Far-infrared spectroscopy of quantum dots

Far-infrared spectroscopy of quantum dots

PHYSICA Physica B 189 (1993) 165-175 North-Holland SDh 0921-4526(93)E0022-7 Far-infrared spectroscopy of quantum dots Ulrich M e r k t lnstitut f i...

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Physica B 189 (1993) 165-175 North-Holland

SDh 0921-4526(93)E0022-7

Far-infrared spectroscopy of quantum dots Ulrich M e r k t lnstitut f iir Angewandte Physik, Universitdt Hamburg, Hamburg, Germany

Quantum dots on semiconductors are few-electron systems with discrete energy spectra. They are fabricated from semiconductor structures with two-dimensional electron gases like metal-oxide-semiconductor (MOS) capacitors or GaAs/GaAIAs heterostructures by laterally confining the electrons. Here some important fabrication schemes employing nanostructure technologies to create isolated quantum dots are outlined and the present status of far-infrared spectroscopy on these atomic-like systems is discussed.

I. Introduction

Advances in lithography and etching techniques make it possible to laterally confine quasi two-dimensional (2D) electron systems in semiconductors [1] into dots of diameters below 100 nm [2]. The energy spectrum of electrons in such dots is totally discrete, i.e. there is no free motion with continuous dispersion but we have a quasi-zero-dimensional (0D) system. Quantum dots have some features in common with shallow donors in semiconductors which may be regarded as natural 0D electron systems with low electron numbers, say one. Both are embedded in a medium of dielectric constant and in both the electron motion is characterized by the effective band-structure mass m~. The diameter of a dot and the effective Bohr radius a* of a shallow donor both are much larger than the lattice constant of the crystal. But there are also differences. Unlike for a donor, we no longer have a more or less isotropic hydrogenic potential with an effectively positive charge in its center. Instead of this, the lateral potential is of Correspondence to: Prof. Dr. Ulrich Merkt, lnstitut fiir Angewandte Physik, Universitfit Hamburg, Jungiusstr. 11, 2000 Hamburg 36, Germany. Tel.: +49-40 4123 4685; telefax: +49-40 4123 6368. 0921-4526/93/$06.00

approximately parabolic shape and the potential well in the z-direction usually is much narrower than the lateral one. In quantum dots we can externally adjust the electron number by a gate voltage Vg in a wide range from 10 4 in classical electron discs to just one electron in the quantum limit. A particularly exciting possibility is offered by the choice of the dot radius R. The quantized kinetic energy is proportional to R -2 whereas the Coulomb interaction energy between electrons scales as R -I. This eventually provides us with the opportunity to study fewelectron systems in the nearly independent-particle limit in small dots (R ~ a * ) and correlated systems in larger ones (R-> a*). Here, the present state of far-infrared spectroscopy and their interpretation is described guided by our own work on InSb [3-5], but also covering recent work on GaAs done by other groups [6,7]. The advantage of InSb in studies of low-dimensional electron systems is its small effective conduction-band mass m*0 = 0 . 0 1 4 m , , which leads to comparatively large energy spacings of up to 25 meV at dot dimensions feasible with present high-resolution semiconductor technologies. On the other hand, in InSb the electron mobility of the originally 2D electron gas (p, ~> 104 cm2W-ls -l) is much less than, e.g. in GaAs/GaAIAs heterojunctions (/z

© 1993- Elsevier Science Publishers B.V. All rights reserved


U. Merkt / Far-infrared spectroscopy of quantum dots

Is-l) eventuating as much sharper line shapes in GaAs. As a result, fine structure in the far-infrared spectra can presently only be detected in experiments with quantum dots on GaAs. 106 c m 2 V

above the valence band edge. Underneath the narrow regions between the NiCr mesh, mobile inversion electrons are induced by the gate voltage Vg. The resulting lateral potential at the InSb surface (z = 0) is also sketched in fig. 1.

