R33] [274,149,150, 279281] coupled in series. Resonant 0D,to 0D PAT occurs when the 0D levels in the neighboring dots are separated exactly by the photon energy.
2.11
Noncircular dots
In this section, we discuss the properties of quantum dots which are somewhat different from the circular shape discussed so far. Two cases are of particular interest" elliptical dots and stadium shaped dots.
2.11.I Elliptical quantum dots Experimental work on the ellipticalquantum dots discussed above [126] inspired us to look at the energy spectrum in those systems [282]. The confinement potential of the dots studied by McEuen et al. [87][R13], [88][R14] were also anisotropic. Theoretically, anisotropy in quantum dots has been treated earlier as a perturbation [283] to the isotropic parabolic quantum dot. However, that is not expected to be correct for large anisotropy. Just like for a a circular dot, one can derive the singleelectron results analytically for anisotropic dots. Let us consider a lone electron in a lateral anisotropic parabolic confinement potential in the presence of a quantizing perpendicular magnetic field. The Hamiltonian is then written 1
eA
~ ~conf (x, Y)
(2.25)
where the confinement potential is ~conf(X, y) __ ~m~ 1 ((.dx2X2 "t Cdyy ),
Cdx r COy
1 , O) and make the Let us choose the symmetric gauge vector potential A  ~B(y,x following transformations
Quantum
dots
101
x
=
Y

X2  ~P2 sinx, X X2 q2 c o s x    P l s i n x , X
qlcosx
Px
=
Pl COS X ~ m X1 q2 sin X,
Py

P2 cos X + X21ql sin X. X
T h e s e are c o n s i s t e n t w i t h t h e c o m m u t a t i o n
X1X2


r e l a t i o n s [p~, qj] 
ih6~j
a n d [q~, qj] 
0 if
X 2.
Accordingly,
we
~_{
rewrite the Hamiltonian
1 2 2?Tte [p2x _It ~21X2 _it py ..jr..~ 2 y 2 + ?TZeCOc (YPx  X p y ) ] ,
__
2
(022x,y._~_ 1
(2.27)
2
a n d w~ is t h e u s u a l c y c l o t r o n f r e q u e n c y . It i s d i a g o n a l if _ [~1 (~2 _.]_ ~2)] 89
X
[Xl 1_
x2 x tan2 X
~o~c [2 (a~ + a~)]~ / (al~  a ~ )

L e t us define
~ = [(~ 
a~) ~ + 2 ~ e ~ ( ~ + a~)] ~
T h e H a m i l t o n i a n is t h e n f u r t h e r t r a n s f o r m e d as
= ~
1
22
22
(~P~ + ~p~ + ~q~ + &q~),
(2.28)
with t212 + 3 ~
~
=
+ ~
2 (~1 + ~ ) m l~ + a ~  a~ 2 ( ~ + a~)
(2.29)
~i _ ~I (a~ + 3 ~  n ~ )
T. Chakraborty
102 15
10 ;>. o v
E
0
2
''''
25
I'''
(b)
_
4
i"]",,,
6
~ I''''
1'''
'
_
20 ,,~..._..
_ ID
E 15
. ~
~10
.~
~
,
.~
.
. ', .
:

.
....
.
_
.
~~

, .,,
i
O r ....
0
..
,,
I,,,,i
1
2
3 hr
4
5
c
F i g u r e 67 (a) Energy levels of an anisotropic quantum dot as a function of the magnetic field (haJc in meV) for a~x  1.0 meV and a~y  1.1 meV. The lines are drawn in ascending order of (nx, ny), as indicated. (b) The Chemical potential (in the CI approximation) for the energy levels of (a) [282].
T h e e n e r g y eigenvalues are t h e n easily o b t a i n e d to be [284,285] 1 1 f'n=,nu  (?~x + ~) ~Cdl 't (ny Jr ~) ~Cd2,
(2.30)
where (M1   Ctl/~l/TYt e a n d aJ2  a2/32/me. T h e energy Eq. (2.30) has t h e following limiting behavior" at zero m a g n e t i c field, t h e s y s t e m b e h a v e s like a pair of h a r m o n i c oscillators in t h e x a n d y directions. For a large m a g n e t i c field (~c >> a~x,ay), we get Enx = 1 (nx + 3) hCOc,i.e., L a n d a u levels form as in t h e case of isotropic p a r a b o l i c confinement. W h e n a~  coy, i.e., t h e c o n f i n e m e n t is isotropic parabolic, nx  n + ~1 l l l  11 a n d
Q u a n t u m dots
103
25 20
:: :~

L
,
.
.
.
.
~ _ _ _ _ x
.
.
.
 
