COMPUTATIONAL MATERIALS SCIENCE
ELSEVIER
Computational Materials Science 3 (1994) 7894
Quantum ballistic transport in semiconductor nanostructures; effect of smooth features in confining potentials Tomas Lundberg Department of Physics and Measurement Technology Linki~ping University, S581 83 Linki~ping, Sweden
Received 14 March 1994; accepted 26 April 1994
Abstract Using a modematching method, we investigate the ballistic transport properties of noninteracting electrons in two types of semiconductor nanostructures in which the confining potential has smooth, rounded corners. The rounded corners are simulated by dividing the structures into narrow strips and varying the width of each strip. The two structures we have investigated are a straight channel connecting two reservoirs of twodimensional electrons and a crossbar structure consisting of two perpendicular channels, one of which connects to the reservoirs. In the case of the straight channel we find that the rounded corners decreases oscillations in the conductance. For the crossbar structure we have compared with results from Berggren et al. (K.F. Berggren, C. Besev and Z.L. Ji, Physica Scripta T 42 (1992) 141) who used a model with sharp corners. For low values of the Fermi energy our results are in good agreement with theirs, which validates the simplification of using sharp corners. As the energy increases differences appear, and at certain energies, the rounded corners drastically changes the conductance, transforming a resonance peak to an antiresonance dip.
1. Introduction Semiconductor nanostructures have been given much attention the past few years. In such systems mobile electrons are confined in specific spatial regions. One example of this is a thin quantum well formed by heterojunction layers on the order of 100 ~. in thickness. In these systems, the energy levels of the electrons perpendicular to the heterojunction are quantified into a discrete spectrum, and the difference in energy between two adjacent levels is large enough so that only the lowest levels are populated at low temperatures. The dimensionality of the electrons is thus reduced and the electrons form a twodimensional electron gas (2DEG). It is possible to
achieve extraordinarily long mean free paths (many micrometers) at low temperatures. This opens the possibility of ballistic transport [1,2]. Recent advancements in lithographic techniques have made it possible to reduce the dimensionality even further. By use of spilt gate techniques it is possible to make e.g. (quasi) 1dimensional quantum channels and 0dimensional quantum dots. If the typical dimensions of the systems are smaller than the mean free path, the electrons maintain their phase coherence as they pass through the system. Experimentally, it was found by Wharam et al. [3] and van Wees et al. [4] that the conductance in a short channel connecting two 2 D E G reservoirs is quantized into G = 2 e 2 N / h , where N is the number of occupied
09270256/94/$07.00 © 1994  Elsevier Science B.V. All rights reserved SSDI 09270256(94)00046F
T. Lundberg / Computational Materials Science 3 (1994) 7894
1D subbands. Recently, more complicated geometries have also been studied, both experimentally [5] and theoretically [610]. Most of theoretical studies cited above have studied geometries with sharp corners for reasons of simplicity and computational convenience. In a real electron waveguide the assumption of sharp corners will not be valid. Firstly, the metal gates which cause the electrostatic confinement cannot be made with atomically sharp corners but will have radii in the order of 100 A. Secondly, even if the metal gates could be made with sharp corners, the fact that the electron gas is separated from the metal gates by a spacer layer will smooth the confinement down in the 2DEG. In this paper we will study the effect of this smoothening of the confinement. Using a modematching technique, we will examine a channel similar to the channel studied by Kirczenow [11] (see Fig. l(a)) and a cross bar structure similar to the one studied by Berggren and Ji [12,13] (see Fig. l(b)), but use rounded corners instead of sharp. We will focus on the effect of the roundness of the corners, but the same method could also easily be used to examine interface roughness and other imperfections. Using rounded corners will increase the complexity of the problem and the computational effort required, and we will therefore investigate the convergence of the method in some detail.