2. Fabrication of dot arrays EXPANDED

In quantum dots we have typical quantization energies of 10meV, corresponding to wavelengths of 100 I~m. The intensity of appropriate light sources like CO2-1aser pumped far-infrared lasers is weak and we overcome the intensity problem with laterally periodic arrays of dots on macroscopic areas. In the following, we first discuss the basic idea of our InSb structures and than describe their fabrication. Subsequently, we outline quantum dot structures on GaAs which are produced employing similar techniques. For exciting spectroscopic results obtained for dots on Si, we must refer the reader to the references

[8-101. The idea of the InSb quantum dots is sketched in fig. 1. Essentially we have a MOS structure with the alloy NiCr evaporated onto the lnSb substrate as a Schottky barrier. This pins the Fermi energy E v at the NiCr/InSb interface






"-4"- Q=2sin,-ff Fig. 2. Setup of the holographic lithography utilizing the superposition of an expanded beam of an argon laser (;t = 458nm). From ref. [11].

resist after developing

plasma etching in oxygen


Si0 2


Si0 2


'..~ ~-----,~ #

~ 66"


J j



:8oo ~



I NiCr shadowing


7Fig. 1. Schematic cross-section of the microstructured fieldeffect device on InSb with its lateral band structure. The dashed line near a conduction-band minimum indicates the parabolic approximation of the bare potential.


J lift-off

Fig. 3. Principal preparation steps for microstructured fieldeffect devices on InSb.

U. Merkt / Far-infrared spectroscopy of quantum dots

T h e r e is virtually no tunneling between adjacent dots since the barrier height between dots is of order of the band-gap energy (Eg = 236meV) and the distance is of order of the grating constant (a = 250 nm) of the dot array. As outlined in the next section, the lateral potential may be approximated by a parabolic potential, hence, the energy spectrum is the one of the harmonic oscillator. For a sample of grating constant a = 250 nm and active area 10 mm 2 we must fabricate about 109 dots. Each of them has a radius of about 100 nm with little variation allowed in the whole array. The desired structures are obtained by holographic lithography [11] whose setup is depicted in fig. 2. After a first exposure of the photoresist, the sample is rotated by an angle of 90 ° and exposed for a second time. After development there is a sinusoidal resist pattern as visualized in fig. 3. In the next preparation step, NiCr is shadowed two times under angles of 66 ° from two sides. Subsequently, the resist dots are


dissolved in acetone (lift-off). This lift-off process only works at longer grating constants (a = 400 nm), at lower ones ( a - 250 nm), we evaporate the NiCr Schottky barrier perpendicularly onto the sample. In the latter case, the resist is used as a gate insulator between InSb and the NiCr metalization. Finally, a SiO 2 gate insulator is deposited by plasma-enhanced chemical vapor deposition ( P E C V D ) and the homogeneous t o p gate is evaporated. An electron micrograph of a monitor sample without SiO 2 insulator and top gate is shown in fig. 4. Via field effect, the dots can be charged without direct contacts to the inversion electrons, since the InSb substrate has a finite resistivity in the megaohm regime even at liquid helium temperatures. A threshold voltage Vt is determined from the onset of absorption and we use the voltage difference AVg = V g - V , as a measure of the number no of electrons in a dot. Two examples of quantum-dot arrays on GaAs are depicted in figs. 5(a) and (b), namely deep-

Fig. 4. Micrograph of a monitor sample with resist dots shadowed with gold for contrast enhancement. The marker is 100 nm long. From ref. [4].


