Potential and Current Distributions Calculated Across a Quantum Hall Effect Sample at Low and High Currents.

The potential and current distributions are calculated across the width of a quantum Hall effect sample for applied currents between 0 μA and 225 μA. For the first time, both a confining potential and a current-induced charge-redistribution potential are used. The confining potential has a parabolic shape, and the charge-redistribution potential is logarithmic. The solution for the sum of the two types of potentials is unique at each current, with no free parameters. For example, the charge-depletion width of the confining potential is determined from a localization experiment by Choi, Tsui, and Alavi, and the spatial extent of the conducting two-dimensional electron gas across the sample width is obtained from the maximum electric field deduced from a high-current breakdown experiment by Cage and Lavine, and from the quantum Hall voltage. The spatial extent has realistic cut-off values at the sample sides; e.g., no current flows within 55 magnetic lengths of the sides for currents less than 215 μA. The calculated potential distributions are in excellent agreement with contactless electro-optic effect laser beam measurements of Fontein et al.

1. Introduction

The potential and current distributions within quantum Hall samples are important aspects of the integer quantum Hall effect [1-3]. In this effect the Hall resistance, RH, of the ith plateau of a fully quantized two-dimensional electron gas (2DEG) has the value RH(i)=h/(e2i), where h is the Planck constant, e is the elementary charge, and i is an integer. Early attempts to measure potential distributions across samples [4-7] used electrical contacts to the two-dimensional gas that were placed within the sample interior. The potentials were found to vary throughout the entire sample. There was concern, however, that the electrical contacts themselves significantly altered the potential distributions. Fontein et al. [8] have made contactless measurements of potential distributions using a laser beam and the electro-optic Pockels effect. They observed major fractions of the quantum Hall voltage occurring near the sides of the sample, but also significant contributions within the interior. Valid predictions of the potential distribution across quantum Hall samples should agree with their results.

In this paper we calculate the potential distributions across the sample for applied currents ISD between 0 μA and 225 μA by: (a) assuming a parabolic confining potential for the charge carriers and using parameters of the parabola obtained experimentally by Choi, Tsui, and Alavi [9]; (b) assuming an applied current-induced logarithmic charge-redistribution potential for the charge carriers that is similar to that of Balaban, Meirav, and Shtrikman [10], but with very different cut-off values for the spatial extent of the potential; (c) assuming that the width of the conducting region varies with applied current because a Lorentz force deflects the conducting electrons slightly towards one side of the sample; (d) using the maximum electric field deduced by Cage and Lavine [11] from a breakdown experiment at high currents to determine the cut-off value for the spatial extent on one side of the sample; and (e) using the quantum Hall voltage to determine the cut-off value of the spatial extent on the other side of the sample. The calculated potential distributions are in excellent agreement with the measurements of Fontein et al. [8].

2. Coordinate System

The coordinate system is shown in Fig. 1. For convenience in writing the equations, the origin is located at the source S and is halfway across the sample width w. The sample labeling is chosen to be consistent with previous work [11]. Potential probes 1 and 2, and the drain D, are not shown. The positive x axis points along the sample in the general direction of the externally applied current ISD. The positive y axis is chosen as indicated. Therefore the positive z axis points downward for a right-handed coordinate system. The magnetic field B also points downward, simply to be consistent with results from the breakdown experiment [11] that will be used in Secs. 4 and 5.

Fig. 1

The coordinate system used in this paper. Only part of the sample is shown. The origin is located at the source S, halfway across the sample width w. The dotted curves indicate the electron flow pattern for this magnetic field direction. FL is the Lorentz force on the conducting electrons and FC is the Coulomb force. B is the magnetic field, v is the electron velocity, and VA and VB are the potentials on either side of the sample.

Note that the conducting charges are electrons with charge q = − e. This is taken into account throughout the paper; it is necessary to do so because the signs of both the confining potential and the charge-redistribution potential depend on the sign of the charge carriers.

