Available online at http://www.idealibrary.com on doi:10.1006/spmi.2001.1013
An explicit model for a quantum channel in 2DEG
V
ALERIA. FEDIRKO
†, MEI
-HSIN
CHEN, CHIUNG-CHUAN
CHIU
, SHI-FONG
CHEN,
KAUNG-HSIUNG
WU, JENH-YIH
JUANG, TZENG-MING
UEN
, YI-SHUN
GOU
Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan 30010, Republic of China (Received 22 November 2001)
A two-parametric model for a channel in a two-dimensional electron gas (2DEG) is pro-posed which allows for the explicit analytical solution for the problem of quantum electron transport through the constriction. Conductance step smearing appears naturally and a sim-ple criterion for the step occurrence in terms of channel parameters is given.
c
2002 Elsevier Science Ltd. All rights reserved. Key words: quantum electron transport, quantum constriction, conductance quantization, two-dimensional electron gas.
1. Introduction
An interest has persistently been growing in micro- and nano-structures in which an electron can move ballistically through some active region [1]. Quantization of the conductance of a narrow ballistic channel in a two-dimensional electron gas (2DEG) by G0 = 2e2/h, first discovered by van Wees et al. [2] and Wharam et al. [3], is, perhaps, one of the most exciting phenomena which can be realized in such structures. Moreover, the differential conductance may also be quantized in such structures as a function of applied bias [4, 5]. Though the principal physical reason for the quantization effects has been clear since the earliest works [2–5], various theoretical approaches have been exploited for treating the problem (see, e.g. [6–23]). Smooth [6–9, 17, 18] and abruptly [10–15] changing channel boundaries were assumed and some channel wall roughness [20, 21] was taken into account. The differential conductance quantization and other effects of non-zero bias [4, 5, 19, 22, 23] were described.
The so-called ‘adiabatic’ approximation based on the Born–Oppenheimer approach [23] for a hard-wall smooth channel [6–9] is, perhaps, one of the most developed and widely used. Nevertheless no explicit model of a constricting quantum channel has yet been analysed which allows for an exact analytical adia-batic solution for electron quantum transmission. Such a model however, though not a general one in itself, could permit a good physical insight and provide physical criteria for experimental observation of quan-tization effects in terms of model structure parameters.1It is useful for experimental structure evaluation and for experimental data analysis and can be easily generalized. In this paper we propose and analyse a two-parametric model of that kind.
†Permanent address: Moscow State University of Technology “Stankin”, Moscow 101472, Russia.
1Actually, an attempt of that kind was undertaken in [17] but a parabolic potential used there can hardly be brought into correspondence to some real structure and is in fact only an approximation near the bottleneck.
12 10 8 6 4 2 – 15 – 10 – 5 0 5 10 15 y / w0 x / w0 3 2 4 1
Fig. 1. Schematic picture of model channel boundaries for various values of dimensionless parameter(βw0): (1) βw0=0.1; (2) βw0=
0.2; (3) βw0=π; (4) boundary defined by (10) forβw0=0.5, W/w0=10.
2. Model
One needs at least two parameters as a minimum set to describe a constricting channel, i.e. the bottleneck width,w0, and the characteristic channel length, l. Assuming that the 2DEG leads to the channel lies far away from its bottleneck at a distance much longer than l and is much wider thanw0one can consider a channel, which is broadening smoothly from the bottleneck to infinite 2DEG reservoirs. For a stiff-wall channel model one thus has to find a proper and convenient two-parametric function y = f(x) for an x-dependent channel border, where x is the coordinate along the channel and y is in the perpendicular direction to the plane of the 2DEG sheet. We put f(x) = 0 for one border, while for the other we choose f (x) in the form
f(x) = w0·cosh(βx), (1)
which, as one shall see, allows for an exact analytical adiabatic solution for the scattering problem of quantum electron transmission throughout the channel. For such borders the variable width of a channel is:
w(x) = w0·cosh(βx). (2)
Channel boundaries described by eqn (1) are drawn schematically in Fig. 1 for various values of dimension-less parameterβw0. One can easily see that whenβw0 1 we have a long channel, while forβw0 1 we have a short one. Varying parametersβ and w0one can adjust such a model channel very close to a number of experimental structures. It is also worth mentioning that the same parameterβ−1 determines the spatial scale of channel boundary variation soβ can be considered as an inverse characteristic length of the channel:
β ∼ l−1.