U. Merkt / Far-infrared spectroscopy of quantum dots



GaAs ± n-GaAtAs GaAIAs spacer

, .. - _ ~ - - - ' ~

i. p o s i t i v e l y c h a r g e d

disc jellium



U(r) R




(b) _

Oateelectrode Phoforesisf

2. remote jellium in d e e p - m e s a

etched dots


Fig. 5. Sketch of dots deep-mesa etched (a) and dots fieldeffect confined (b) quantum dots on GaAs. The etched dots may be charged by visible light pulses via the persistent photoeffect, the field-effect confined dots by a doped contact layer (dashed line) underneath the dots. After refs. [6,7].

mesa etched [6] and field-effect confined [7] dots. They are prepared by similar methods, including holographic lithography. In case of the deepmeas etched dots, an optimized reactive ion etching process with a SiF 4 plasma was employed [6,12].


electron dot

3. application






to field-effect defined dots










. . . . . . .

c o n t a c t layer

Fig. 6. Simple models for the electrostatic lateral potential in quantum dot samples. Details are explained in the text.

3. Simple electrostatic models A full calculation of the dot potentials for real samples is cumbersome and essentially only one work has been published on this subject [13]. In order to arrive at simple approximate potentials for the lateral confinement in quantum dots, we first deal with an electron on a positively charged disc in the x - y plane as shown in the upper part of fig. 6. In the z-direction, the electron is considered to be strongly confined, e.g., by the band offset in a G a A s / G a A 1 A s heterojunction or by the oxide barrier and the surface potential in a MOS capacitor. The potential of the jellium can be given analytically [14] in terms of elliptic integrals E and K:

U(r) ( -e2N+

|~ 4



2N÷ r

r <~R , R 2

(1) In this equation, we have the vector r = (x, y) in the plane, the radius R of the disc, the number N÷ of positive charges, and an effective dielectric constant ~ that accounts for the presence of different dielectric constants of semiconductors and insulators. The minimum of the potential

U. Merkt / Far-infraredspectroscopyof quantum dots energy is U(0) = -e2N+/2~r&oR. In the approximation r/R ~ 1, we obtain the harmonic oscil1 • 2 2 with lator potential 7motoor

2 e2N+ too- 4w&om,R 3 •


As is evident from fig. 6, this is in fact a very good approximation on the whole of the disc (r/R <~1). Next, we consider in fig. 6 a remote jellium as realized, e.g., in a deep-mesa etched dot. For this situation, one obtains in the parabolic approximation

2 e2N+ [1+{d~2] -3/2 too- 4~r&om,R 3 [ ~) ]


with the distance d. This result also can be applied to the field-effect confined dots in the bottom part of fig. 6 when we mentally put in an equal number of positive and negative charges into the resist dots as depicted in the figure. Since laterally homogeneous charge distributions do not contribute to a lateral force, the lateral potential results from the remote negative disc jellium on top of the photoresist dots plus the fictitious positive jellium in the plane of the GaAs surface. In order to get a simple model for the InSb sample depicted in fig. 1, we consider in fig. 7 two conducting plates with a geometrical hole of radius R [15]. The gate voltage can thought to be

® d>>R R

\ R U(r)


Q /lm,~0Zr z

Fig. 7. A parallel-plate capacitor with a geometrical hole in the lower plate provides a model to calculate the lateral potential for our quantum dots on InSb sketched in fig. 1. The parabolic potential is a good approximationto the exact solution given in eq. (4).


applied between the two plates rather than between the top plate and the InSb substrate (z = 0), since the resistivity of the lnSb is much less compared to the one of the gate oxide. Inside the hole (z = 0, r<~R), the potential is given by the expression

U ( r )-- e 2 n += R ~ /

1 -

(R) 2


,rl-~E o

and zero outside [15]. Provided that the oxide thickness dox exceeds the radius of the hole (dox >>R), the areal charge density n~ = geoVgate/ edox on the top plate is homogeneous. The parabolic approximation gives 2



e ns

too- rr&om* R .


