- Email: [email protected]
Where is the potential drop in a quantum point contact?
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
Where is the potential drop in a quantum point contact?∗ Stefan Ulreich, Wilhelm Zwerger Sektion Physik, Ludwig-Maximilians-Universit¨at, Theresienstr. 37, D-80333 M¨unchen, Germany (Received 29 September 1997) A scattering theory calculation is presented for the potential distribution and the associated current–voltage characteristic in a ballistic quantum point contact. Following Landauer’s idea, the electric field is calculated as the response to a given incoming current. The field is finite only near the entrance to or exit from the constriction, while no potential drop occurs in the device itself, where the motion is adiabatic. As a result dissipation is restricted to the reservoirs, whose detailed description, however, requires inelastic scattering. c 1998 Academic Press Limited
Key words: quantum point contacts, local field distributions, nonlinear ballistic transport.
1. Introduction Quantum point contacts have been investigated quite extensively since the remarkable discovery by van Wees et al. [1] and Wharam et al. [2], that they exhibit quantized longitudinal conductances [3]. The initially surprising fact that even a system with perfect transmission exhibits a finite resistance has been understood in terms of the inevitable contact resistance between the constriction and the reservoirs. Unfortunately a detailed microscopic calculation of the local potentials and the dissipation in such a ballistic system is still not available except for a qualitative discussion based on a simple kinetic equation with a finite relaxation time τ [4]. Now as far as the linear conductance is concerned, the latter can be calculated without knowing the precise local fields [3]. In order to go beyond linear response and also from a more general perspective, however, it is of considerable interest to calculate the local potentials in a real transport situation. This problem was first addressed in a seminal paper by Landauer [5]. Turning around the usual procedure where the current is determined as the response to a given electric field, Landauer formulated transport as a scattering problem in which the voltage appears as the response to a given incident current. Indeed such a formulation is quite natural from the experimental point of view since there one usually imposes an external current and measures the resulting potential drop due to scattering. For the case of localized impurities, Landauer showed that the current induces a dipolar field around each of the scatterers [5–7]. In a system with a finite concentration of randomly distributed impurities, an incoherent superposition of these so-called Landauer resistivity dipoles gives rise to a finite spatially averaged field hE(x)i. The residual resistivity % defined in response to hE(x)i then agrees precisely with that obtained from the standard linear response calculation [6, 8]. In this work Landauer’s basic idea of treating transport as a response to a given incident current is applied to the case of a ballistic quantum point contact. By a careful treatment of the purely geometric scattering at the boundaries we are able to calculate the field distribution near the constriction and the associated nonlinear current–voltage characteristic. It turns out that the electric field is nonzero only in a finite range between ∗
This work is dedicated to Rolf Landauer on the occasion of his 70th birthday.
0749–6036/98/030719 + 12 $25.00/0
sm970535
c 1998 Academic Press Limited
720
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
the source contact and the constriction and also between there and the drain contact. The field is due to backscattering, which occurs both at the constrictions entrance and exit. In contrast there is essentially no potential drop near the constriction itself, where the electronic motion is adiabatic. This result confirms previous qualitative discussions of the field distribution in a ballistic device [6, 9]. Such distributions are in strong contrast to the behaviour expected in a classical conductor, where the dissipation is local. Indeed in a classical conductor the electric field E(x) is directly proportional to the local current density j(x). Taking a quantum point contact of width b(x) and a fixed total current I = j (x)b(x) in the longitudinal x-direction, the resulting local field is E cl (x) = %I /b(x). The classical field is therefore strongest in the immediate vicinity of the constriction, where b(x) has a minimum. In contrast, in a ballistic system with a smooth boundary, the electric field essentially vanishes there as we will see below.
2. Scattering theory in waveguides of variable width As a first step in a fully quantum-mechanical transport theory, we need to solve the stationary single particle Schr¨odinger equation h¯ 2 2 (1) ∇ + V (x, y) 9(x, y) = E9(x, y) − 2M in a given confining potential V (x, y). For simplicity V (x, y) is assumed to be harmonic in the transverse coordinate, i.e. M 2 b0 2 2 ω0 y . (2) V (x, y) = 2 b(x) The function b(x) is a measure of the effective constriction width, which has a minimum b0 = b(x = 0) at the origin and takes a finite value b∞ = b(x → ±∞) > b0 at |x| larger than a characteristic constriction length L. The source and drain reservoirs are therefore replaced by perfect waveguides of finite constant width as in the standard derivation of the B¨uttiker–Landauer theory for the conductance coefficients [10, 11]. It is important to note that these waveguides are not real reservoirs even in the limit of infinitely many transverse channels, because there is no inelastic scattering and thus no equilibration there [6]. They allow us, however, to define a well-defined quantum scattering problem and thus to relate the linear conductances to corresponding transmission probabilities without explicitly taking into account the reservoirs [8]. As we will see, this procedure fails for the nonlinear transport regime, where a proper solution of the screening problem and inelastic scattering are required to determine the actual local field and the nonlinear IV-characteristic. Nevertheless, in a ballistic system, this approach allows us to determine qualitatively the field distribution in the device itself and the regime over which the linear relation between current and field is valid. In the asymptotic regions, where b(x) is constant, the Schr¨odinger equation (1) trivially separates into discrete transverse modes and a plane wave longitudinal motion exp(ikx) with a continuous wavevector k. The transverse modes n = 0, 1, 2, . . . have discrete energies εn = h¯ ω∞ (n + 12 ) and wavefunctions 8n (y). An electron incident from the left asymptotic region in mode n with wavevector kn > 0 then has a wavefunction whose asymptotic form is P ikn x −ikm x 8m (y) x → −∞ e 8n (y) + rmn e m (3) 9nk (x, y) = P tmn eikm x 8m (y) x → +∞. m
The wavevectors kn of the asymptotically propagating modes follow from the condition of fixed total energy E = h¯ 2 k 2 /2M + εn . The amplitudes rmn and tmn are related to the reflection and transmission probabilities Rmn and Tmn via km km |rmn |2 , Tmn = |tmn |2 (4) Rmn = kn kn
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
721
since the R’s and T ’s are defined via the ratio between the scattered and the incoming probability current. In order to calculate the exact scattering amplitudes and wavefunctions for a given width function b(x), we use a novel method which we have recently developed [12]. This method is based on expanding the wavefunction X χn (x)8nx (y) (5) 9(x, y) = n
in terms of the local transverse modes 8nx (y) with
x-dependent frequency ω(x) = ω0 b0 /b(x). In the adiabatic approximation, where b(x) changes slowly on scales of order λ F [13], the equation for the vector χ (x) = (χ1 (x), χ2 (x), . . .) of longitudinal wavefunctions then decouples into the separate eigenmodes n, since the matrix ε(x) in 2 pˆ χ (x) 1l + ε(x) χ (x) = Eχ (6) 2M is diagonal ε(x) ≈ εad (x) = diag (εn (x))
(7)
with εn (x) = h¯ ω(x)(n + 12 ) the local energy of the nth subband. Now it turns out [12] that even the full problem including arbitrary intersubband scattering can be written in the form (6). The energy matrix ε(x), however, is in general nondiagonal and given by ad ˆ (x) Sˆ + (x) ε(x) = S(x)ε
(8)
ˆ with a unitary operator S(x), which can be written as a local squeeze operator, known from quantum optics [14] b(x) 1 ˆ ln aˆ +2 − aˆ 2 (9) S(x) = exp 4 b∞ (aˆ + and aˆ are the standard oscillator creation and annihilation operators). For our specific calculation we use a width function [15] h π x i b0 b(x) =1− 1− ; |x| ≤ L (10) cos2 b∞ b∞ 2 L and solve the coupled equations (6) numerically via a discretization procedure. For this purpose it is useful to write (6) in the form of two coupled first-order equations ! d χ χ 1l 0 = (11) 2M χ0 χ0 0 ε(x) − E1l dx h¯ 2 for the vectors χ and χ 0 (the prime denotes differentiation with respect to the longitudinal coordinate x). With discrete steps of size a and χ (x = ia) = χ i , a numerically stable recursive solution of (11) is obtained from −1 + ˆ + ˆ 2 Sˆi+1 (12) Si χ i − Sˆi+1 Si−1 χ i−1 , χ i+1 = (1 + E)1l − εad i+1
where we have used dimensionless energies in units of h¯ 2 /2Ma 2 . This equation determines the explicit local wavefunctions in a general waveguide with arbitrary width function b(x). It requires knowledge of the reflection amplitudes rmn for the initial conditions χ 0 , χ 1 of the recursive relation. For the calculation of the transmission and reflection amplitudes, we use the method of recursive Green functions [16] which gives the asymptotic form (3) of the exact scattering states. This method is reliable even in cases with b0 b∞ , where many evanescent modes have to be included. For our numerical calculations below we have included up to 24 transverse channels.