• r',I
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79
Berggren and Ji found that there exists bound states in the center of the cross bar structure. Bound states in the intersection of two infinitely long channels had been found earlier by Schult et at. [14]. These bound states give rise to resonant tunneling, similar to the resonant tunneling through a double barrier. A significant difference is that the cross bar structure is classically an open system, unable to trap any particle. Berggren and Ji also examined the electron density and the quantum mechanical current distribution. They found that the flow of carriers is laminar at low energies, but as the energy increases vortices appear in the current. When the energy was increased even further, the vortices formed increasingly complex and volatile patterns [13]. An interesting question here is if these patterns arise due to the sharp corners used, and if the bound states are affected. We will therefore examine, in some detail, the current distribution and the electron density for the cross. An interesting analogy can be drawn between an electron waveguide and an ordinary waveguide for electromagnetic fields [15]. It is possible to make a correspondence between the solution of the threedimensional Maxwell equations in a waveguide and the twodimensional Schr6dinger equation in an electron waveguide. In the case of electromagnetic waveguides it is possible to make corners sharp since the wavelength is longer in
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Fig. 1. Schematic figure of the models (without rounded corners). The potential is zero inside the channel and in the two reservoirs, and infinite elsewhere. W h e n a potential is applied the electrons flow from the left reservoir to the right. (a) The straight channel. (b) T h e crossbar structure.
80
T Lundberg/ComputationalMaterialsScience3 (1994) 7894
the electromagnetic case. In Section 3 we will elaborate somewhat this analogy. The model and the theoretical framework is presented in Section 2. The results of the numerical calculations are given in Section 4 and finally a short summary is given in Section 5.
2. Theoretical model
In an ideal sample, the channels form an electron waveguide where the electronic transport properties are only determined by the geometry of the channel and the wavelike nature of the electrons. The detailed shape of the potential that confines the electrons should thus not be important if we only look for qualitative effects. Selfconsistent calculations [16] show that a square well potential should be close to reality when several subbands in the channel are populated, while a parabolic well is better when only the lowest one or two subbands are populated. Here we will use an infinitely high square well potential for its simplicity.
2.1. Sharp comers The first model device is shown in Fig. l(a). It consists of two 2 D E G regions acting as source and drain, and connected with one straight channel. If a weak potential difference is applied between source and drain, an electrical current will flow through the channel. The length of the channel considered here is in the order of 1000 which is smaller than the mean free path of the 2DEG. Thus, at low temperatures the transport through the channel is ballistic, i.e. no collisions occur. It is therefore not unreasonable to consider only oneelectron wave functions and ignore interactions between the electrons. We will assume that the confining potential is an infinite square well. This makes it possible to solve the Schr6dinger equation exactly and obtain simple analytical expressions for the wave function. The device is divided into three parts (see Fig. 1), namely the left reservoir (the source), the interconnecting channel and the right reservoir
(the drain). The Schr6dinger equation in two dimensions reads 2m*
~
~b(x, y) = E $ ( x , y ) ,
+
(1)
where m* is the effective mass of the electron; m* 0.067m, for GaAsGaA1As. The choice of confining potential makes it easy to solve Eq. (1) since the wave function has to vanish at the infinite walls. For the wave function in the left reservoir we get
I]1k = e ikxx+ikry + f +:A L(ky) ' e ik'x+ik'yy dky,
(2) where the first term corresponds to an electron incident from the left and the second term to an electron being backscattered. A L relates to the probability for backscattering, k x and k r are the wave vectors in x and ydirection, respectively, of the incident electron and are related through E = h2k2/2m *, where k = kxf + krJ3. The component k'x is defined as
/ 2m*E k'= w ~
k~
(3)
and may be either real or imaginary, i.e. evanescent waves are included. Similarly for the wave function in the right reservoir, we get ~bff .[_~ AR (ky' ) e ik'xx+ik'yy d k y, ,
1 ~
p
(4)
"xp+l
ly
Xp _1
Xp
Fig. 2. The pth strip and its neighbours. The width of the strip is denoted wp and the xcoordinate is translated to X   X p _ _ 1.