The Lorentz force L = e × is in the positive y direction. This force deflects the electrons slightly to the right until it is matched by the Coulomb repulsive force C = − e [12]. A charge-redistribution of the 2DEG results from this deflection. Also because of the Lorentz and Coulomb forces, the electrons enter the corner of the source at y = − w/2 for this magnetic field direction and exit at the corner +w/2 of the drain—in agreement with the experiment of Klass et al. [13]. We assume that the electrons spread out across the sample interior in agreement with the experiment of Fontein et al. [8]. Potential probes 4 and 6 are near the potential of the source. Probes 3 and 5 are near the potential of the drain, and have a positive potential relative to the source for these current and magnetic field directions. The chemical potential ϕA = VA is therefore positive relative to the chemical potential ϕB = VB on the opposite side of the sample.

3. Confining Potential

We begin the calculations with a confining potential to prevent the 2DEG from spilling out the sides of samples. Choi, Tsui, and Alavi [9] performed an experiment on mesa-etched GaAs/AlGaAs heterostructure samples in zero magnetic field. They then used one-dimensional localization theory to deduce the charge-depletion widths, , of the confining potentials, and found that was (0.5 ± 0.2) μm for a 2DEG of surface number density ns = i(eB/h) = 1.5 × 1011/cm2. We will use their results to define the depletion width of the confining potential for a mesa-etched sample.

3.1 Charge-Depletion Region

Figure 2(a) shows a schematic of the charge distribution in the GaAs/AlGaAs interface region near one side of the mesa when there is no applied magnetic field. The GaAs layer of our sample [11] has a residual donor density of about 1 × 1014/cm3, while the donor concentration in the AlGaAs layer is about 1 × 1018/cm3 and ns = 5.94 × 1011/cm2. There is an ionized donor atom in the AlGaAs layer for every electron in the 2DEG but, unlike Choi, Tsui, and Alavi [9], we assume the ionized donor atoms are distributed over a volume rather than in a surface sheet with density ns. The confining potential is generated from electron surface charges on the side of the mesa, as indicated in the figure. There is an ionized donor atom or ionized impurity site in the charge-depletion region for every surface charge.

Fig. 2

(a) Schematic diagram of the GaAs/AlGaAs interface region near one side of the mesa. See Sec. 3.1 for further explanation. (b) The ionized donor charge-depletion density distribution ND. (c) The confining potential Vc for negatively charged surface states.

We assume a homogeneous charge-depletion region in Fig. 2(b). The depletion width for a homogeneous three-dimensional material is [14] where s = 0 is the dielectric permittivity of the semiconductor, is the dielectric constant ( = 13.1 for GaAs), 0 is the permittivity of vacuum, Vm is the value of the confining potential at ±w/2 and ND is the average density of ionized donors and impurity sites in the charge-depletion region. We selected the value of the charge-depletion width to be = 0.5 μm [15]. This value is consistent with the results of Choi, Tsui, and Alavi [9]. We chose the value of Vm to be one-half the 1.50 V separation between the valence and conduction bands of GaAs at 1 K [14], or Vm = 0.75 V. The value of the average charge-depletion density from Eq. (1) is thus ND = 4.3 × 1015/cm3, which seems quite reasonable.

3.2 Confining Potential Equation

A homogeneous charge-depletion region results in a parabolic confining potential Vc, with the origin at y = λ = w/2−, as indicated schematically in Fig. 2(c). The confining potential is negative because the charges on the side of the mesa are electrons.

The equations for the confining potential Vc and its electric field Ec = − Vc are where a = Vm/2 = 3.0 × 1012 V/m2 for = 0.5 μm and Vm = 0.75 V, and

3.3 Confining Potential at ISD = 0 μA

Given the values of and Vm, there is a surprising amount that can be deduced about the electron states of the confining potential when the magnetic field is adjusted to be at the center of the i = 2 quantum Hall plateau and ISD = 0 μA. Since there is no applied current, and therefore no Hall voltage, the Fermi energy F is constant across the sample width and is located halfway between Landau levels. Under these conditions, states of the lowest (N = 0) Landau level are occupied up to the Fermi energy F = ħωc/2, no states are occupied in the second (N = 1) Landau level, ωc = eB/m* is the cyclotron angular frequency, m* is the reduced mass of the electron (0.068 times the free electron mass in GaAs), and ħ ≡ h/2π. References [11,16-18] describe how these states can be defined in the Landau gauge.