722
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
3. Local potential distribution In order to determine the local potential from the exact single particle scattering states, we follow Landauer’s idea of calculating the induced potential for a given incident current from infinity. In practice this is done by assuming that the incoming momentum distribution is an equilibrium Fermi sphere which is shifted by a fixed wavevector k0 6= 0. With this approach, which was also used in [7] for the case of localized scatterers, the total electron density at zero temperature is then simply given by (the factor of 2 is due to spin degeneracy) k0 +kn X Z dk |9nk (x, y)|2 , n(x, y) = 2 2π n F
(13)
k0 −knF
where knF > 0 is determined by ε F = h¯ 2 (knF )2 /2M + εn . This density can be split into an equilibrium part n 0 (x, y) which is obtained from (13) at k0 = 0, and a current-induced perturbation δn(x, y) which is linear in k0 to lowest order. Now in equilibrium the electrochemical potential µ + e8(x, y) is constant and thus the spatially varying equilibrium density n 0 (x, y) does not give rise to a finite average field. Instead it is screened out self-consistently by the actual confining potential of the device, giving a constant interacting electron density and potential in the absence of current on scales larger than the screening length. In contrast, the current induced part δn(x, y) which is calculated from the noninteracting electron approximation, may be considered as a small external perturbation to the real interacting electron system. The spatially dependent part of the full potential δ8 is then related to δn by a linear response relation which—in two dimensions—has the form Z 2 (14) d x 0 dy 0 K (x y, x 0 y 0 ) δn(x 0 y 0 ). − e8(x, y) = 2πe The relevant kernel K depends on x y and x 0 y 0 separately due to the lack of translation invariance. It is only far from the constriction and boundaries, where the system is spatially homogeneous and the Fourier transform of K (x − x 0 , y − y 0 ) is then equal to [q2d (q)]−1 with 2d (q) the dielectric function in two dimensions. In order to avoid the complications associated with the nonlocal nature of (14) and a proper calculation of the kernel K (e.g. within a generalized RPA-approach) we restrict ourselves to variations of the potential on scales larger then the two-dimensional screening length rs = h¯ 2 /Me2 , which is about 10 nm for GaAs. On these length scales charge neutrality requires that the potential ∂µ δn(x, y) (15) − e8(x, y) ≈ ∂n eq is directly proportional to the external density modulation and thus 2π e2 K (x y, x 0 y 0 ) is effectively replaced by a δ-function with strength ∂µ/∂n. This quasineutrality approximation was previously used by Levinson [4] in the quantum point contact case and also for the potential variations around localized scatterers in [7]. The current-induced potential distribution is therefore directly reflected in the deviation δn of the noninteracting electron density (13) from its value at zero current. In particular in the limit k0 → 0 of linear response, the resulting electric field is determined by the scattering states right at the Fermi energy, in agreement with the more familiar Kubo formula for the linear conductivity or the standard Landauer–B¨uttiker results for the conductance coefficients (see also section 4 below). For the numerical calculation of the transport potential in a quantum point contact we use a constriction of the form (10) with b0 /b∞ = 0.5. The characteristic length L between the narrowest point x = 0 and the perfect waveguide at |x| > L is taken to be L = 250 nm. Finally we choose a Fermi energy such that one is near the end of the second plateau G = 2 · 2e2 / h in the quantized linear conductance. The result for the current induced density δn(x, y) in units of the equilibrium P density n is shown in Fig. 1 for the case k0 = 0.2k F , corresponding to a total incoming current I = 2eh¯ k0 n k nF /π M which is of the order of a few µA if typical parameters for GaAs are chosen.
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
723
y (nm) –100
0
100
0.003 0 Density –0.003 –0.006 –1500 –1000 –500
0
500
1000
x (nm)
1500
Fig. 1. Current-induced density in a quantum point contact with an incoming current from the left. The form of the width function b(x) is indicated below.
As is evident from the figure, the induced density is negative and essentially constant along the current direction in the vicinity of the constriction, say for |x| smaller than about 3 or 4 times the characteristic length |L|. Further away from that, the current-induced density in- or decreases roughly linearly up to the maximum distance |x|max = 8L to which we have extended the numerical calculations. The dependence in the transverse direction is smooth, with a tendency to weaken the built-up density variations near the boundary of the device. Physically these results may be understood in the following way. Part of the incoming current is reflected by quantum-mechanical backscattering. As a result of that there is an enhanced density in front of the ∇8 constriction whose roughly linear dependence on x is equivalent to a locally constant electric field E = −∇ via the simple proportionality (15). Remarkably this field extends far back into the regime of the incoming perfect waveguide. The enhancement of the density due to backscattering entails a corresponding depletion in the forward direction. This depletion is already present in the vicinity of the constriction itself, however since the current-induced density δn(x, y) is essentially flat there, the associated electric field vanishes. Thus, no potential drop appears near the narrowest point in striking contrast to a classical, diffusive conductor, where the voltage drop would be concentrated there. In the ballistic system studied here a second drop in the potential appears farther away from the constriction, where δn(x, y) decreases again roughly linearly with x. The physical origin of the associated field is the backscattering which arises from the change in width b(x) near the exit to the perfect waveguide. Similar to the behaviour near the entrance to the point contact, the regime where the field is nonzero extends far into the (quasi)reservoir. This demonstrates the strongly nonlocal nature of the relation between current and field in such a ballistic situation. Finally, the fact that |δn(x, y)| is smaller near the boundaries than in the centre of the waveguide, indicates that the field is concentrated near the centre. This is an example of a collimation effect [3], which becomes more important for larger incoming currents and smaller width ratios b0 /b∞ . An extreme case of the collimation effect is shown in Fig. 2, where the total electron density is plotted for a waveguide of the form given in (10) with b0 /b∞ = 0.1 and a large incoming current corresponding to k0 = 1.25 k F .
4. Linear conductance In order to determine the current–voltage characteristic within our method, it is necessary to evaluate the asymptotic difference 1n in density for a given incident current I and to translate 1n into a resulting total potential drop V . Using the general expression (13) for the total density and the asymptotic behaviour (3) of the exact scattering states, it is obvious that for any given I , the current induced asymptotic density difference
724
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 750
500
x (nm)
250
0
–250
–500
–200
–100
0
100
200
y (nm) Fig. 2.
Collimation of the electron density in a waveguide of width b(x) as indicated, with a large incoming current in the x-direction.