T~ Lundberg / Computational Materials Science 3 (1994) 7894
where B n and C, are expansion coefficients. The index n = 1, 2 , . . . runs over all sublevels with energies E n = h2(nar/w)2/2m *. For E > E , the longitudinal states are traveling waves with wave number qn = i~2m*/h2(E  E , ) . For E < E~ the solutions become exponential with q,
where A R relates to the probability for transmission. W i t h i n t h e c h a n n e l , t h e w a v e f u n c t i o n is exp a n d e d in e x a c t s o l u t i o n s to E q . (1). F o r c h a n n e l regions of width w and length 2d we have
o c = ~ { B. eq'~ + C~ eq~x} sin
w)l
y + 2
81
,
= (2m*/h2(E.  E).
n
Applying the usual technique of matching 0 and OO/Ox at the interfaces, we arrive at a non
(5) (a)
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y (AngstrSm) Fig. 3. The matching of the wave function and its derivative at the interface between the channel and the left reservoir at E = 1.3 meV. (a) The matching of 4/ at the interface is virtually perfect. Here the real parts of the wave functions are shown. It is impossible to separate them by eye. (b) An extreme case of mismatch for Ot~/Ox. The two curves oscillates around a common mean value and spikes develops at the channel boundary. As the number of modes is increased, the oscillations diminish and the two curves converge except at the boundary where the spikes remain. Note that the derivatives are not required to match outside the channel. Here the real parts of a$/Ox are shown. ( ) the derivative in the channel; ( . . . . . . ) the derivative in the reservoir.
82
T Lundberg/Computational
homogeneous
system of equations [ 11,121
(6)
Materials Science 3 (I 994) 7894
ing $ and a$/& between the pth and p + lth strip gives us a matrix relating the expansion coefficients of the two strips
where Tm,n * 4,;s,,m
S*m,n=
erqnd
Pa)
3
A, = 2ik, eeikxdMky,,, Tm,n
P)
+m dk; =1 ’
/
m
(7c)
k:Mk;,mMk;,n,
2T
(9)
The total transfer matrix is then obtained multiplying the transfer matrices
by
T = Ti,2T2,3 . so that
. * TNI,N
(10)
(:j=$:)
and Mk,n
I.
I
=
dy sin[y(y+t)] 1w;:’
eiky.
Solving this system numerically Cl*
(7d)
gives us B, and
Eq. (6) now reads (“d’
*:p = c (B&n n
e4,JPx,l)
S$:)+($
a,(::)
=[(“o: ‘;)+(;
2.2. Rounded comers The effect of rounded corners will be included in an approximate way by dividing the channel into strips and varying the width and the length of each strip to simulate smooth rounded corners. This will be a good approximation if the change of the width in each strip is much smaller than the Fermi wavelength. Fof~ GaAsGaAlAs, A, is in the order of 3001000 A. Hence, if the change of width is N 10 A or less, the curvature will appear smooth to the electrons. The form of the rounded corners is chosen to be elliptical. We will use a transfer matrix method similar to the method used by Wu et al. [9] and Xu et al. [lo]. We also refer to Ref. [17] for a more detailed description. We divide the channel into N strips (see Fig. 2). The wave function in the pth strip is written as in Eq. (5) except that the xcoordinate is translated for each step + C, n eq,,,(xx,l)
1
xsin[%(y+:)]. In Eqs. (2) and (4) the xcoordinate is also translated to x x0 and x xN, respectively. Match
(11)
$)T](;;)=(:).