The symmetric channel with boundaries |y| = f(x) can obviously be treated in the same way.
For a particular (usually the lowest) two-dimensional (2D) subband the problem of electron transmission through a stiff-wall channel is that of a solution of a 2D Schrödinger equation for stationary states with zero boundary condition at the channel walls and finite asymptotic at infinity for wavefunction8(x, y)
¯ h2 2m d28 d x2 + ¯ h2 2m d28 d y2 =E8, 8(x, y = 0) = 0, 8(x, y = f (x)) = 0, (3)
where the energy E is measured relative to the edge of the 2D subband. In the adiabatic approximation [6– 9, 20] one searches for a stationary solution of (3) as a product
assuming thatφ(x, y) satisfies the equation ¯ h2 2m ∂2φ d x2+ ¯ h2 2m ∂2φ d y2 +ε(x)φ = 0, (5)
with the boundary conditions
φ(x, y = 0) = φ(x, y = f (x)) = 0, (50)
where x is considered as a parameter, but not as a variable. For that one easily finds a discrete spectrum
ε(x) = εn(x) = π 2n2 w2(x), (6) with φ(x, y) = s 2 w(x)sin πn w(x)y, (7)
and n = 1, 2, . . . being a transverse mode quantum number.
Substituting (4), (6), (7) into eqn (3) and neglecting slowly (‘adiabatically’) varying terms [6–9, 20] one finds the one-dimensional (1D) Schrödinger-like equation for a single nth mode wavefunction ψn(x) of electron motion along the channel in the effective quasi-potentialεn(x):
¯
h2 2m
d2ψn
d x2 + [E −εn(x)]ψn=0. (8)
Transitions between transverse modes are not thus taken into account by the adiabatic approximation. Intermode scattering breaks the adiabatic approximation when a channel is not smooth enough [6]. How-ever intermode scattering results in a change of a longitudinal momentum of the order ofπ/w(x) for a constant energy, so it is essential only when the spatial scale of channel boundary variation is of the order of
w(x)/π. The latter can be locally characterized by a logarithmic derivative w0(x)/w(x) which for a model
under consideration is:
w0(x)/w(x) = β · tanh(βx).
When the dimensionless parameter
λ−1=βw/π < 1, (9)
intermode scattering is negligible in the region
|x|< β−1lnλ
unless a ballistic regime holds.1That is a condition for local adiabacity in the vicinity of the bottleneck. However, whenλ−1is small (i.e. a long channel), one has
β−1lnλ > 1,
so such a channel can be considered as adiabatic at the length longer than its characteristic length, l, i.e. almost ‘globally’ adiabatic [8].
Equation (1) may look unsatisfactory because the width of the channel becomes infinite when x tends to infinity. To limit the channel width by some finite value and to formally include into the model the leads of the width ∼ W w0at x → ∞ one can use, for example
f(x) = (1 + w0/W) w0
cosh(βx)
1 +(w0/W) cosh(βx)
(10) 1Adiabatic approximation fails near turning points, however the asymptotic results hold.
ε / ε1 ε / ε1 εF εF 8 6 4 2 8 10 6 4 2 – 7.5 – 5 – 2.5 2.5 5 7.5 – 7.5 – 5 – 2.5 2.5 5 7.5 x/w0 x/w0 A B
Fig. 2. Effective quasi-potentialεn(x) for the first three transversal modes (n = 1, 2, 3): A λ = 2π; B λ = 3π; Fermi-energy, εF is shown in A and B for the same value of(kF/β) = 2π.
instead of (1). We show a channel boundary(10) in Fig. 1 for illustration. The channel length can then be estimated as L ≈ l ln(W/w0). It, however, has no effect on electron transmission through the channel when L l so eqns (6)–(8) with w(x) in the form of (2) are valid anyway. Yet this may be useful, say, for transversal mode counting.