1n is completely determined by the energy-dependent transmission and reflection probabilities Tmn (k) and Rmn (k), provided that the spatially oscillating contributions in |ψnk |2 are averaged out. Simple analytical results may be obtained in the linear response limit k0 → 0 of a very small incident current 2eh¯ X F k0 kn . (16) I = πM n In this limit, the current-induced density at zero temperature is quite generally given by the exact scattering states at the Fermi energy in terms of k0 X |ψn,k F (x, y)|2 − |ψn,−k F (x, y)|2 . (17) δn(x, y) = π n Here ψn,k F or ψn,−k F are wavefunctions at the Fermi energy, which are incident in the nth subband from left or right towards the constriction. Using the asymptotic form (3) of the exact scattering states and integrating δn(x, y) over the transverse coordinate y, it is straightforward to evaluate the asymptotic difference in density # " X 2k0 X 2 2 |rmn | − |tmn | (18) 1+ 1n = π n m to linear order in the incident current. The corresponding reflection and transmission amplitudes are now those at the Fermi energy. Due to the orthogonality of the transverse wavefunctions 8n (y) the spatially oscillating terms in the asymptotic density vanish, except for the diagonal reflection terms proportional to rnn cos(2kn x), which—we assume—are averaged out. In a second step we need to translate the asymptotic density difference 1n into a resulting potential drop V in our wide–narrow–wide-type geometry. Now physically what happens is that the density 1n is screened out by moving charges such that the induced density δn ind essentially neutralizes the current-induced charge pile up. The screening requires delocalized electrons and thus for instance in the case of almost total reflection at the constriction the voltage drop V becomes very large although 1n remains finite. The relevant quasi-onedimensional thermodynamic density of states ∂n (1) /∂µ, which enters into (15) for translating 1n into V has
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
725
therefore only contributions from those states which are actually extended from one reservoir to the other. In our quasi one-dimensional situation this is given by 2M X0 1 ∂n (1) , (19) = ∂µ π h¯ 2 n knF P where the summation 0n is over the transverse modes n at the narrowest point only and we have assumed that at least one mode is open there (in the pinch-off regime the relevant thermodynamic derivative is proportional to the exponentially small transmission probability instead of being zero, however this limit will not be considered further here). Calculating the potential drop eV from the density difference (18), we obtain a linear relation V = G −1 I between induced voltage and the incoming current with a linear zero temperature conductance −1 P P 2 knF 0 knF 2 2e n . n (20) G= P h P 2 2 1 + (|rmn | − |tmn | ) n
m
This result is very similar to a multichannel generalization by B¨uttiker et al. [17] of the original Landauer result G L = 2e2 / h ·T /R for the conductance of a one-dimensional quantum wire with transmission probability T as calculated from the voltage difference between two ideal conductors bordering the sample. It differs slightly from the B¨uttiker et al. result since no transmission probabilities appear in the numerator because the relevant density of states of the extended modes in (19) was that of a ballistic constriction with unit transmission probability for the open channels. In a general, adiabatic wide–narrow–wide geometry with N N or N W open modes at the constriction or in the asymptotic regions, the conductance (20) is approximately given by 2e2 N W N N 2e2 N N −1 NN 1 − = . (21) GW NW ≈ h NW − N N h NW Such an expression was first derived by Landauer [18] with a slightly different prefactor of the N N /N W correction. It shows that the ideal waveguides mimic a true reservoir in the limit N W N N , where the conductance reduces to the well-known result G = 2e2 / h · N N of a perfect constriction with N N open modes between two real reservoirs. As emphasized by Landauer [18], the larger conductance entailed by the factor (1− N N /N W )−1 is a consequence of the fact, that the expression (20) does not contain the boundary resistance between the asymptotic perfect waveguides and the true reservoirs. It is one of the great virtues of B¨uttiker’s formulation [10] of the scattering approach to the conductance that the much simpler relation 2e2 (22) Tr tt + h between the conductance and the matrix t of the transmission amplitudes correctly describes the conductance measured from the voltage difference between points far inside the reservoirs which is the quantity most easily accessible in experiments. As a final point in our discussion of the regime of linear transport, we show that the appearance of quantized steps in the conductance of a quantum point contact can also be seen in a formulation with a given incident current. Calculating numerically the asymptotic current-induced density difference 1n for a given k0 = 0.01k F , we find clear plateaus in 1n as the constriction is made narrower (see Fig. 3). These plateaus arise from the sum over n in (18), which was replaced just by N W − N N in our crude approximation (21). Since −1 P in the conductance formula (20) exhibits sharp steps as a function of the constriction the factor 0 knF G=
n
width whenever an additional transverse mode is opened, the associated conductance is not a smooth function of b0 /b∞ and moreover—as pointed out above—it neglects the nonzero boundary resistance between the perfect waveguide and the reservoir. For a comparison we have therefore used the simple expression (22)
726
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 0.012
k0 = 0.01 kF
1n/n
0.008
0.004
0 0.4
0.5
0.6
0.7
0.8
0.9
1
b0/b∞ Fig. 3. Plateaus in the current-induced asymptotic density difference 1n as a function of the constriction width b0 /b∞ . The dotted line is the associated conductance, including the boundary resistance.
which includes the boundary resistance and thus gives a finite maximum conductance G max = N W · 2e2 / h for a waveguide of constant width b0 = b∞ , with N W being the number of open modes in the asymptotic regime. Since N W = 10 is still small in our example, the plateaus in the reservoir to reservoir conductance do not precisely coincide with those in the current-induced density, however it is clear that the steplike structure in 1n is closely connected to that of the conductance. It is also important to note that a study of the dependence of both 1n and G as a function of b0 /b∞ is directly relevant to the experimental situation, where the incoming current is fixed and the constriction width is changed by varying the gate voltage [1, 2].
5. Nonlinear current–voltage characteristic The method of prescribing the incident current I and calculating the induced voltage drop in response to that by straightforward scattering theory, can in principle be applied for arbitrary values of I , thus determining the nonlinear current–voltage characteristic. In practice, for the ballistic device considered here, where only the scattering at the boundary of the system is included, this approach is valid only locally on scales shorter than the mean free path for additional scattering processes of a different origin. Since there are no localized impurities near the constriction in a sufficiently clean system, only inelastic scattering processes are relevant for instance due to phonons [19]. These will occur with essentially equal probability at any point of the sample and thus will lead to a uniform field distribution for a given incident current. Now in order to really evaluate the electric field from the current-induced density δn, one needs the effective density of states as in (15). This density of states will be very large in the reservoirs and thus the field will eventually vanish there. A proper solution of the nonlocal screening problem (14), possibly extended to situations in which δn is not small enough for a linear relation between 8 and δn to be valid, is therefore necessary to quantitatively calculate the precise field distribution. Neither this nor the effects of inelastic background scattering are included in our present calculations. As a consequence, the restriction to pure boundary scattering in the vicinity of the
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
727
2
k0/kF
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
eV/εF Fig. 4. Normalized current–voltage characteristic of a quantum point contact with pure boundary scattering only.