When the electron system of the dot is remote from the hole by a distance d, a factor [1 + (d/ R)2] -2 has to be taken into account. In our InSb dots, the distance is virtually zero, since the electron system results from an inversion layer right at the InSb surface. The above calculations may justify the harmonic oscillator approximation for the lateral dot potential. However, one h a s to be careful about the direct application of the formulas to a real sample. To give an example, additional depletion and surface charges in the deep-mesa etched dots on GaAs in fig. 5 form a negatively charged tube of fixed carriers around the dot system. While this does not alter the approximately parabolic nature of the potential, it may significantly enhance the characteristic frequency too. In the quantum limit of independent electrons (R ,~ a*), the electronic radius of a quantum dot can be defined by the distance between the dot center and the classical turning point of the wave function of the highest occupied state. For large electron numbers no, one obtains in the classical limit (R >>a*) the radial distribution [16] n(r)=

3n° ,//r\Vl_~) 2 2~rR 2


U. Merkt / Far-infrared spectroscopy of quantum dots


with the classical electronic radius



H _ 2 m ; Z [Pi + eA(ri)] 2 + 17mow 0 , 2 E r~ (





R = \4~geom,w~ /

+ 4ve,,------g~

4. Single-electron picture and Kohn's theorem As discussed above, we can describe the external lateral potential by a two-dimensional 6scillator. Its single-particle eigenenergies in the presence of a magnetic field along the z-direction are given by the expression [3]

E,m=E, o+(2n+)mJ+ h~ + --~- m .



In this equation we presume that the motion in z-direction is frozen out into the lowest 2D subband of energy Eg=0 and we ignore electron spin. The lateral motion is described by the radial (n = 0, 1 , . . . ) and the azimuthal (m = 0, + 1 . . . . ) quantum number. The magnetic-field strength enters via the cyclotron frequency wc = e B / m ~ . There are only two allowed dipole transitions which are excited with the two circular polarizations of light and have frequencies [3,4]



2 '

The corresponding real part of the classical highfrequency conductivities [4]


eZnor/ m *o 1 + (w~/w Wc)2"r2 -




contains the electron number per dot n o and a phenomenological relaxation time 7. The many-electron Hamiltonian of the lateral motion in a perpendicular magnetic field described by the vector potential A = ½ ( - y , x, 0)B reads



1 Iri

rj[ "


Introducing center-of-mass coordinates R and P and relative coordinates r i - r j , this Hamiltonian separates into a sum of two:


1 I • 2n2 ( P + 0 A ) 2 ~- ~- Jvlf.o0/l[ + H r e ' .

H ~---~


The first two terms represent the center-of-mass motion and the third one the intrinsic or relative motion. We have the total charge O = no e and the total mass M = n 0 m ~. It is clear that this separation is true for any kind of electron-electron interaction as long as it depends only on the relative distances between electrons. Consequently, the wave functions are products O ( R ) ~ o ( r i - r j) and the eigenenergies are sums E,m + Erd. The eigenenergies Enm of the centerof-mass motion and the single-electron values of eq. (8) are identical since the corresponding Hamiltonians are of the same form and the eigenvalues depend only on the ratio e / m *o -- Q~ M. The dipole operator no

H~ = e ~ E ' r i e -i~ = O E ' R e -i~



yields nonvanishing matrix elements only between different center-of-mass states 0(R). The relative motion of electrons described by the wave function ¢ remains rigid in this excitation. The squared matrix elements differ by the factor n o from the single-electron values, selection rules and transition energies are identical to them. This result is called the generalized Kohn theorem [17-20]. In the application of the generalized Kohn theorem to experimental situations one must proceed with some caution: the lateral potential is only approximately parabolic, it is certainly not strictly so. Also, the band structure of the semiconductors is only approximately accounted for by a constant effective mass, i.e. a quadratic energy dispersion h2k2/ 2m~. Further, the dots are not strictly isolated.

u. Merkt / Far-infraredspectroscopy of quantum dots Besides Coulomb interaction between the dots, there is also coupling of the dot electrons to excitations of the host semiconductor like LOphonons.