~
................... (~,) (~x=
0
2


.
~io; ~,=s.o) :I 4
6
30
~. 2O > v
10
0L, 0
"
, .... 2
,
t
I
i
I
4 he0c
Figure 68 Same as in Fig. 67, but for a~y = 5.0 meV [282].
1 ny = n + ~lll + 1l, where n and I are the principal and azimuthal q u a n t u m numbers, respectively. Also, when cox ~ coy, the energy levels are very similar to that of the isotropic case except that the (2n + Ill + 1)fold degeneracies at B = 0 are lifted [282] as a result of breaking of the circular symmetry. A similar situation arises when the circular symmetry is broken by Coulomb coupling between two neighboring dots [71][R19]. The selection rules for the transition to higher energy levels can also be calculated from the dipole transition matrix elements [71][R19], and are as follows: polarization along the x or yaxis, (i) An~ = 0, Any = +1, (ii) An~ = +1, An v = 0. There are just two modes as in the case of isotropic parabolic confinement [30][R7]. The only major
104
T. Chakraborty
difference here is that at B = 0 the two modes split, A E = h(cox COy). This mode splitting has indeed been observed experimentally by Dahl et al. [126]. In Fig. 67 (a), we show the magnetic field dependence of the singleelectron energy levels for a quantum dot with COx = 1.0 meV and COy = 1.1 meV. For this choice of COx and COy, the deviation from the circular dot is minimal and therefore, the energy levels are very similar to those of a circular dot except at the origin where the degeneracies are lifted. Qualitatively similar results were obtained in perturbation calculations [283]. In this figure, we also present the chemical potential calculated in the CI model discussed in Sect. 2.3.1 [Eq. (2.8)]. In those results, we include the Zeeman energy with a gfactor to be 0.44 and the effective mass of rn* = 0.067me, appropriate for GaAs. We have also used U = 0.6 meV, taken from the work by Ashoori et al. [76][R21]. The energies and chemical potentials for COx = 1 meV and COy = 5.0 and 10.0 meV are plotted in Fig. 68 and Fig. 69 respectively. Clearly, the level crossings shift to higher energies as COy is increased, and oscillations in chemical potentials are also suppressed at lower energies. For example, when COy = 5, the oscillations are suppressed for ne = 1  12 and for COy = 10, it happens for ne = 1  22. In addition, the amplitude of the oscillations decreases considerably with increasing anisotropy. On the other hand, the magnetic field threshold beyond which the oscillations in chemical potentials cease to exist increases with increasing coy. With this increase of the magnetic field threshold, the oscillations also move to higher magnetic fields like the observed experimental results of Ashoori et al. It is also to be noted that the confinement potential for the q u a n t u m dots in the experiments of McEuen et al. [88][R14] is anisotropic with COy/COx~ 4.4, which lie within the range of COx,y considered here. Madhav and Chakraborty [282] studied the twoelectron states in an elliptical quant u m dot interacting via the potential of the form v(r) cv 1 / r 2, which has the advantage that most of the analysis can be performed analytically. The anisotropic system with interacting electrons were investigated via perturbation theory. For the Coulomb potential, Maksym [286] studied the eigenstates of two and three interacting electrons in an elliptical dot (2.26) by using the basis states of a circular dot [Sect. 2.5.2]. He found that the ground state in this case can undergo transitions similar to those of a circular dot but some of those in a circular dot do not survive the lowering of the symmetry. Transitions corresponding to cases where the angular momenta on either side of the transition in the circular dot differ by an odd integer do survive. However, transitions between states of the same spin whose orbital angular momenta differ by two are forbidden in the elliptical dot. Very recently, Austing et al. [287, 288] have reported their studies of ellipsoidally deformed fewelectron vertical QDs. From measurements of the addition energy they found that even a small deformation of the circular dot radically alters the shell structure that they observed earlier in a circularly symmetric QD (see Sect. 2.4.6). As the deformation