(12) Solving this system of equations gives us B, and C, and all the subsequent coefficient vectors Bi and Cj can be obtained through Eq. (9). 2.3. Derivation of transport parameters Given a certain energy, it is obvious that the expansion coefficients, and hence also the wave functions, depend on the direction of the incident wave (this is indicated by the subscript k on I+!J above). For each incident wave k and each energy E the quantum mechanical current j can be written as j(k)
=j,(k)P
+j,(k)E
= $(Re[$*iG$]i+Re[$*i&d]?), (13) where the subscript k has been dropped from $ for simplicity. The electrical current carried through the channel can then be found by integrating j, over the width of the channel j(k)
= 2/_;,;
dyj,(k),
T. Lundberg~Computational Materials Science 3 (1994) 7894
with a b o u n d a r y ~ and translating the surface in the zdirection. If a scalar field q,(x, y) satisfies
w h e r e a factor 2 derives f r o m the spin. T h e total current J at T = 0 K is given by the sum of contribution of all incident waves [11]. If we apply an infinitesimally small potential difference V b e t w e e n the source and the drain, only electrons in the energy interval [ E F  e V , E v] will contribute to the current because of the Pauli principle. T h e c o n d u c t a n c e G = J / V at E = E F is then given by a
~

83
(v +
k2)q,(x,
y) = 0 (x, y)
0(x, Y)[y=0,
(16)
then E and B fields of the form E = i k 0 ; L B =  i X
[email protected] will satisfy Maxwell's equations and the b o u n d a r y conditions for T E m o d e s in the waveguide. Calculating the Poynting vector P, we find
2e 2 J~r/2 dqbj(k) l Ikl=kF'
1
where k = kv(cos q52 + sin ~b~), k v is the Fermi wave n u m b e r and ~b is the angle of incident for the incoming wave. In the same way, the total current J and the electron density p can be f o u n d by summing the contribution of all incident waves.
k
P= Re{E 21z
xS*} = =Re{6iV0*} z/x 
k = Re{[g,*( 2t*
k iV0)]*} =  Re{0*iV6}. 2t* (17)
A p a r t for numerical factors, the Poynting vector has the same structure as the q u a n t u m mechanical current in Eq. (13). T h e r e is thus a close c o r r e s p o n d e n c e between the electromagnetic and the q u a n t u m mechanical waveguide. It may be easier to observe the current distribution in the electromagnetic waveguide than in the q u a n t u m mechanical.
3. Comparison with electromagnetic waveguides W e will now briefly c o m m e n t on the similarity between the q u a n t u m mechanical waveguide and the electromagnetic waveguide. Following Carini et al. [15], the electromagnetic waveguide is modeled by constructing a surface o in the x  y plane
t'~
2" e~O) tO_
tt~ c5
1
i 2
I 3
I 4
E
i 5
I 6
I 7
(meV)
Fig. 4. The conductance G (in units of 2e2/h) at different Fermi energies (in meV) for the straight channel at T = 0 K. ( ) is for sharp corners; ( . . . . . ) is for r, = ry = 1000 A; (     ) is for r x = 1000 A, ry = 500 A; ( . . . . . ) is for r x = 500 .~, ry = 1000 ,~ (see insert for definition of rx and ry).
T. Lundberg / Computational Materials Science 3 (1994) 7894
84
~.¢
.,..¢
~t2
m
d d
w
~
Fig. 5. Schematic figure of the crossbar structure connecting the left and right reservoir. T h e upper part shows a typical appearance when rounded corners are used, while the lower part gives the definition of the parameters used. W e have used the following values: w = 1000 ?, and d = 500 A.
4. Results and discussion
The method presented here is formulated in an infinite basis [see Eq. (5)], and in this basis the formulation is exact. However, Eqs. (6) and (12) must be solved numerically by limiting the number of elements in the basis. Due to numerical instabilities for certain geometries, the maximum number of basis elements cannot be chosen too large. This will of course affect the convergence. Usually it is enough to include at least two evanescent modes to obtain convergence. For the
ER1

energies and geometries studied here, this is equivalent to including at least five transverse modes. The matching of the wave function and its derivative was examined by plotting 0 and OO/Ox at a crosssectional cut of the channel. The matching of the wave function was found to be excellent, even with a low number of transverse modes, while for the derivative the matching was not as good (see Fig. 3). The derivative is only matched at the opening of the channel and not outside the opening. This makes O0/Ox discontinuous across the interfaces (except at the opening), and spikes caused by Gibbs phenomenon appeared at the channel boundary. Furthermore, the two functions oscillates around a common mean value. By increasing the number of modes the matching could be made better. This 'mismatch' does not however seem to have any observable effect. As another test of the convergence, the conductance was calculated at both ends of the channel and the two values were required to agree within a few percent. Except for a few geometries this was easily acquired. In a straight channel with sharp corners (as the one presented in Fig. l(a)), the conductance exhibits regular oscillations as a function of the Fermi energy (see Fig. 4). These oscillations are due to longitudinal resonant states in the chan
ER2
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E (meV) Fig. 6. The conductance G (in units of 2e2/h) at different Fermi energies E (in meV) for the crossbar structure at T = 0 K.