15 3

5. Spectroscopic results

3 t ' 0,



AT 2o-(w, B)/a2Yo T = l+x/~+crD/Y o '


which contains the vacuum admittance Y0 = (/%c) - t , the grating constant a, and the sheet resistivity o-n of the metalization. This relation allows us to determine the electron number from the integrated line shape [22] when we assume, e.g., the classical conductivity of eq. (10). The first spectra on quantum dots were obtained on InSb [3]. The resonance energies versus magnetic-field strength, i.e., the Zeeman splittings are depicted in fig. 8. The inset shows the quantization energy hw 0 extrapolated from the Z e e m a n splitting (B--->0) as a function of gate voltage AVg above threshold and as a function of electron number n o . In marked contrast to the situation in real atoms, the quantization energy does not noticeably depend on electron number. This is explained by the





tY I /

~ 1"

ol 0



o Savg(v).~}., ~ -


In the spectroscopic experiments, the transmittance of normally incident radiation of an optically pumped far-infrared laser or of a Fourier transform spectrometer [21] is recorded at liquid helium temperatures. This radiation has the proper polarization to excite transitions between laterally confined conduction-band states. Interference effects in plane-parallel samples are avoided by wedging the backs of the samples. A magnetic field applied perpendicularly to the samples provides the Zeeman splitting of their atomic-like states. In Fourier and laser spectra, the relative change of transmittance AT~T= [T(Vg) T(Vt)]/T(Vt)<~ 1 is plotted as a function of the wave number 13 and the strength B of the magnetic field, respectively. The relative change of transmittance is related to the conductivity o'(w,B) of the dot electrons by',the Fresnel formula [4]

~6 no

,ol ~2o'~



, ----T1.0 B(T)


................. " .........

" p- InSb(1111 T=t,K =AVg =BV







Fig. 8. Zeeman splitting of the resonance energy of quantum dots on InSb. The inset gives the dependence of the quantization energy hwo at zero magnetic field on gate voltage and electron number. The solid line is calculated from the EMA result in eq. (9), the dotted line from a k.p estimate. From ref. [4],

generalized Kohn theorem for parabolic potentials. According to this theorem, which was derived above, one only sees the external or bare potential characterized by the frequency % in the absence of a magnetic field. Hence, the experimental result means, that the potential created by all charges except for the dot electrons themselves is parabolic and is not noticeably changed by the gate voltage. At first glance, the latter point again is surprising particularly in view of eq. (5) which predicts an increase of the frequency % proportional to the square root of the gate voltage. Also in contrast to eq. (5) is the experimental finding, that the electron number in our first series of dot arrays on InSb saturated at a low electron number n o = 20 (see inset of fig. 8). A qualitative explanation for the constancy of the characteristic frequency and for the saturation may be provided by the idea, that the bare potential is created in essence by fixed surface charges and fixed charges in the Schottky contact around a dot. It is hardly influenced by the gate voltage and the accompanying positive charges on the top gate that compensate for the electrons in the dots. The technological reason may have been, that the lift-off process in fig. 3 did not


U. Merkt / Far-infrared spectroscopy of quantum dots

work properly and that there was photoresist and metal left above the dots as mentioned in section 2. Since then, we have improved our technology and have obtained samples in which the characteristic frequency w0 and concomitantly the electron number no is tuned by the gate voltage [5]. The spectra in fig. 9 are obtained for such a sample. The grating constant of the dot array was chosen a = 4 0 0 n m and the geometrical radius of an individual dot R - 90 nm was estimated from micrographs of samples processed in the same run. The resonance position shifts to higher magnetic field strengths which is characteristic of the w - m o d e according to eq. (9). The integrated signal strength is found to increase in proportion to the gate voltage above threshold and there is no saturation of the electron number [5]. Instead, their number is only limited by the breakdown voltage of the gate oxide. We have verified the polarization selection rules with the aid of wire-grid polarizers and quarter-wave plates made from y-cut quartz of appropriate thickness [5]. An example is given in figs. 10 (a) and (b). For comparative reasons, the two spectra in fig. 10(a) were recorded for a homogeneous 2D electron gas on InSb for the cyclotron resonance active (B +) and inactive ( B - ) direction of the magnetic field, respectively. An example for a dot sample is given in fig.