3. Results and discussion
The problem of electron transmission through a constricted channel is thus reduced to a 1D scattering problem with a potential barrierεn(x). When the channel width w(x) is given by formula (2), eqn (8) leads to the following equation
ψ00+ k2+ β 2λ2n2 cosh2βx ψ = 0, (10) where we put E = h¯ 2k2 2m (10 0)
andλ is defined by (9). Equation (10) is similar to that considered by Pöschel and Teller [24]. The effective
quasi-potentialεn(x) for the first three modes (n = 1, 2, 3) is shown in Fig. 2 for two values of parameter λ:
With the substitution ψ = coshs(βx) u, where s(s − 1) = λ2n2, s =12 1 + i √ 4λ2n2−1,
eqn (10), by introducing the new variable z = sinh2(βx), is reduced to a hypergeometric equation (see, e.g. [25])
z(1 − z)u00+ [c −(a + b + 1)z]u0−abu = 0, (11)
with a = (s + iκ)/2, b = (s − iκ)/2, c = 1/2, κ = k/β. As follows from (11) the general solution of eqn (10) is of the form
ψ = C1ψ1+C2ψ2, (12)
where
ψ1=coshs(βx) ·2F1(a, b; 1/2; − sinh2(βx)),
ψ2=coshs(βx) · sinh(βx) ·2F1(a + 1/2, b + 1/2; 3/2; − sinh2(βx)),
2F1being the hypergeometric function of corresponding arguments, and C1and C2are arbitrary constants. So, one can find the exact analytical solution of eqn (10).
To solve the scattering problem one should match the general solution (12) to the boundary conditions at x = ±∞
ψ(x)x→∞=ei kx+r e−i kx, ψ(x)x→−∞=t ei kx,
where r and t are the reflection and transmission amplitude, respectively. That can easily be done using the asymptotic expansion for the hypergeometric function, which leads to the known expression for the transmission coefficient T = |t|2(see, e.g. [26]) for the nth transversal mode:
Tn(k) =
sinh2(πk/β)
sinh2(πk/β) + cosh2 πp4λ2n2−1/2. (13)
The reflection coefficient is found immediately as R = |r |2=1 − T .
For the above scattering problem we also find the coefficients C1and C2as
C1= 0 1 4− i 2γ− 0 1 4− i 2γ+ 20(1/2)0(−iκ)eiκ ln 2 , C2= − 0 3 4− i 2γ− 0 3 4− i 2γ+ 20(3/2)0(−iκ)eiκ ln 2 (14)
for any n, where0 is the gamma function of corresponding argument and we introduce the notations γ± =
κ ± δ, δ =p
4λ2n2−1/2. Formulae (13) and (14) give full analytical solution of the problem.
Forκ, δ 1 the transmission coefficient T for any n can be approximated by the simplified expression:
Tn= 1 1 + cosh2πδ/ sinh2πκ ≈ 1 1 + e2π(δ−κ). (15) Whenλ > 1, δ ≈ λn, so κ − δ ≈k −πn w0 β,
and eqn (15) shows that Tnare step-like functions of a variable
ξn=(kw0/π − n)/β
with a step-smearing width of the order ofβw0/π2 =1/πλ. Steps are thus well-defined for the parameter of eqn (9),λ ≥ 1. Transmission coefficients Tn for some n calculated numerically as a function ofξ at
g, 2e2/h g, 2e2/h g, 2e2/h g, 2e2/h 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 – 1.5 – 0.2 – 0.1 0.1 0.2 – 0.06 – 0.04 – 0.02 0.02 0.04 0.06 – 1 − 0.5 0.5 1 ζ – 0.6 – 0.4 – 0.2 0.2 0.4 0.6 ζ ζ ζ A C B D
Fig. 3. Dependence of the transmission coefficient Tnon electron energy through the variableξn=(kw0/π − n)/β for various values of parameterλ: A λ = 0.5; B λ = 1; C λ = π; D λ = 3π; dash-dot line—n = 1; dashed line—n = 2; solid line—n = 5; dotted line—n = 10.
various values of parameterλ are plotted in Fig. 3 (due to eqn (100) this is, in fact, Tndependence on electron energy). One can see that whenλ > 1 they have a sharp enough stepwise shape, and the step smearing is still less than 1 even forλ = 1, but it becomes more than 1 when λ = 0.5. Step smearing is consistent with estimation (15). Forλ ≤ 1 the form of a step changes drastically with the mode number, n, but for a few lowest modes only. Forλ ≥ 1 the form of a step only slightly depends on a mode number even for low n and this dependence tends to vanish whenλ increases.