Figure 3 shows a schematic drawing of the energy of the confining potential for ISD = 0 μA, with greatly exaggerated values of and ħωc, and only a small fraction of allowed states. The occupied/unoccupied states are indicated as solid/open circles, and the occupied (filled) states lie between ymax and ymin = − ymax. In the presence of the magnetic field, electrons of the 2DEG occupy Landau level states that penetrate into the charge-depletion regions near the mesa edge, and current circulates around the sample periphery. Under these conditions where λ = w/2 − Δ.

Fig. 3

Schematic drawing of the energy of the confining potential Vc across the sample when ISD = 0 μA. Values of the charge-depletion width Δ and the Landau energy level spacing ħωc are greatly exaggerated. The occupied/unoccupied states of the first two Landau levels are shown as solid/open circles. The occupied (filled) states lie between the locations ymax = − ymin.

The occupied states of the right-hand side (rhs) confining potential generate a total current Ic (rhs) that is where Jc(y) is the current density, σ is the off-diagonal conductivity tensor component, Vc(ymax) = − a(ymax − λ)2, and Vc(λ) = 0. In the absence of significant dissipative scattering on the quantum Hall plateau, σxy = 1/RH [12]. Similarly, where Vc(ymin) = − a(ymin +λ)2.

It follows from Eqs. (2) to (6) for the 12 906.4 Ω, i = 2 plateau at 12.3 T, for the 400 μm wide sample of Ref. [11], and for ISD = 0 μA that and Thus, a rather large 0.81 μA current circulates around the sample at 12.3 T when ISD = 0 μA, = 0.5 μm and Vm = 0.75 V. The maximum extent of this current is 60 times farther from the sides of the sample than that produced by skipping orbits bouncing off of a hard wall with a cyclotron radius or magnet length l = (ħ/eB)1/2 of 7.3 nm.

4. Charge-Redistribution Potential

Section 2 noted that the Lorentz force exerted on the conducting electrons causes deviations − eδσ(y) from the average surface charge density − eσave = − ens = − ie2B/h of the 2DEG charge-redistribution across the sample width. The resulting charge-redistribution potential, Vr(y), arising from applied currents would be a linear function of y if the mobile electrons occupied a three-dimensional volume. They occupy a two-dimensional sheet, however, and MacDonald, Rice, and Brinkman [19] expressed this charge-redistribution self-consistently in terms of a charge-redistribution potential as where as shown in Appendix A. Riess [20] extended this potential to a 2DEG with finite thickness. Thouless [21] then found an analytic logarithmic approximation of this potential far from the sample sides, and Beenakker and van Houten [22] then approximated the near-edge behavior by introducing a cut-off at a distance ξ from the sample side, and a linear extrapolation for from |y| = w/2 − ξ to |y| = w/2. The characteristic length ξ is , where l = (ħ/eB)1/2 is the magnetic length and a* = 4πκε0 ħ2/m*e2 is the effective Bohr radius in SI units. Our values of ξ, lB, and a* for the i = 2 plateau at 12.3 T are 3.3 nm, 7.3 nm, and 10.2 nm, respectively.

Balaban, Meirav, and Shtrikman [10] used a nonlinear (quadratic) extrapolation near the sample sides and obtained the charge-redistribution potential for |y| < w/2 − δ, where δ = l for the i = 2 plateau, and δ is not the differential δ of Eq. (10). They successfully used this potential to describe the sample-width dependence for breakdown at small currents, but could not account for the larger breakdown currents observed in other experiments [11, 23–28]. Their geometry factor is for our values of ξ and δ at w = 400 μm.

4.1 Charge-Redistribution Potential Equation

The charge-redistribution potential described by Eq. (13) was calculated for an infinite square-well confining potential, and must be modified for use with a more realistic confining potential. To do this correctly would require a numerical solution of Eq. (12), with the confining potential included, as is discussed in Appendix A. We approximated this numerical solution (and then tested the approximation) by using the form of the potential in Eq. (13) but introducing two parameters, ymin and ymax, that alter the charge-redistribution potential due to the presence of the quadratic confining potential.