constriction is not sufficient to calculate the actual value of the local electric field and thus also the nonlinear IV-characteristic. However provided that (a) background scattering does not introduce a particular dependence of the associated field on position or total current and (b) reservoirs are sufficiently well separated from the device itself our method is able to describe at least qualitatively the spatial dependence of the field as well as the current dependence of the total voltage drop. With these limitations in mind we have calculated the nonlinear IVcharacteristic by evaluating the asymptotic density difference 1n for a given current, which is specified by the shift k0 of the incident momentum distribution. In order to convert 1n to a corresponding voltage drop, we take ∂µ/∂n as that of a bulk two-dimensional electron gas, which implies the simple relation 1n eV . (23) = εF n The resulting normalized IV-characteristic is shown in Fig. 4 for a quantum point contact of the form (10) with b0 /b∞ = 0.3 and characteristic length L = 250 nm. The Fermi energy was chosen such that there are three open subbands at the narrowest point, i.e. the linear conductance is 6e2 / h. Evidently the relation between current and voltage is linear up to potential drops eV of the order of the Fermi energy ε F , beyond which the IV-curve bends back to smaller voltages with increasing current. In our purely elastic scattering calculation the backbending is a simple consequence of the fact that by increasing k0 , the average electron energy increases. Therefore the backscattering probability which gives rise to the density difference 1n decreases and the induced voltage drop eventually vanishes as k0 k F . Experimentally such a behaviour is not observed, instead the voltage increases monotonically with current up to voltage drops of almost 20ε F [20]. This behaviour may be explained by the strong deformation of the bare subband energies εn (x) with increasing voltage drop, provided inelastic processes lead to an essentially classical field distribution in which E(x) ∼ 1/b(x) [20]. As we have found above, such a distribution is not appropriate in a ballistic system. The
728
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
fact however, that such a model agrees with the experimental results at large voltage drops indicates that the inelastic background scattering neglected here becomes the dominant factor in the nonlinear IV-characteristic for voltage drops exceeding the Fermi energy. In the regime of small voltages, however, the linear behaviour observed up to eV ' ε F is in agreement with a purely boundary scattering calculation as performed here. Finally it should be pointed out that simple generalizations of the B¨uttiker–Landauer formula (22) to the nonlinear regime [21] are also unable to describe the full IV-characteristic observed experimentally. Indeed using the fact that the transmission probabilities are bounded by one, the current in such a formulation would saturate at a finite value 2e (24) Imax = N · ε F = N · 0.078 µA · ε F [meV] h where N is the number of subbands which contribute to transport. Taking into account the fact that electrons are reflected even over the barrier at a rapid drop of the local potential, one obtains a negative differential resistance [22]. In marked contrast with the experimentally observed behaviour, the current is therefore predicted to decrease with increasing voltage. The reason why the nonlinear generalizations of the Landauer formula fail to describe nonlinear transport in ballistic systems is probably related to the fact that they can only be applied in situations with a well-localized potential drop like in tunnel junctions. As discussed above, however, this is far from the actual situation in a ballistic device.
6. Conclusion In this paper, we have calculated the potential distribution in a ballistic quantum point contact, using Landauer’s basic idea of determining the field as the response to a given incident current. Including only the purely geometrical scattering at the boundary, we have found that the voltage will drop near the entrance to and exit from the point contact but not in the vicinity of the constriction, as would be the case for a classical conductor. Unfortunately our results are still of a rather qualitative nature because the quantitative calculation of the actual potential from the current-induced density δn requires a proper solution of the inhomogeneous screening problem. This problem is quite involved already for small values of δn, but even more so in a regime where δn and the potential are no longer linearly related. In addition we have completely neglected inelastic background scattering which is certainly important in the strongly nonlinear regime, where a simple theory for the IV-characteristic based on a classical field distribution apparently describes the experimental data rather well [20]. Comparing our results with earlier approaches to the problem, we mention that Levinson [4] has split the electronic density into two contributions δn ∞ and δn e . The former is obtained in principle by solving a quantum kinetic equation with a finite relaxation time, corresponding to a uniform background scattering. It is this component which is associated with the current flow and which is precisely equivalent to our current-induced contribution δn except for the fact that we do not include background scattering here and moreover, instead of fixing the asymptotic voltage difference, we impose a given incident current as suggested by Landauer. The second contribution δn e introduced by Levinson is the induced density in response to the perturbation δn ∞ . It is approximately equal to −δn ∞ , guaranteeing charge neutrality on sufficiently long length scales. Obviously this screening charge δn e does not induce any current flow [4]. This is evident in our formulation where δn e is merely the change in electron density required to effectively neutralize the current-induced density variations δn. Unfortunately no quantitative results for δn ∞ were given in [4] and thus no comparison is possible with our numerical results in Fig. 1, which only includes boundary but no background scattering. More recently the problem of the voltage drop in mesoscopic devices was reconsidered by McLennan et al. [23], using quantum kinetic theory. They calculated the local chemical potential δµ which is essentially equivalent to our δn and then determine the electrostatic potential by a linearized Thomas–Fermi screening. Similar to our present results, δµ turns out to be rather flat in the vicinity of the constriction [23], varying strongly only near the entrance to or exit from the point contact. This behaviour is smoothed out in the potential 8,
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
729
however the dominant potential drops are still outside the constriction itself and near the transition to the reservoirs. Now the precise potential distribution in their approach depends on how the complete device is split into a constriction and the reservoirs, which are modelled by semi-infinite conducting sheets at a given potential with a sharp boundary. In our present approach we have fixed the current instead and used perfect waveguides away from the constriction with no background scattering. As a result, both approaches cannot really calculate reliably the length scale over which the electric field will eventually go to zero in an actual device in which there is no sharp boundary between the point contact and the reservoirs. A quantitative theory for the local fields, which would require both the inclusion of inelastic scattering and a realistic reservoir model, is therefore still lacking. From an experimental point of view it would be very interesting to measure the local currents and fields via modern scanning probe techniques. In fact it has recently been possible [24] to verify the collimation effect in quantum point contacts created in a subsurface two-dimensional electron gas. With the rapid improvement in these techniques, Landauer’s ground-breaking ideas about local fields in quantum transport are therefore eventually accessible to experiments, as is demonstrated by the recent direct observation of residual resistivity dipoles [25]. Acknowledgements—It is a pleasure to acknowledge many very helpful discussions with David Wharam. This work was supported by the SFB 348 Nanometer-Halbleiterbauelemente.
References [1] }B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhouwen, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988). [2] }D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. C21, L209 (1988). [3] }C. W. J. Beenakker and H. van Houten, Solid State Physics 44, p. 1–228, ed. H. Ehrenreich and D. Turnbull, Academic Press, London (1991). [4] }Y. Levinson, Sov. Phys. JETP 68 1257 (1989) [5] }R. Landauer, IBM J. Res. Develop. 1, 223 (1957) [6] }R. Landauer, Z. Phys. B68, 217 (1987) [7] }W. Zwerger, L. B¨onig, and K. Sch¨onhammer, Phys. Rev. B43,6434 (1991) [8] }W. Zwerger in Quantum Transport and Dissipation, T. Dittrich et al., VCH-Wiley, Weinheim (1997). [9] }M. C. Payne, J. Phys. Condens. Matter 1, 4931 (1989) [10] }M. B¨uttiker, Phys. Rev. Lett. 57, 1761 (1986) [11] }H. Baranger and A. D. Stone, Phys. Rev. B40, 8169 (1989) [12] }S. Ulreich and W. Zwerger, Europhys. Lett., in print (1998). [13] }A. Yacoby and Y. Imry, Phys. Rev. B41, 5341 (1990) [14] }H. P. Yuen, Phys. Rev. A13, 2226 (1976) [15] }A. Brataas and K. Chao, Mod. Phys. Lett. 7, 1021 (1993) [16] }F. A. Maaø, I. V. Zozoulenko, and E. H. Hauge, Phys. Rev. B50, 17320 (1994) [17] }M. B¨uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B31, 6207 (1985) [18] }R. Landauer, J. Phys.: Condens. Matter 1, 8099 (1988). [19] }V. L. Gurevich, V. B. Pevzner, and K. Hess, Phys. Rev. B51, 5219 (1995). [20] }T. Heinzel, D. A. Wharam, F. M. de Aguiar, J. P. Kotthaus, G. B¨ohm, W. Klein, G. Tr¨ankle, and G. Weimann, Semicond. Sci. Tech. 9 1220 (1994). [21] }P. F. Bagwell and T. P. Orlando, Phys. Rev. B40, 1456 (1989). [22] }R. J. Brown, M. J. Kelly, M. Pepper, H. Ahmed, D. G. Hasko, D. C. Peacock, J. E. F. Frost, D. A. Ritchie, and G. A. C. Jones, J. Phys. Condens. Matter 1, 6285 (1989).
730
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
[23] }M. J. McLennan, Y. Lee, and S. Datta, Phys. Rev. B43, 13846 (1991). [24] }M.A. Eriksson et al., Appl. Phys. Lett. 69, 671 (1996). [25] }B. G. Briner, R. M. Feenstra, T. P. Chin, and J. M. Woodall, Phys. Rev. B54, R5283 (1996).