With the help of eqn (13) one can estimate the ballistic conductance G of a quantum channel in a degen-erate 2DEG of geometry (1) at the low temperature limit as
G = e 2 π ¯h ∞ X n=1 Tn(kF), (16)
where kFis the Fermi wave-number of 2DEG in the leads (see, e.g. [27]).
In Fig. 4 we present the conductance G, calculated numerically in accordance with (16), as a function of a variableξ = (kFw0/π) for a set of values of parameter λ. We sum as many terms K ξ in (16) as necessary to reach the smallest estimated error due to the summation of a finite number of terms for all values of the variableξ under consideration.
One can see that the step-like dependence of G onξ survives even when λ = 1 but only a few (two to three) first steps are apparent enough. However step structure is obviously absent atλ = 0.5. For λ > π conductance steps show no marked dependence on the step number up to quite a high number of steps.
G, 2e2/h 20 15 10 5 2 4 6 8 10 6 5 4 3 2 1
Fig. 4. Conductance, G, of a channel calculated numerically as a function of a variableξ = (kFw0/π) for a set of values of parameter
λ: (1) λ = 0.5; (2) λ = 1; (3) λ = 1.5; (4) λ = π; (5) λ = 2π; (6) λ = 5π. (Every subsequent curve is shifted up by 1 for clarity). Actually our calculations show that N ≥ 20 steps are perfectly apparent, almost without smearing, say, for
λ = 2π, i.e. βw0 = 0.5. It evidently indicates out that if considerable step smearing, as well as higher
order step smearing, is observed in a long smooth channel it likely results from some other reason, probably from scattering by channel wall roughness [20, 21]. The above results, therefore, enable one to evaluate the ‘quality’ of a channel.
Atξ < 1 no free propagating mode exists and the contact behaves as a peculiar tunnelling junction. When
λ > 1 its conductance for the values of ξ not very close to 1 can be estimated in accordance with (15) as:
G ≈ e
2πλ(ξ−1) 1 − e−2πλ.
It decays exponentially with kFw0decreasing as seen in Fig. 5A where some numerical results are presented for a low-mode channel. Low modes dominate the contact conductance ifλ 1 but for smaller λ higher modes contribute significantly due to tunnelling through the quasi-potential barrier. If, e.g.λ = π one can see at Fig. 5A (2) that G ≈ 0.1 even when ξ is as small as 0.8.
Whenξ < 2 a particular case of a single-mode channel is realized. Nevertheless the higher modes may
as well contribute significantly to contact conductance by tunnelling through the quasi-potential barrier ifλ is not very large. This is well illustrated in Fig. 5A where the first transmission coefficient T1is shown in comparison with the whole conductance and with the contribution of higher modes.
In Fig. 5B the conductance of a contact with two propagating modes compared to the contribution of higher modes is shown for variousλ which demonstrate typical features for a channel with more propagating modes. Tunnelling modes contribute near the step pointsξ = m. If λ ≈ 2π their contribution is substantial far enough from the step point, i.e. at(ξ − m) ∼ ±1. For smaller values of λ the contribution of tunnelling modes, with n> m, to the channel conductance at (m − 1) < ξ < m is comparable with that of propagating modes, and steps are smeared markedly. Whenλ > 3π the conductance for ξ < m is determined mostly by Tnwith n < m, except for the small vicinity just near the mth step point and step smearing is small, which means that only propagating modes actually contribute to the conductance when parameterλ is large enough.