Q u a n t u m dots
105
40
30 :>
E v 20
10
0 0
40
5
i
' '
I '
.... " '
10
i ' '
'i
~
15
'
'
'
I~
'
"
____:
30 >
E
20
10 .....
,
2
,,
!hi .....
4
6 he0
F i g u r e 69
,
.....
,,
8
e
Same as Fig. 67, but for aJy  10.0 [282].
of the circular dot is introduced, the shell structure is either disrupted or smeared out. This was a t t r i b u t e d directly to lifting of degeneracies of singleparticle states present in a circular dot, as discussed above.
2.11.2 Other asymmetric potentials As we noted earlier, experiments of Tarucha et al. [4850] on the tunneling of electrons through a q u a n t u m dot and the observation that the addition energy of an electron in the fewelectron dot reveals the existence of shell structure, p r o m p t e d theoretical
T. Chakraborty
106
L_ ___.~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
F i g u r e 70 Schematic diagram of the quantum dot stadium [294].
investigations by several groups. Ezaki et al. [54] reported exact diagonalization of a fewelectron Hamiltonian with various asymmetric potentials. They modeled the experiment of Tarucha et al. by the following form of the confinement potential,
Y) 
+
) [1 +
cos a r
(2.31)
where c~ = 0, 1 and r is the angle with respect to the specific axis in the xyplane. For = 0, we get the elliptical dot (~x ~~y). Setting c~ = 1 and ~x = ~y, one gets a triangular shaped confinement potential. The heelectron eigenstates were obtained by diagonalizing the neparticle interacting Hamiltonian. The basis states were constructed from the Slater determinants composed from singleelectron eigenstates of the isotropic and harmonic system. The calculated addition energy, or the energy needed to add one more electron to the q u a n t u m dot, was plotted as a function of the electron number in the dot. In a circular dot, the addition energy was found to be quite large for ne = 2 and 6. This was a t t r i b u t e d to complete shell filling. A somewhat weak peak was seen at ne = 4, which was interpreted as due to a spin polarized halffilled shell. In an elliptical dot, the addition energy does not have a clear structure (except at ne = 6) which was a t t r i b u t e d to a s y m m e t r y of this system. The a s y m m e t r y leads to splitting of the degenerate singleelectron eigenstates and mixing of many eigenstates with various angular momenta. In a triangular dot, slightly stable states due to localization of electrons at the corners were observed.
Quantum dots
2.11.3
107
Quantum dot stadium
It is well known [289] that the motion of a classical particle in a closed stadium is chaotic. Studies of quantum analog of such systems showed the eigenvalue spectrum follows the Gaussian orthogonal ensemble [290], and the most stationary states usually concentrate around narrow channels called scars which resemble the classical periodic orbits [291]. Experimental work on chaotic scattering has been reported on ballistic GaAsheterostructures [292]. The magnetoresistance across a stadiumshaped dot has been measured. In the tunneling regime, i.e., a stadium that is classically isolated, the magnetoresistance shows periodic oscillations at high magnetic fields [293]. Theoretical results of singleelectron states of a quantumdot stadium in a magnetic field have been reported recently by Ji and Berggren [294]. They calculated the energy spectrum, twodimensional spatial distribution of the charge and current density to look at the transition from the chaotic to regular behavior as the magnetic field is increased. Here the model is a stadium shaped wall (Fig. 70) where two semicircles of radius R are connected by two parallel, rectilinear intervals of length 2L = 2R. For spinless electrons, the singleelectron Hamiltonian is 7{ = ~