Where is the potential drop in a quantum point contact?∗ Stefan Ulreich, Wilhelm Zwerger Sektion Physik, Ludwig-Maximilians-Universit¨at, Theresienstr. 37, D-80333 M¨unchen, Germany (Received 29 September 1997) A scattering theory calculation is presented for the potential distribution and the associated current–voltage characteristic in a ballistic quantum point contact. Following Landauer’s idea, the electric field is calculated as the response to a given incoming current. The field is finite only near the entrance to or exit from the constriction, while no potential drop occurs in the device itself, where the motion is adiabatic. As a result dissipation is restricted to the reservoirs, whose detailed description, however, requires inelastic scattering. c 1998 Academic Press Limited
Key words: quantum point contacts, local field distributions, nonlinear ballistic transport.
1. Introduction Quantum point contacts have been investigated quite extensively since the remarkable discovery by van Wees et al. [1] and Wharam et al. [2], that they exhibit quantized longitudinal conductances [3]. The initially surprising fact that even a system with perfect transmission exhibits a finite resistance has been understood in terms of the inevitable contact resistance between the constriction and the reservoirs. Unfortunately a detailed microscopic calculation of the local potentials and the dissipation in such a ballistic system is still not available except for a qualitative discussion based on a simple kinetic equation with a finite relaxation time τ [4]. Now as far as the linear conductance is concerned, the latter can be calculated without knowing the precise local fields [3]. In order to go beyond linear response and also from a more general perspective, however, it is of considerable interest to calculate the local potentials in a real transport situation. This problem was first addressed in a seminal paper by Landauer [5]. Turning around the usual procedure where the current is determined as the response to a given electric field, Landauer formulated transport as a scattering problem in which the voltage appears as the response to a given incident current. Indeed such a formulation is quite natural from the experimental point of view since there one usually imposes an external current and measures the resulting potential drop due to scattering. For the case of localized impurities, Landauer showed that the current induces a dipolar field around each of the scatterers [5–7]. In a system with a finite concentration of randomly distributed impurities, an incoherent superposition of these so-called Landauer resistivity dipoles gives rise to a finite spatially averaged field hE(x)i. The residual resistivity % defined in response to hE(x)i then agrees precisely with that obtained from the standard linear response calculation [6, 8]. In this work Landauer’s basic idea of treating transport as a response to a given incident current is applied to the case of a ballistic quantum point contact. By a careful treatment of the purely geometric scattering at the boundaries we are able to calculate the field distribution near the constriction and the associated nonlinear current–voltage characteristic. It turns out that the electric field is nonzero only in a finite range between ∗
This work is dedicated to Rolf Landauer on the occasion of his 70th birthday.
0749–6036/98/030719 + 12 $25.00/0
sm970535
c 1998 Academic Press Limited
720
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
the source contact and the constriction and also between there and the drain contact. The field is due to backscattering, which occurs both at the constrictions entrance and exit. In contrast there is essentially no potential drop near the constriction itself, where the electronic motion is adiabatic. This result confirms previous qualitative discussions of the field distribution in a ballistic device [6, 9]. Such distributions are in strong contrast to the behaviour expected in a classical conductor, where the dissipation is local. Indeed in a classical conductor the electric field E(x) is directly proportional to the local current density j(x). Taking a quantum point contact of width b(x) and a fixed total current I = j (x)b(x) in the longitudinal x-direction, the resulting local field is E cl (x) = %I /b(x). The classical field is therefore strongest in the immediate vicinity of the constriction, where b(x) has a minimum. In contrast, in a ballistic system with a smooth boundary, the electric field essentially vanishes there as we will see below.
2. Scattering theory in waveguides of variable width As a first step in a fully quantum-mechanical transport theory, we need to solve the stationary single particle Schr¨odinger equation h¯ 2 2 (1) ∇ + V (x, y) 9(x, y) = E9(x, y) − 2M in a given confining potential V (x, y). For simplicity V (x, y) is assumed to be harmonic in the transverse coordinate, i.e. M 2 b0 2 2 ω0 y . (2) V (x, y) = 2 b(x) The function b(x) is a measure of the effective constriction width, which has a minimum b0 = b(x = 0) at the origin and takes a finite value b∞ = b(x → ±∞) > b0 at |x| larger than a characteristic constriction length L. The source and drain reservoirs are therefore replaced by perfect waveguides of finite constant width as in the standard derivation of the B¨uttiker–Landauer theory for the conductance coefficients [10, 11]. It is important to note that these waveguides are not real reservoirs even in the limit of infinitely many transverse channels, because there is no inelastic scattering and thus no equilibration there [6]. They allow us, however, to define a well-defined quantum scattering problem and thus to relate the linear conductances to corresponding transmission probabilities without explicitly taking into account the reservoirs [8]. As we will see, this procedure fails for the nonlinear transport regime, where a proper solution of the screening problem and inelastic scattering are required to determine the actual local field and the nonlinear IV-characteristic. Nevertheless, in a ballistic system, this approach allows us to determine qualitatively the field distribution in the device itself and the regime over which the linear relation between current and field is valid. In the asymptotic regions, where b(x) is constant, the Schr¨odinger equation (1) trivially separates into discrete transverse modes and a plane wave longitudinal motion exp(ikx) with a continuous wavevector k. The transverse modes n = 0, 1, 2, . . . have discrete energies εn = h¯ ω∞ (n + 12 ) and wavefunctions 8n (y). An electron incident from the left asymptotic region in mode n with wavevector kn > 0 then has a wavefunction whose asymptotic form is P ikn x −ikm x 8m (y) x → −∞ e 8n (y) + rmn e m (3) 9nk (x, y) = P tmn eikm x 8m (y) x → +∞. m
The wavevectors kn of the asymptotically propagating modes follow from the condition of fixed total energy E = h¯ 2 k 2 /2M + εn . The amplitudes rmn and tmn are related to the reflection and transmission probabilities Rmn and Tmn via km km |rmn |2 , Tmn = |tmn |2 (4) Rmn = kn kn
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
721
since the R’s and T ’s are defined via the ratio between the scattered and the incoming probability current. In order to calculate the exact scattering amplitudes and wavefunctions for a given width function b(x), we use a novel method which we have recently developed [12]. This method is based on expanding the wavefunction X χn (x)8nx (y) (5) 9(x, y) = n
in terms of the local transverse modes 8nx (y) with
x-dependent frequency ω(x) = ω0 b0 /b(x). In the adiabatic approximation, where b(x) changes slowly on scales of order λ F [13], the equation for the vector χ (x) = (χ1 (x), χ2 (x), . . .) of longitudinal wavefunctions then decouples into the separate eigenmodes n, since the matrix ε(x) in 2 pˆ χ (x) 1l + ε(x) χ (x) = Eχ (6) 2M is diagonal ε(x) ≈ εad (x) = diag (εn (x))
(7)
with εn (x) = h¯ ω(x)(n + 12 ) the local energy of the nth subband. Now it turns out [12] that even the full problem including arbitrary intersubband scattering can be written in the form (6). The energy matrix ε(x), however, is in general nondiagonal and given by ad ˆ (x) Sˆ + (x) ε(x) = S(x)ε
(8)
ˆ with a unitary operator S(x), which can be written as a local squeeze operator, known from quantum optics [14] b(x) 1 ˆ ln aˆ +2 − aˆ 2 (9) S(x) = exp 4 b∞ (aˆ + and aˆ are the standard oscillator creation and annihilation operators). For our specific calculation we use a width function [15] h π x i b0 b(x) =1− 1− ; |x| ≤ L (10) cos2 b∞ b∞ 2 L and solve the coupled equations (6) numerically via a discretization procedure. For this purpose it is useful to write (6) in the form of two coupled first-order equations ! d χ χ 1l 0 = (11) 2M χ0 χ0 0 ε(x) − E1l dx h¯ 2 for the vectors χ and χ 0 (the prime denotes differentiation with respect to the longitudinal coordinate x). With discrete steps of size a and χ (x = ia) = χ i , a numerically stable recursive solution of (11) is obtained from −1 + ˆ + ˆ 2 Sˆi+1 (12) Si χ i − Sˆi+1 Si−1 χ i−1 , χ i+1 = (1 + E)1l − εad i+1
where we have used dimensionless energies in units of h¯ 2 /2Ma 2 . This equation determines the explicit local wavefunctions in a general waveguide with arbitrary width function b(x). It requires knowledge of the reflection amplitudes rmn for the initial conditions χ 0 , χ 1 of the recursive relation. For the calculation of the transmission and reflection amplitudes, we use the method of recursive Green functions [16] which gives the asymptotic form (3) of the exact scattering states. This method is reliable even in cases with b0 b∞ , where many evanescent modes have to be included. For our numerical calculations below we have included up to 24 transverse channels.