15 ha~ =10.4 meV p-InSb(111)


T= 1.4~ Io





m*= O.023rr~







Ia~ = 1 0 . 4

I meV

p-lnSb(111) T=I.4K














B(T) Fig. 10. Far-infrared resonances observed with circularly polarized light for a homogeneous 2D electron gas (a) and for a quantum-dot array (b) on lnSb. For the homogeneous 2D gas, cyclotron resonance is excited in the B ~-direction of the magnetic field. On the quantum dot sample (h~o < h%), one observes the o) mode excited in the B direction.

10 a


h ~ =10.4 meV p - l n S b ( l l i) T= 1.4K

noise I







B(T) Fig. 9. Far infrared laser spectra of quantum dots on lnSb for various gate voltages. These to resonances shift to higher resonance magnetic fields as a consequence of the increase of the quantization energy he%.

10(b). Generally, the w+ and o) modes are excited in the B ÷ and B - configuration, respectively. This agrees with the theoretical selection rules underlying eq. (9). The Zeeman splitting for a particular gate voltage is shown in fig. 11, in which we also have plotted the LO and TO phonon frequencies of InSb. This makes clear that we observe modes of a totally confined electron system that are comparable in frequency to the frequencies of these

U. Merkt I Far-infrared spectroscopy of quantum dots 3 0

I / Lo i




/ / / / / /











Ix w,= 16.6 meV p-InSb(111) T=I.4K


Av, = 3ov

1.00 >

3 +~












Fig. 11. Zeeman-splitting of the quantum dot sample from fig. 9. The reststrahlen band of InSb is indicated by the hatched regime.

important lattice excitations. Possible implications of this will be discussed in the last section. T h e extrapolated (B---~0) quantization energies h~o0 for this sample are summarized in fig. 12. The solid line represents the theoretical description according to eq. (5). When we take the m e a n ~ = (Cox + ~)/2 of the dielectric constant of the oxide %x and semiconductor • as the effective constant, we obtain a radius R = 80 nm. This is consistent with the geometrical value R = 90 nm from the micrographs.





T= 1.4K

~ 15











4~0 610 B~O 9.()0



vg(v) Fig. 12. Quantization energy at zero magnetic field of the

sample from fig. 9 as a function of the gate voltage. The increase of the quantization energy is reasonably well described by the solid line calculated from eq. (5). From ref.













4b '6b

Frequency (cm -1 ) Fig. 13. Fourier transform spectra of field-effect confined dots on GaAs. From ref. [23].

Fourier spectra on field-effect confined dots on G a A s [23] are reproduced in fig. 13. The period of the dot array is a = 500 nm in this sample and there are about 50 electrons in a dot at a gate voltage Vg = - 0 . 8 V. In magnetic fields, the resonance ~o0 splits into only two modes indicating a parabolic confining potential. In fig. 14, the Z e e m a n splitting is shown for two distinct electron numbers. For both numbers, the dispersions of the two modes are well described by eq. (9). The quantization energy at zero magnetic field hw o = 3 . 6 meV is only very little d e p e n d e n t on the electron n u m b e r indicating that surface charges play an important role in creating the lateral potential. For a lattice constant a = 200 nm, dots with very few electrons could be realized [24]. In these experiments, the discrete electron n u m b e r becomes apparent as a step-like increase of the integrated line as shown in fig. 15(b). It is, on a first glance, surprising that for the large n u m b e r of dots essentially all of them are charged with the same n u m b e r of electrons. The reason is the high C o u l o m b charging energy e2/2C = 15 meV which can be estimated from the


U. Merkt / Far-infrared spectroscopy of quantum dots 80




25e / ' d o t i







> b-.