G / G0 G / G0 G / G0 G / G0 ∆ ∆G / G0 ∆G / G0 G / G0 G / G0 G / G0 G / G0 ∆ ∆G / G0 ∆G / G0 1. 2. 3. 1. 2. 3. T1 T1 T1 ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.65 1.7 1.75 1.80 1.85 1.90 1.95 2.00 1.86 1.88 1.90 1.92 1.94 1.96 1.98 2.00 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.5 1.0 1.5 2.0 1.2 1.4 1.6 1.8 2.0 0.0 0.5 1.0 1.5 2.0 2.2 2.4 2.6 2.8 3.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 2.90 2.92 2.94 2.96 2.98 3.00 2.70 2.75 2.80 2.852.90 2.95 3.00 0.25 0.50 0.75 1.00 1.25 1.50 0.1 0.2 0.3 0.4 0.5 0.5 1.0 1.5 2.0 2.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 A B
Fig. 5. A Conductance, G, of a contact with none and single propagating mode calculated numerically; also the transmittance T1for
the lowest mode and the contribution of higher(n ≥ 2) modes, 1G(2)are shown for comparison; G—solid line, T1—dashed line;
B conductance, G, of a contact for the two lowest propagating modes compared to the contribution of higher(n ≥ 3) modes, 1G(3); (1)λ = 1, (2) λ = π, (3) λ = 3π.
We also draw attention to the fact that at the step points(ξ = m) the tunnelling component contribute about 0.5G0.
Whenλ 1 for ξ < (m + 1) far enough from the (m + 1)th step their contribution can be estimated,
using (15), as: ∞ X n=m Tn(kF) ≈ e2πλ(ξ−m−1) 1 − e−2πλ .
With the help of (13), (15) one can also estimate the differential conductance of a long channel under some, not very high, bias, V , between the 2DEG reservoirs. There are reasons to believe that in a long narrow channel the electric potential varies monotonously along the x-axis (see also [28]). In that case the approach developed in [4, 21] can be applied using Tn given by eqns (13), (15) instead of the step-like transmission coefficient used in [4, 21]. Assuming that a bias is small enough so one can neglect the change of quasi-potential, we find for the current, I , through a channel
I = e π ¯h Z εF εF−eV T(E) d E, (17) g , 2e2 / h g , 2e2 / h A B 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 2 4 6 8 10 ξ ξ
Fig. 6. Low bias voltage dependence of differential conductance, g, of a long narrow channel with m propagating modes: A m = 2;
where: T(E) = ∞ X n=1 Tn(k), E = ¯ h2k2 2m . Differential conductance, g(V ) = d I/dV , then is:
g = e 2
π ¯hT(εF−eV). (18)
Bias dependence of the differential conductance of a narrow channel with only a few, m, propagating modes calculated for various values of λ is shown in Fig. 6. It decreases step-like with increasing bias at [4, 21]
V ≈ εF−εn(0)
e , n = m, m − 1, m − 2, . . . , 1 (19)
that is a particular feature of 1D flow. The channel reaches its ‘saturated’ regime when:
V > Vm = εF
−ε1(0)
e . (20)
Nevertheless the differential conductance never reaches zero due to higher modes tunnelling. The steps are quite sharp whenλ ≥ 3π, smear vastly for smaller λ, and disappear when λ ≤ 1.
We should emphasize that the above approach is valid for a long channel and for quite low bias. It means that the step-wise voltage dependence and saturation of differential conductance can be observed in a long channel with only a few propagating modes, when Vmis not very high. For higher bias and for a short channel the potential distribution along the channel should be considered more carefully. We found earlier that space-charge effects at the mouths of a channel became essential at higher bias [21, 22] and results in substantial non-zero differential conductivity at V > Vm. So some background is actually always present resulting in non-zero differential conductance at a bias V higher than Vm, and the higher is Vm, the more pronounced are the background effects. Voltage dependence of a conductance for a short channel was estimated in [18, 19, 21, 22] and the results also differ markedly from those given by formulae (18)–(20).
4. Summary
In this paper we consider the problem of electron transport through a constricted quantum channel in a 2DEG. We propose and analyse a convenient two-parametric model for such a channel, which allows for an analytical solution of the scattering problem in the adiabatic approximation. Conditions for observation of channel conductance quantum steps and step-like differential conductance dependence on bias voltage are expressed in terms of model parameters. We found that intrinsic smearing of the quantum steps is not crucial for even quite a short channel—almost up to the limit of adiabatic approximation. The model, by fitting of the parameters, can be used for a good variety of experimental structures, and thus enables one to estimate the quality of a channel and to analyse experimental data.
Acknowledgements—The work was supported by the National Science Council of Taiwan, ROC, under the grants: NSC89-2112-M009-049 and NSC89-2112-M009-051.
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