722
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
3. Local potential distribution In order to determine the local potential from the exact single particle scattering states, we follow Landauer’s idea of calculating the induced potential for a given incident current from infinity. In practice this is done by assuming that the incoming momentum distribution is an equilibrium Fermi sphere which is shifted by a fixed wavevector k0 6= 0. With this approach, which was also used in [7] for the case of localized scatterers, the total electron density at zero temperature is then simply given by (the factor of 2 is due to spin degeneracy) k0 +kn X Z dk |9nk (x, y)|2 , n(x, y) = 2 2π n F
(13)
k0 −knF
where knF > 0 is determined by ε F = h¯ 2 (knF )2 /2M + εn . This density can be split into an equilibrium part n 0 (x, y) which is obtained from (13) at k0 = 0, and a current-induced perturbation δn(x, y) which is linear in k0 to lowest order. Now in equilibrium the electrochemical potential µ + e8(x, y) is constant and thus the spatially varying equilibrium density n 0 (x, y) does not give rise to a finite average field. Instead it is screened out self-consistently by the actual confining potential of the device, giving a constant interacting electron density and potential in the absence of current on scales larger than the screening length. In contrast, the current induced part δn(x, y) which is calculated from the noninteracting electron approximation, may be considered as a small external perturbation to the real interacting electron system. The spatially dependent part of the full potential δ8 is then related to δn by a linear response relation which—in two dimensions—has the form Z 2 (14) d x 0 dy 0 K (x y, x 0 y 0 ) δn(x 0 y 0 ). − e8(x, y) = 2πe The relevant kernel K depends on x y and x 0 y 0 separately due to the lack of translation invariance. It is only far from the constriction and boundaries, where the system is spatially homogeneous and the Fourier transform of K (x − x 0 , y − y 0 ) is then equal to [q2d (q)]−1 with 2d (q) the dielectric function in two dimensions. In order to avoid the complications associated with the nonlocal nature of (14) and a proper calculation of the kernel K (e.g. within a generalized RPA-approach) we restrict ourselves to variations of the potential on scales larger then the two-dimensional screening length rs = h¯ 2 /Me2 , which is about 10 nm for GaAs. On these length scales charge neutrality requires that the potential ∂µ δn(x, y) (15) − e8(x, y) ≈ ∂n eq is directly proportional to the external density modulation and thus 2π e2 K (x y, x 0 y 0 ) is effectively replaced by a δ-function with strength ∂µ/∂n. This quasineutrality approximation was previously used by Levinson [4] in the quantum point contact case and also for the potential variations around localized scatterers in [7]. The current-induced potential distribution is therefore directly reflected in the deviation δn of the noninteracting electron density (13) from its value at zero current. In particular in the limit k0 → 0 of linear response, the resulting electric field is determined by the scattering states right at the Fermi energy, in agreement with the more familiar Kubo formula for the linear conductivity or the standard Landauer–B¨uttiker results for the conductance coefficients (see also section 4 below). For the numerical calculation of the transport potential in a quantum point contact we use a constriction of the form (10) with b0 /b∞ = 0.5. The characteristic length L between the narrowest point x = 0 and the perfect waveguide at |x| > L is taken to be L = 250 nm. Finally we choose a Fermi energy such that one is near the end of the second plateau G = 2 · 2e2 / h in the quantized linear conductance. The result for the current induced density δn(x, y) in units of the equilibrium P density n is shown in Fig. 1 for the case k0 = 0.2k F , corresponding to a total incoming current I = 2eh¯ k0 n k nF /π M which is of the order of a few µA if typical parameters for GaAs are chosen.
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
723
y (nm) –100
0
100
0.003 0 Density –0.003 –0.006 –1500 –1000 –500
0
500
1000
x (nm)
1500
Fig. 1. Current-induced density in a quantum point contact with an incoming current from the left. The form of the width function b(x) is indicated below.
As is evident from the figure, the induced density is negative and essentially constant along the current direction in the vicinity of the constriction, say for |x| smaller than about 3 or 4 times the characteristic length |L|. Further away from that, the current-induced density in- or decreases roughly linearly up to the maximum distance |x|max = 8L to which we have extended the numerical calculations. The dependence in the transverse direction is smooth, with a tendency to weaken the built-up density variations near the boundary of the device. Physically these results may be understood in the following way. Part of the incoming current is reflected by quantum-mechanical backscattering. As a result of that there is an enhanced density in front of the ∇8 constriction whose roughly linear dependence on x is equivalent to a locally constant electric field E = −∇ via the simple proportionality (15). Remarkably this field extends far back into the regime of the incoming perfect waveguide. The enhancement of the density due to backscattering entails a corresponding depletion in the forward direction. This depletion is already present in the vicinity of the constriction itself, however since the current-induced density δn(x, y) is essentially flat there, the associated electric field vanishes. Thus, no potential drop appears near the narrowest point in striking contrast to a classical, diffusive conductor, where the voltage drop would be concentrated there. In the ballistic system studied here a second drop in the potential appears farther away from the constriction, where δn(x, y) decreases again roughly linearly with x. The physical origin of the associated field is the backscattering which arises from the change in width b(x) near the exit to the perfect waveguide. Similar to the behaviour near the entrance to the point contact, the regime where the field is nonzero extends far into the (quasi)reservoir. This demonstrates the strongly nonlocal nature of the relation between current and field in such a ballistic situation. Finally, the fact that |δn(x, y)| is smaller near the boundaries than in the centre of the waveguide, indicates that the field is concentrated near the centre. This is an example of a collimation effect [3], which becomes more important for larger incoming currents and smaller width ratios b0 /b∞ . An extreme case of the collimation effect is shown in Fig. 2, where the total electron density is plotted for a waveguide of the form given in (10) with b0 /b∞ = 0.1 and a large incoming current corresponding to k0 = 1.25 k F .
4. Linear conductance In order to determine the current–voltage characteristic within our method, it is necessary to evaluate the asymptotic difference 1n in density for a given incident current I and to translate 1n into a resulting total potential drop V . Using the general expression (13) for the total density and the asymptotic behaviour (3) of the exact scattering states, it is obvious that for any given I , the current induced asymptotic density difference
724
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 750
500
x (nm)
250
0
–250
–500
–200
–100
0
100
200
y (nm) Fig. 2.
Collimation of the electron density in a waveguide of width b(x) as indicated, with a large incoming current in the x-direction.