.::T'':" ii!i:.-?".iI





.,!" . : . . - 0 . 4 8


"2 > Fx

,o ,° "°.,•o








1.00 ~



12 e - / d o t 60 QO

°o °°











48 ( c m -1 )


B (T)

Fig. 14. Dispersions of the to+ modes for the sample from fig. 13 for two distinct electron numbers. The dotted lines are calculated from eq. (9). From ref. [23].




,'-capacitance C -- 5 x 10 TM F of the 2D disc. This value of the Coulomb energy is significantly larger than local fluctuations of the threshold voltage. Interesting fine structure and splitting has been observed in the detailed resonance positions as shown in fig. 15(a). It seems, that the confining potential deviates from a parabolic shape when the next electron is squeezed into the dot [24].








I --








-0.74 -0.72 -0.70 Gate voltage (V)



Fig. 15. Fourier spectra (a) of field-effectconfined dots with few electrons. The stepwise increase of the integrated line shape in a laser experiment (b) indicates the incremental occupation of the dots with one, two and three electrons. From ref. [241.

Far-infrared spectroscopy yields important parameters of isolated quantum dots like the electron number per dot and the strength of the lateral potential. In quantum dots studied so far, this potential is of approximately parabolic shape. This imposes an important restriction on the far-infrared spectroscopy formulated by the generalized Kohn theorem: in a strictly parabolic potential only the electron number n 0 and the characteristic frequency too can be determined. This theorem can be extended to an anisotropic oscillator [20], where one observes [25] just the

two frequencies to0x and to0y. However, there are various routes out of this dilemma. Since the depth of the lateral potential well is finite, there are deviations from a parabolic shape. In fact, the influence of higher-order terms, e,g., proportional to r 4 already have been studied theoretically in detail for the two-electron problem. This problem is commonly addressed as quantum-dot helium and is discussed by Pfannkuche in her contribution to this volume. The same holds for deviations induced by the nearly quadratic geometrical shape of deep-




U. Merkt / Far-infrared spectroscopy of quantum dots

m e s a e t c h e d dots. T h e K o h n t h e o r e m is also o v e r c o m e by t h e b a n d s t r u c t u r e o f the host semiconductor, in p a r t i c u l a r by the nonp a r a b o l i c i t y o f t h e c o n d u c t i o n b a n d a n d by t h e s p i n - o r b i t i n t e r a c t i o n . It is c l e a r , t h a t d e v i a t i o n s f r o m a p a r a b o l i c b a n d in t h e H a m i l t o n i a n o f eq. (11) i n j u r e t h e K o h n t h e o r e m as d o d e v i a t i o n s f r o m a p a r a b o l i c p o t e n t i a l . First signs o f b a n d n o n p a r a b o l i c i t y w e r e a l r e a d y o b s e r v e d for d o t s o n o u r n a r r o w - g a p s e m i c o n d u c t o r I n S b as d e v i a tions o f t h e d i s p e r s i o n of the w+ m o d e f r o m the r e s u l t of the e f f e c t i v e - m a s s a p p r o x i m a t i o n ( E M A ) in fig. 8. A l s o , c h a r a c t e r i s t i c line splittings i n d u c e d b y b a n d n o n p a r a b o l i c i t y a n d s p i n o r b i t i n t e r a c t i o n h a v e b e e n p r e d i c t e d [26], howe v e r , c o u l d n o t b e o b s e r v e d until now. O n I n S b , t h e r e l a t i v e l y b r o a d s p e c t r a l lines p r e v e n t this. A s l o n g as b e t t e r I n S b s a m p l e s a r e n o t a v a i l a b l e , s e m i c o n d u c t o r s like H g C d T e with s t r o n g e r b a n d nonparabolicity and spin-orbit interaction may a l l o w t h e s t u d y o f b a n d s t r u c t u r e effects in q u a n t u m dots. Q u a n t i z a t i o n e n e r g i e s that a r e c o m p a r a b l e to the energy of LO phonons open another interesting field for the F I R s p e c t r o s c o p y of i s o l a t e d quantum dots, namely the study of few-polaron s y s t e m s . This is e x p e c t e d to e n h a n c e o u r u n d e r standing of the interplay between Coulomb and e l e c t r o n - L O - p h o n o n i n t e r a c t i o n t h a t has b e e n s t u d i e d p r e v i o u s l y in 2D e l e c t r o n systems [27].