It was necessary to do this because the potential distribution of Eq. (13), with a cut-off distance δ = l, gave the correct quantum Hall voltage VH = RHISD across the sample, but the electric field Er = − ∇Vr did not increase quickly enough for increasing current to satisfy the ISD = 0 μA conditions of Sec. 3.3 and then reach the electric field values necessary for quasi-elastic inter-Landau level scattering (QUILLS) transitions [11, 16–18,25–28] at high currents.

We use the same form for the charge-redistribution potential as Balaban et al. [10], but with a different geometrical factor and very different cut-off values, ymin and ymax, which vary with applied current. Our charge-redistribution potential is Ic(rhs) and Ic(lhs) are defined by Eqs. (5) and (6), and the geometry factor G in Eq. (15) is We assume G is current-independent, and assign the value to Eq. (17) by using the value of ymax = 199.559 μm found in Sec. 3.3 for ISD = 0 μA and w = 400 μm. Our value of G is thus somewhat larger than the value G = 0.08 that would be used by Balaban et al. [10]. The cut-off values will be determined in Sec. 5. Appendix B discusses the agreement between our Eq. (15) and the self-consistent Eqs. (10) and (11).

The electric field Er = − ∇Vr due to redistribution of the 2DEG with applied current is We now have nearly all the information necessary to determine the potential and current distributions.

5. Calculations

Figure 4 shows the confining potential − Vc(y) and the charge-redistribution potential − Vr(y) across the sample for greatly exaggerated values of , δmax, and δmin, and for an arbitrary value of Ir, where Ir is defined by Eq. (16). Vr becomes infinite at ±w/2, but that is of no concern because it is only the occupied states which contribute to the Hall voltage, and those states occur only between ymax and ymin. The potentials are therefore finite and well-behaved in the region of interest.

Fig. 4

The confining potential − Vc(y) and the charge-redistribution potential − Vr(y) across the sample for greatly exaggerated values of , δ/2 − ymax, and δmin = w/2 + ymin.

5.1 Total Potential

Of course the electrical transport properties depend on the total potential Vt(y), but we can unambiguously separate Vt(y) into the confining and charge-redistribution potential components We have uniquely defined the potentials Vc(y) and Vr(y) in Eqs. (2) and (3) of Sec. 3.2 and Eqs. (15) to (18) in Sec. 4.1, plus Eqs. (5) and (6) in Sec. 3.3. The current-independent parameters for the confining potential and the charge-redistribution potential are: = 0.5 μm, Vm = 0.75 V, and G = 0.147. For a given sample we know the applied current ISD and the sample width w, but there are still two free parameters: ymax and ymin.

Ordinarily, it would not be possible to uniquely determine the values of ymax and ymin since the only other piece of information is that the quantum Hall voltage VH is and there is a range of values for ymax that satisfies this equation. It is possible, however, to determine the value of ymax for a particular type of experiment, and we believe that the results are representative of most other experiments since our calculations agree with the experimental data of Fontein et al. [8]. We first note that E (y) = − V(y). Therefore In an experiment described in Ref. [11] we measured the quantized longitudinal voltage drops along a GaAs/AlGaAs sample between potential probes 4 and 6 of Fig. 1 at high currents, and deduced the maximum electric field Emax from a quasi-elastic inter-Landau level scattering model. The results were and The value Emax = 1.1×106 V/m at ISD = 215 μA was just sufficient to excite the lowest, M = 1, QUILLS transitions [11,25-28]. It is clear from Fig. 4 that Emax will occur at ymax, so We can therefore use Eqs. (23) and (24) to determine ymax, and then Eq. (22) to obtain ymin for the sample of Ref. [11]. Note that changing the values of ymax and ymin also alters the values of Ic(rhs), Ic(lhs), and thereby the value of Ir in Eqs. (5), (6), and (16). Thus there are no free parameters, and one can obtain unique solutions to the total potential and other transport properties.

5.2 Results

Relevant values for the solution at ISD = 0 μA are shown in Table 1. Most were calculated in Sec. 3.3; the remainder were found from Eqs. (2), (3), and (15) to (22). Note that ymax and ymin are predicted to be about 60 magnetic lengths from the sides of the sample.