1n is completely determined by the energy-dependent transmission and reflection probabilities Tmn (k) and Rmn (k), provided that the spatially oscillating contributions in |ψnk |2 are averaged out. Simple analytical results may be obtained in the linear response limit k0 → 0 of a very small incident current 2eh¯ X F k0 kn . (16) I = πM n In this limit, the current-induced density at zero temperature is quite generally given by the exact scattering states at the Fermi energy in terms of k0 X |ψn,k F (x, y)|2 − |ψn,−k F (x, y)|2 . (17) δn(x, y) = π n Here ψn,k F or ψn,−k F are wavefunctions at the Fermi energy, which are incident in the nth subband from left or right towards the constriction. Using the asymptotic form (3) of the exact scattering states and integrating δn(x, y) over the transverse coordinate y, it is straightforward to evaluate the asymptotic difference in density # " X 2k0 X 2 2 |rmn | − |tmn | (18) 1+ 1n = π n m to linear order in the incident current. The corresponding reflection and transmission amplitudes are now those at the Fermi energy. Due to the orthogonality of the transverse wavefunctions 8n (y) the spatially oscillating terms in the asymptotic density vanish, except for the diagonal reflection terms proportional to rnn cos(2kn x), which—we assume—are averaged out. In a second step we need to translate the asymptotic density difference 1n into a resulting potential drop V in our wide–narrow–wide-type geometry. Now physically what happens is that the density 1n is screened out by moving charges such that the induced density δn ind essentially neutralizes the current-induced charge pile up. The screening requires delocalized electrons and thus for instance in the case of almost total reflection at the constriction the voltage drop V becomes very large although 1n remains finite. The relevant quasi-onedimensional thermodynamic density of states ∂n (1) /∂µ, which enters into (15) for translating 1n into V has
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
725
therefore only contributions from those states which are actually extended from one reservoir to the other. In our quasi one-dimensional situation this is given by 2M X0 1 ∂n (1) , (19) = ∂µ π h¯ 2 n knF P where the summation 0n is over the transverse modes n at the narrowest point only and we have assumed that at least one mode is open there (in the pinch-off regime the relevant thermodynamic derivative is proportional to the exponentially small transmission probability instead of being zero, however this limit will not be considered further here). Calculating the potential drop eV from the density difference (18), we obtain a linear relation V = G −1 I between induced voltage and the incoming current with a linear zero temperature conductance −1 P P 2 knF 0 knF 2 2e n . n (20) G= P h P 2 2 1 + (|rmn | − |tmn | ) n
m
This result is very similar to a multichannel generalization by B¨uttiker et al. [17] of the original Landauer result G L = 2e2 / h ·T /R for the conductance of a one-dimensional quantum wire with transmission probability T as calculated from the voltage difference between two ideal conductors bordering the sample. It differs slightly from the B¨uttiker et al. result since no transmission probabilities appear in the numerator because the relevant density of states of the extended modes in (19) was that of a ballistic constriction with unit transmission probability for the open channels. In a general, adiabatic wide–narrow–wide geometry with N N or N W open modes at the constriction or in the asymptotic regions, the conductance (20) is approximately given by 2e2 N W N N 2e2 N N −1 NN 1 − = . (21) GW NW ≈ h NW − N N h NW Such an expression was first derived by Landauer [18] with a slightly different prefactor of the N N /N W correction. It shows that the ideal waveguides mimic a true reservoir in the limit N W N N , where the conductance reduces to the well-known result G = 2e2 / h · N N of a perfect constriction with N N open modes between two real reservoirs. As emphasized by Landauer [18], the larger conductance entailed by the factor (1− N N /N W )−1 is a consequence of the fact, that the expression (20) does not contain the boundary resistance between the asymptotic perfect waveguides and the true reservoirs. It is one of the great virtues of B¨uttiker’s formulation [10] of the scattering approach to the conductance that the much simpler relation 2e2 (22) Tr tt + h between the conductance and the matrix t of the transmission amplitudes correctly describes the conductance measured from the voltage difference between points far inside the reservoirs which is the quantity most easily accessible in experiments. As a final point in our discussion of the regime of linear transport, we show that the appearance of quantized steps in the conductance of a quantum point contact can also be seen in a formulation with a given incident current. Calculating numerically the asymptotic current-induced density difference 1n for a given k0 = 0.01k F , we find clear plateaus in 1n as the constriction is made narrower (see Fig. 3). These plateaus arise from the sum over n in (18), which was replaced just by N W − N N in our crude approximation (21). Since −1 P in the conductance formula (20) exhibits sharp steps as a function of the constriction the factor 0 knF G=
n
width whenever an additional transverse mode is opened, the associated conductance is not a smooth function of b0 /b∞ and moreover—as pointed out above—it neglects the nonzero boundary resistance between the perfect waveguide and the reservoir. For a comparison we have therefore used the simple expression (22)
726
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 0.012
k0 = 0.01 kF
1n/n
0.008
0.004
0 0.4
0.5
0.6
0.7
0.8
0.9
1
b0/b∞ Fig. 3. Plateaus in the current-induced asymptotic density difference 1n as a function of the constriction width b0 /b∞ . The dotted line is the associated conductance, including the boundary resistance.
which includes the boundary resistance and thus gives a finite maximum conductance G max = N W · 2e2 / h for a waveguide of constant width b0 = b∞ , with N W being the number of open modes in the asymptotic regime. Since N W = 10 is still small in our example, the plateaus in the reservoir to reservoir conductance do not precisely coincide with those in the current-induced density, however it is clear that the steplike structure in 1n is closely connected to that of the conductance. It is also important to note that a study of the dependence of both 1n and G as a function of b0 /b∞ is directly relevant to the experimental situation, where the incoming current is fixed and the constriction width is changed by varying the gate voltage [1, 2].
5. Nonlinear current–voltage characteristic The method of prescribing the incident current I and calculating the induced voltage drop in response to that by straightforward scattering theory, can in principle be applied for arbitrary values of I , thus determining the nonlinear current–voltage characteristic. In practice, for the ballistic device considered here, where only the scattering at the boundary of the system is included, this approach is valid only locally on scales shorter than the mean free path for additional scattering processes of a different origin. Since there are no localized impurities near the constriction in a sufficiently clean system, only inelastic scattering processes are relevant for instance due to phonons [19]. These will occur with essentially equal probability at any point of the sample and thus will lead to a uniform field distribution for a given incident current. Now in order to really evaluate the electric field from the current-induced density δn, one needs the effective density of states as in (15). This density of states will be very large in the reservoirs and thus the field will eventually vanish there. A proper solution of the nonlocal screening problem (14), possibly extended to situations in which δn is not small enough for a linear relation between 8 and δn to be valid, is therefore necessary to quantitatively calculate the precise field distribution. Neither this nor the effects of inelastic background scattering are included in our present calculations. As a consequence, the restriction to pure boundary scattering in the vicinity of the
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
727
2
k0/kF
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
eV/εF Fig. 4. Normalized current–voltage characteristic of a quantum point contact with pure boundary scattering only.