Acknowledgements I t h a n k D. H e i t m a n n a n d U. R6131er for v a l u a b l e discussions a n d the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t for financial s u p p o r t .

References [1] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [2] T. Chakraborty, Comm. Condens. Matter Phys. 16 (1992) 35.


[3] Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62 (1989) 2164. [4] U. Merkt, Ch. Sikorski and J. Alsmeier, in: Spectroscopy of Semiconductor Microstructures, eds. G. Fasol, A. Fasolino and P. Lugli (Plenum, New York, 1989) p. 89. [5] P. Junker, U. Kops and U. Merkt, to be published. [6] T. Demel, D. Heitmann, P. Grambow and K. Ploog, Phys. Rev. Lett. 64 (1990) 788. [7] A. Lorke, J.P. Kotthaus and K. Ploog, Phys. Rev. Lett. 64 (1990) 2559. [8] J. Alsmeier, E. Batke and J.P. Kotthaus, Phys. Rev. B 41 (1990) 1699. [9] J. Alsmeier, E. Batke and J.P. Kotthaus, Surf. Sci. 229 (1990) 287. [10] J.P. Kotthaus, A. Lorke, J. Alsmeier and U. Merkt, Springer Series in Solid State Sciences, Vol. 97, eds. F. Kucher, H. Heinrich and G. Bauer (Springer, Berlin, 1990) p. 29. [11] U. Mackens, D. Heitmann and J.R Kotthaus, in: Insulating Films on Semiconductors, eds. J.J. Simonne and J. Buxo (North-Holland, Amsterdam, 1986) p. 11. [12] P. Grambow, T. Demel, D. Heitmann, M. Kohl, R. Schfile and K. Ploog, Microelectronic Eng. 9 (1989) 357. [13] A. Kumar, S.E. Laux and F. Stern, Phys. Rev. B 42 (1990) 5166. [14] D.A. Broido, K. Kempa and P. Bakshi, Phys. Rev. B 42 (1990) 11 400. [15] J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1975) p. 121. [16] V. Shikin, S. Nazin, D. Heitmann and T. Demel, Phys. Rev. B 43 (1991) 11 903. The result for R 3 in their eq. (8), which is given in cgs units, was corrected by a factor 4/n. [17] W. Kohn, Phys. Rev. 123 (1961) 1242. [18] L. Brey, N.F. Johnson and B.I. Halperin, Phys. Rev. B 40 (1989) 10 647. [19] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [20] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [21] E. Batke and D. Heitmann, Infrared Phys. 24 (1984) 189. [22] U. Merkt, in: Festk6rperprobleme/Advances in Solid State Physics, Vol. 30, ed. U. R6ssler (Vieweg, Braunschweig, 1990) p. 77. [23] D. Heitmann, B. Meurer and K. Ploog, preprint (1992). [24] B. Meurer, D. Heitmann and K. Ploog, Phys. Rev. Lett. 68 (1992) 1371. [25] C. Dahl, F. Brinkop, A. Wixforth, J.P. Kotthaus, J.H. English and M. Sundaram, Solid State Commun. 80 (1991) 673. [26] U. R6ssler, private communication. [27] J.T. Devreese, Physica Scripta T 25 (1989) 309.