Table 1

Values of some quantities obtained from the calculations in Sec. 5 for ISD = 0 μA, 25μA, 215 μA, and 225 μA. The quantities common to all the calculations are i = 2, B = 12.3 T, = 13.1, w = 400 μm, = 0.5 μm, Vm = 0.75 V, a = 3.0×1012 V/m2, λ = 199.500 μm, G = 0.147, and l = 7.3 nm. See Secs. 2–5 for the definitions of these symbols

ISD(μA)	Ic(rhs)(μA)	Ic(lhs)(μA)	Ir(μA)	ymax(μm)	ymin(μm)	Vc(ymax)(V)	Vr(ymax)(V)	Ec(ymax)(MV/m)	Er(ymax)(MV/m)	Et(ymax)(MV/m)	δmax/lB
0	0.81	−0.81	0.00	199.559	−199.559	−0.010	0.000	0.354	0.000	0.354	60.3
25	0.94	−0.68	24.74	199.564	−199.554	−0.012	−0.160	0.382	0.054	0.436	59.6
215	2.30	−0.05	212.75	199.599	−199.515	−0.030	−1.392	0.596	0.504	1.100	54.8
225	37.36	−0.00	187.64	199.901	−198.044	−0.482	−1.477	2.405	1.795	4.200	13.5

We calculate the values shown in Table 1 at ISD = 215 μA by increasing the value of ymax until Et(ymax) = 1.1×106 V/m, adjusting the value of ymin to obtain the correct Hall voltage, and remembering that changing the values of ymax and ymin also changes the values of Ic(rhs), Ic(lhs), and Ir. The solution is unique, with no free parameters. The same procedure is done at ISD = 225 μA, except that the value of ymax is increased until Et(ymax) = 4.2×106 V/m. Note in Table 1 that ymax is still about 13 magnetic lengths away from the side of the sample at ISD = 225 μA.

We also calculate the relevant quantities at ISD = 25 μA, which is a current often used in precision quantized Hall resistance measurements. In this case, however, we do not know the value of Et(ymax), so we use a linear interpolation of the value of ymax between its values for ISD = 0 μA and 215 μA. The quantities shown in Table 1 for ISD = 25 μA are relatively insensitive to this choice for ymax.

5.3 Plots

We now plot the potentials, using Eqs. (2), (3), (15) to (19), and (21). Figure 5 shows Vc(y) and Vr(y) for the parameters used in Table 1 at ISD = 215 μA, except that the plot is between ±0.99999 w/2 (±199.998 μm) rather than ymax and ymin in order to show the sharpness of the confining potential and the extent of the charge-redistribution potential at these extreme values of y. Figure 6 shows Vt(y) plotted between ymax and ymin using the parameters in Table 1 at ISD = 215 μA and 225 μA. Other than moving farther to the right, the total potential does not significantly change shape with increasing current.

Fig. 5

Vc(y) and Vr(y) plotted between ±0.99999 w/2 for the parameters used in Table 1 at ISD = 215 μA. The parameters common to all plots in Figs. 5–11 are i = 2 (12 906.4 Ω), B = 12.3 T, w = 400 μm, = 13.1, = 0.5 μm, Vm = 0.75 V, and G = 0.147.

Fig. 6

Vt(y) plotted between ymax and ymin, using the parameters in Table 1 for ISD = 215 μA and 225 μA. The values of ymax and ymin are 199.599 μm and −199.515 μm, and 199.901 μm and −198.044 μm for ISD = 215 μA and 225 μA, respectively.

Figure 7 shows Vt(y) at ISD = 25 μA. The shape of this predicted potential is in excellent agreement with the experimental measurements shown in Fig. 6 of Fontein et al. [8]. It is this agreement which provides the best verification of our results. The “linear” part of the potential distribution within the sample interior, attributed in Ref. [8] to heating effects which cause R = V/ISD to increase, is accounted for by our charge-redistribution potential in a sample which has minimal heating at these currents [24].

Fig. 7

Vt(y) at ISD = 25 μA. This potential is in excellent agreement with the experimental measurements shown in Fig. 6 of Fontein et al. [8].

The electric fields Ec(y) = − ∇c(y) and Er(y) = − ∇Vr(y) are shown in Fig. 8 for ISD = 215 μA; they were determined from Eqs. (2), (3), (18), and (20). The value of ymax = 199.599 μm is such that Et(ymax) = 1.1×106 V/m in equation (23). The contribution to the total electric field at ymax is slightly more for the confining potential than for the charge-redistribution potential at this current. Table 1 shows that the confining potential also provides the dominant contribution to Et(ymax) at other currents.