1
(  i h V + cA) 2 Jr ~conf,
where V~onf is the confinement potential and A  ~1B (  y , x 0) is, as usual, the vector potential in the symmetric gauge. The electrons are confined by infinite walls. Inside the dot Vconf : 0. The Schr6dinger equation was solved numerically and for each eigenstate, the twodimensional current density ~t
J(x, y)   m , I m [r162
e
,
+ ~%A I~(x y)
]2
and the probability density p(x, y)  I~(x, y)l 2 were also numerically evaluated [294]. Looking at the energy spectrum Fig. 71, these authors noticed that [294] at high magnetic fields the states were found to converge to degenerate bulk Landau levels, as seen earlier for a circular dot. At low magnetic fields, there are crossings and anticrossings of the energy levels. The anticrossings or level repulsions arise due to the nonintegrability of the problem and are a signature of quantum chaos. With increasing magnetic field, the level repulsion becomes weaker. From the charge density and the current pattern in the stadium, these authors determined how the electron motion changes from chaotic to regular behavior as the magnetic field is increased.
T. Chakraborty
108
d
.,,.
....
.':.;
:,,~.: ............. .... .
' . .9... . . . . . . . . . :  : . . . .  " ..... .  " :',', ."
10
. . . . ,I . . . . :o.:;;:" " " , s ' " ...":,',p ........ .. .z.:.. ;8eoe:_'" ." "ell: ..... ". ."'.'.;'/ ~ ~ ..... .,,,,'*~ oe'" 3~ 9~ 9 * e" .e... "o ..,o o ' ~ eo ,,e ooo igiioellOOl.._ ..11 ..... . .S * .$'$~," *e** o ..l e . 1 2 ~ II' . . . . . .......:... . "" "+ . . . ....... " " ""+ ..::,, ....... ::,.s::::u ' "....... ....9.0,%8 ,.: l e ..:..:m.tim e**.
....... ;.
........
".... ..... .
, ....
..."
:,"
9
"'.o""
,..: . . . . . . . . . .
O
,,
_~, S s
:,..; ....
,:~."
,,...:a . . . .
..:.. ....
0
::,.: .... ..~;i.':
2 B (T)
.......
...........
: . t ' l ~ +, "'. ........
..,.:,_.~;~,
~
1
:..:,.;j
.:, . . . . . . :  p . . .
! iiiiiii ii ili ::ii+;!!
t2
"',,"  ...... .  0'
:..:,,,:/':"" ::,.:::i;illJP ..... ." S'.. ::,='" . . . . : , j ~ .
:.:
. . . . . :.::::" . . . . . : : : : . , : ~ . , . . ~ : :  :,...,.: ":::'... .... .,
B
"'. ........
"
...: .... ,: . . . . . .
,,,: . . . . . . . .
.... : +:'" 8 . ... ... .. . " ""   'r
.. .:,
4 2
"
"'" 
 . . . . . ,, A ".....
9. . . . . . . . . . . . . .
.................
3
4
F i g u r e T1 Energy levels of the quantum dot stadium as a function of the magnetic field. The dashed lines correspond to the Landau level energies of an ideal twodimensional system [294].
T h e d i s t r i b u t i o n of energylevel spacings is defined as P(s) = s/{s}, where s is the level spacing and {} d e n o t e the m e a n value, and is a statistical m e a s u r e of the s p e c t r u m . P(s)ds is t h e probability of finding a s e p a r a t i o n of neighboring levels b e t w e e n s and s+ds. A t B  0, one finds a G a u s s i a n o r t h o g o n a l ensemble d i s t r i b u t i o n P(s)  (?rs/2) e ~s2/4. At i n t e r m e d i a t e values of the m a g n e t i c field, the level spacing d i s t r i b u t i o n is G a u s s i a n u n i t a r y ensemble type, P ( s )  (32/+r 2) s 2 e (4/~)s2 and at higher m a g n e t i c fields, these a u t h o r s find a d i s t r i b u t i o n close to a Poisson distribution P(s) = e  s as e x p e c t e d for a regular system. T h e s t u d y of q u a n t u m dot s t a d i u m , as r e p o r t e d in [294] could be i m p r o v e d by using a m o r e realistic confinement, such as a parabolic potential. More i m p o r t a n t l y , the role of electronelectron in this s i t u a t i o n should be studied. M a n y e x p e r i m e n t a l results on q u a n t u m  d o t s t a d i u m have been r e p o r t e d in the literature which invites m o r e theoretical work on this system.