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Fig. 7. The relative electron probability distribution corresponding to the antiresonance at EA3. It is associated with a standing wave in the vertical stubs, but the rounded corners allow the wavefunction to extend further out in the leads. The region with highest probability has been designed as red and the lowest region as blue, by scaling down 128 levels from the m a x i m u m relative level 11.69. The jagged outer rim and the nonzero probabilities in the forbidden region is due to interpolation error in the plotting program.
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Fig. 8. The relative electron probability distribution associated with the first quasibound state at energy EBb. Colours have the same meaning as above but the scaling factor is here 57.74.
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Fig. 9. The relative electron probability distribution associated with the second quasibound state at energy EB2. Colours have the same m e a n i n g as above but the scaling factor is 44.03.
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7:. Lundberg/Computational Materials Science3 (1994) 7894
nel, similar to the acoustic standing wave in an open pipe [11]. These types of oscillations are not seen in experiments. When we examine the same channel but with rounded corners, we find that these oscillations become less prominent, and also that the peaks of the oscillations are shifted towards higher energies (see Fig. 4). This is in agreement with the results found earlier by Szafer and Stone [18]. By varying the x and yradius of the corners independently, we find that the oscillations are mainly affected by the yradius only, while changing the xradius changes the steepness of the conductance curve at the steps in the conductance. In principle it would thus be possible to get rough estimate of the true curvature of the electrostatic confinement in the 2 D E G by comparing the shape of the experimental conductance curve with a conductance curve from simulations. For the crossbar structure, we have used the structure illustrated in Fig. 5. The same structure with sharp corners has also been studied by Berggren and Ji [12] and by Kawamura and Leburton [19] who used a recursive Green'sfunction. In general, we have found good agreement when comparing with these two previous results. Although the method presented in this paper is more flexible than the method used by Berggren and Ji, it is also simpler. They have divided the structure into five regions, while we have only one section, albeit with varying width. This means that in order for our method to converge, it will not be possible to make the curvature of the corners arbitrarily small since this would lead to too abrupt changes in the width. We have in the following taken the curvature of the corners in the xdirection to be as large as possible to make the convergence as good as possible. The conductance plot for the crossbar structure is shown in Fig. 6. Comparison with the results of Berggren et al. [13] shows good agreement for lower energies ( E v < 3 meV) while there are discrepancies at higher energies. The two peaks marked E m and EB2 are due to resonant tunneling via the quasibound states at the center of the cross [14] and are shifted downwards in energy compared with the sharp corner case. The two peaks marked ER1 and ER2 appear in addi
tion to E m and E~2 because the cross is finite. They are associated with standing waves mainly in the vertical stubs and are also shifted downwards in energy. The peak at ER2 is broadened while the peak at ER1 remain a very sharply peaked resonance. Due to quantum mechanical interference, the conductance vanish at certain energies (EAI, EA2 , EA3, and higher). The antiresonances at EA1 and EA2 can be found in the data of Berggren et al., but the antiresonance at EA3 is interesting. At the corresponding energy Berggren et al. has instead a peak in the conductance associated with the third standing wave resonance in the stubs. That the peak at EA3 actually corresponds to this situation can be seen in Fig. 7. The standing wave pattern with five lobes can clearly be seen. With rounded corners the center of the cross is more open than in the sharp corner case, and this allows the wave function to extend further out in the leads. As a result of this, the standing wave does not align as neatly as in the sharp corner case, and the effect is an antiresonance instead of a resonance. Although calculations using sharp corners give a qualitatively correct picture, our work shows that a more realistic model with rounded corners might drastically change the conductance at certain energies. This will be of great importance in the future if devices like these are to be used as quantum interference transistors. The method developed in this paper is well suited to investigate the effect of smooth, rounded corners. In this work we have used an infinite square well as the confining potential. This is a simplification, and further work is required to investigate the effect of also introducing confining potentials closer to reality. 4.1. Spatial distribution o f electrons
Figs. 810 show the electron densities associated with the current flow. The first of these, Fig. 8, shows the electron density distribution associated with the lowest bound state in an infinite cross. The wave function is localized in the center of the crossbar structure and is in good agreement with the corresponding calculations by Schult et al. [14].