constriction is not sufficient to calculate the actual value of the local electric field and thus also the nonlinear IV-characteristic. However provided that (a) background scattering does not introduce a particular dependence of the associated field on position or total current and (b) reservoirs are sufficiently well separated from the device itself our method is able to describe at least qualitatively the spatial dependence of the field as well as the current dependence of the total voltage drop. With these limitations in mind we have calculated the nonlinear IVcharacteristic by evaluating the asymptotic density difference 1n for a given current, which is specified by the shift k0 of the incident momentum distribution. In order to convert 1n to a corresponding voltage drop, we take ∂µ/∂n as that of a bulk two-dimensional electron gas, which implies the simple relation 1n eV . (23) = εF n The resulting normalized IV-characteristic is shown in Fig. 4 for a quantum point contact of the form (10) with b0 /b∞ = 0.3 and characteristic length L = 250 nm. The Fermi energy was chosen such that there are three open subbands at the narrowest point, i.e. the linear conductance is 6e2 / h. Evidently the relation between current and voltage is linear up to potential drops eV of the order of the Fermi energy ε F , beyond which the IV-curve bends back to smaller voltages with increasing current. In our purely elastic scattering calculation the backbending is a simple consequence of the fact that by increasing k0 , the average electron energy increases. Therefore the backscattering probability which gives rise to the density difference 1n decreases and the induced voltage drop eventually vanishes as k0 k F . Experimentally such a behaviour is not observed, instead the voltage increases monotonically with current up to voltage drops of almost 20ε F [20]. This behaviour may be explained by the strong deformation of the bare subband energies εn (x) with increasing voltage drop, provided inelastic processes lead to an essentially classical field distribution in which E(x) ∼ 1/b(x) [20]. As we have found above, such a distribution is not appropriate in a ballistic system. The
728
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
fact however, that such a model agrees with the experimental results at large voltage drops indicates that the inelastic background scattering neglected here becomes the dominant factor in the nonlinear IV-characteristic for voltage drops exceeding the Fermi energy. In the regime of small voltages, however, the linear behaviour observed up to eV ' ε F is in agreement with a purely boundary scattering calculation as performed here. Finally it should be pointed out that simple generalizations of the B¨uttiker–Landauer formula (22) to the nonlinear regime [21] are also unable to describe the full IV-characteristic observed experimentally. Indeed using the fact that the transmission probabilities are bounded by one, the current in such a formulation would saturate at a finite value 2e (24) Imax = N · ε F = N · 0.078 µA · ε F [meV] h where N is the number of subbands which contribute to transport. Taking into account the fact that electrons are reflected even over the barrier at a rapid drop of the local potential, one obtains a negative differential resistance [22]. In marked contrast with the experimentally observed behaviour, the current is therefore predicted to decrease with increasing voltage. The reason why the nonlinear generalizations of the Landauer formula fail to describe nonlinear transport in ballistic systems is probably related to the fact that they can only be applied in situations with a well-localized potential drop like in tunnel junctions. As discussed above, however, this is far from the actual situation in a ballistic device.
6. Conclusion In this paper, we have calculated the potential distribution in a ballistic quantum point contact, using Landauer’s basic idea of determining the field as the response to a given incident current. Including only the purely geometrical scattering at the boundary, we have found that the voltage will drop near the entrance to and exit from the point contact but not in the vicinity of the constriction, as would be the case for a classical conductor. Unfortunately our results are still of a rather qualitative nature because the quantitative calculation of the actual potential from the current-induced density δn requires a proper solution of the inhomogeneous screening problem. This problem is quite involved already for small values of δn, but even more so in a regime where δn and the potential are no longer linearly related. In addition we have completely neglected inelastic background scattering which is certainly important in the strongly nonlinear regime, where a simple theory for the IV-characteristic based on a classical field distribution apparently describes the experimental data rather well [20]. Comparing our results with earlier approaches to the problem, we mention that Levinson [4] has split the electronic density into two contributions δn ∞ and δn e . The former is obtained in principle by solving a quantum kinetic equation with a finite relaxation time, corresponding to a uniform background scattering. It is this component which is associated with the current flow and which is precisely equivalent to our current-induced contribution δn except for the fact that we do not include background scattering here and moreover, instead of fixing the asymptotic voltage difference, we impose a given incident current as suggested by Landauer. The second contribution δn e introduced by Levinson is the induced density in response to the perturbation δn ∞ . It is approximately equal to −δn ∞ , guaranteeing charge neutrality on sufficiently long length scales. Obviously this screening charge δn e does not induce any current flow [4]. This is evident in our formulation where δn e is merely the change in electron density required to effectively neutralize the current-induced density variations δn. Unfortunately no quantitative results for δn ∞ were given in [4] and thus no comparison is possible with our numerical results in Fig. 1, which only includes boundary but no background scattering. More recently the problem of the voltage drop in mesoscopic devices was reconsidered by McLennan et al. [23], using quantum kinetic theory. They calculated the local chemical potential δµ which is essentially equivalent to our δn and then determine the electrostatic potential by a linearized Thomas–Fermi screening. Similar to our present results, δµ turns out to be rather flat in the vicinity of the constriction [23], varying strongly only near the entrance to or exit from the point contact. This behaviour is smoothed out in the potential 8,
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
729
however the dominant potential drops are still outside the constriction itself and near the transition to the reservoirs. Now the precise potential distribution in their approach depends on how the complete device is split into a constriction and the reservoirs, which are modelled by semi-infinite conducting sheets at a given potential with a sharp boundary. In our present approach we have fixed the current instead and used perfect waveguides away from the constriction with no background scattering. As a result, both approaches cannot really calculate reliably the length scale over which the electric field will eventually go to zero in an actual device in which there is no sharp boundary between the point contact and the reservoirs. A quantitative theory for the local fields, which would require both the inclusion of inelastic scattering and a realistic reservoir model, is therefore still lacking. From an experimental point of view it would be very interesting to measure the local currents and fields via modern scanning probe techniques. In fact it has recently been possible [24] to verify the collimation effect in quantum point contacts created in a subsurface two-dimensional electron gas. With the rapid improvement in these techniques, Landauer’s ground-breaking ideas about local fields in quantum transport are therefore eventually accessible to experiments, as is demonstrated by the recent direct observation of residual resistivity dipoles [25]. Acknowledgements—It is a pleasure to acknowledge many very helpful discussions with David Wharam. This work was supported by the SFB 348 Nanometer-Halbleiterbauelemente.
References [1] }B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhouwen, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988). [2] }D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. C21, L209 (1988). [3] }C. W. J. Beenakker and H. van Houten, Solid State Physics 44, p. 1–228, ed. H. Ehrenreich and D. Turnbull, Academic Press, London (1991). [4] }Y. Levinson, Sov. Phys. JETP 68 1257 (1989) [5] }R. Landauer, IBM J. Res. Develop. 1, 223 (1957) [6] }R. Landauer, Z. Phys. B68, 217 (1987) [7] }W. Zwerger, L. B¨onig, and K. Sch¨onhammer, Phys. Rev. B43,6434 (1991) [8] }W. Zwerger in Quantum Transport and Dissipation, T. Dittrich et al., VCH-Wiley, Weinheim (1997). [9] }M. C. Payne, J. Phys. Condens. Matter 1, 4931 (1989) [10] }M. B¨uttiker, Phys. Rev. Lett. 57, 1761 (1986) [11] }H. Baranger and A. D. Stone, Phys. Rev. B40, 8169 (1989) [12] }S. Ulreich and W. Zwerger, Europhys. Lett., in print (1998). [13] }A. Yacoby and Y. Imry, Phys. Rev. B41, 5341 (1990) [14] }H. P. Yuen, Phys. Rev. A13, 2226 (1976) [15] }A. Brataas and K. Chao, Mod. Phys. Lett. 7, 1021 (1993) [16] }F. A. Maaø, I. V. Zozoulenko, and E. H. Hauge, Phys. Rev. B50, 17320 (1994) [17] }M. B¨uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B31, 6207 (1985) [18] }R. Landauer, J. Phys.: Condens. Matter 1, 8099 (1988). [19] }V. L. Gurevich, V. B. Pevzner, and K. Hess, Phys. Rev. B51, 5219 (1995). [20] }T. Heinzel, D. A. Wharam, F. M. de Aguiar, J. P. Kotthaus, G. B¨ohm, W. Klein, G. Tr¨ankle, and G. Weimann, Semicond. Sci. Tech. 9 1220 (1994). [21] }P. F. Bagwell and T. P. Orlando, Phys. Rev. B40, 1456 (1989). [22] }R. J. Brown, M. J. Kelly, M. Pepper, H. Ahmed, D. G. Hasko, D. C. Peacock, J. E. F. Frost, D. A. Ritchie, and G. A. C. Jones, J. Phys. Condens. Matter 1, 6285 (1989).
730
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
[23] }M. J. McLennan, Y. Lee, and S. Datta, Phys. Rev. B43, 13846 (1991). [24] }M.A. Eriksson et al., Appl. Phys. Lett. 69, 671 (1996). [25] }B. G. Briner, R. M. Feenstra, T. P. Chin, and J. M. Woodall, Phys. Rev. B54, R5283 (1996).