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Fig. 10. The relative e l e c t r o n p r o b a b i l i t y d i s t r i b u t i o n a s s o c i a t e d with the second r e s o n a n c e at e n e r g y ER2. In a d d i t i o n to the four i n n e r lobes, r e m i n i s c e n t of the s e c o n d b o u n d state, t h e r e are also two o u t e r lobes, localized in the stubs, which are r e l a t e d to a vertical, s t a n d i n g wave r e s o n a n c e . C o l o u r s have the s a m e m e a n i n g as above but the s c a l i n g f a c t o r is 11.69.
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Fig. 11. The relative quantummechanical current distribution associated with resonant tunneling via the quasibound state at energy E m and the density distribution in Fig. 8. The highest velocity has been designed as red and the lowest velocity as blue, by scaling down 32 levels from the maximum relative velocity value 0.092.
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T. Lundberg/ Computational Materials Science 3 (1994) 7894 The electron density distribution in Fig. 9 is associated with the second bound state in an infinite cross. The wave function is now mainly situated at the corners of the cross. Again, this is in good agreement with the result by Schult et al. The third figure, Fig. 10, shows the electron density distribution related to the second resonance peak at energy ER2. It has four inner lobes in the center of the cross, somewhat similar to the density distribution for the second lowest bound state of an infinite cross. In addition there are two outer lobes, localized in the stubs, which are associated with a vertical standing wave resonance.
4.2. Spatial distributions of velocities The quantummechanical probability current distribution at three different energies is shown in Figs. 1113. Fig. 11 shows the current distribution associated with the lowest bound state at energy Em. The electron flow is clearly laminar. The corresponding density distribution is shown in Fig. 8. As the energy increases, there is a transition from laminar to vortex flow. Fig. 12 shows the current distribution at the first antiresonance at energy EA1. The conductance here is low, G = 0.08 [2e2/h]. On further increase of the energy, the current distributions form increasingly complex and volatile patterns. Fig. 13 illustrates the current distribution at an energy ~ 0.015 meV lower than the energy for the third antiresonance, EA3. Small changes in energy may result in large, global changes in the current flow. When comparing with the data from Berggren et al. [13], we find that most of the complex flow patterns do survive the transition from sharp to rounded corners. However, the crossbar structure with rounded corners seems to be less sensitive to small changes in energy compared with the same structure with sharp corners.
5. Conclusions We have developed a method for investigating the effect of smooth features in the confining
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potential of semiconductor nanostructures. The method is based on a modematching method combined with a transfer matrix method. We have used this method to examine the effect of rounded corners in the confining potential in two types of quantum ballistic channels. For a straight channel we find that the effect of the rounded corners is to wash out the oscillations coming from longitudinal standing waves in the channel. For the crossbar structure we find that most of the features in the conductance survive the transition from sharp corners to rounded corners with only a few changes, e.g. small shifts in energy or a slight broadening of peaks. This validates the previously used simplification of using sharp corners when modeling various structures. However, our results shows that a more realistic modeling using rounded corners might drastically change the conductance at certain energies, transforming resonance to antiresonances.
Acknowledgements I would like to thank K.F. Berggren for many helpful discussions. This work has been supported by the Swedish Natural Science Research Council.
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