Fig. 8

Electric fields Ec(y) = − ∇Vc(y) and Er(y) = − ∇Vr(y) for ISD = 215 μA.

The location, ymax, of the last-filled state on the right-hand side of the sample increases with applied current ISD. We can use Eq. (A-3) and Table 1 to determine what part of this increase in ymax is due to the increase in the total electric field at ymax. The percentage contributions, relative to the values of ymax and Et(ymax) at ISD = 0 μA, are 4 %, 5 %, and 3 %, for ISD = 25 μA, 215 μA, and 225 μA, respectively. Therefore, most of the increase in ymax is due to the Lorentz force pushing the electrons closer to the side of the sample.

The current density Jt(y) for electrons moving in the positive x direction is Figure 9 shows Jt(y) for ISD = 25 μA, 215 μA, and 225 μA. The maximum two-dimensional current density is at ymax, and is 85 A/m and 325 A/m, respectively at ISD = 215 μA and 225 μA. There is current in the negative x direction in the vicinity of ymin at small currents due to the dominance of the confining potential. When ISD = 215 μA and 225 μA, however, Er(ymin) > |Ec(ymin)| and no current flows in the −x direction anywhere across the sample.

Fig. 9

Current density Jt(y) for ISD = 25 μA, 215 μA, and 225 μA.

The current I (y) for electrons moving in the positive x direction is where and We divide the sample width into 20 equal segments in Fig. 10 and determine the percentage of current flowing through each segment for ISD = 25 μA, 215 μA, and 225 μA. We do not show a plot for ISD = 0 μA, but ΔI would be −0.81 μA and +0.81 μA for the left-hand side and right-hand side segments, respectively, and zero for the other 18 segments because Ic(rhs) = –Ic(lhs) = 0.81 μA.

Fig. 10

Percentage of current flowing through 20 equal segments across the sample width for ISD = 25 μA, 215 μA, and 225 μA.

The current distributions in Fig. 10 are virtually identical between 25 μA and 215 μA, even though large numbers of electrons are being excited into higher Landau levels at 215 μA. The left and right side distributions are nearly symmetric. There is, however, a significant transfer of current from the left-hand side segment to the right-hand side segment at 225 μA. We saw in Sec. 5.2 that no current flows within 60, 55, and 13 magnetic lengths of the sample side for ISD = 25 μA, 215 μA, and 225 μA, respectively. Also, 68 %, 70 %, and 51 % of the current is in the 19 segments to the left of the right-hand side segment where the edge channel current would flow for these three applied currents. The current density was negative in the left-hand side of Fig. 9 at ISD = 25 μA because electrons were flowing in the –x direction at ymin −199.554 μm, but that contribution to ΔI in the left-hand side segment of Fig. 10 is so small that the net current is positive.

Finally, we investigate the charge-redistribution –eδσ (y) of the electrons in the 2DEG in terms of the deviation δσ (y) in the number of electrons/cm2 from the average number ns = 5.94×1011/cm2 on the i = 2 plateau at 12.3 T, where from Eq. (A-5). Figure 11 is a logarithmic plot of |δσ (y)| versus y for ISD = 215 μA. There is an excess of electrons on the +y side of the sample, and a depletion on the −y side.

Fig. 11

Logarithmic plot of the charge-redistribution |δσ (y)| across the sample for ISD = 215 μA, where δσ (y) is the deviation from the average number density n The “−” region represents an excess of electrons, the “+” region a depletion of electrons.

An assumption made in deriving Eq. (A-5) was that the charge density varies slowly across the sample, i.e., that ρ (y) ≈ ρ (y+dy), or δσ (y) << ns. This assumption is valid here because the largest value of δσ (y) occurs at ymax, and is 2 %, 2 %, and 6 % of ns at ISD = 25 μA, 215 μA, and 225 μA, respectively.

One of the consequences of our approximate form of the charge-redistribution potential is that the net charge does not vanish when the charge-redistribution –eδσ (y) sample width. The area under the curves in Fig. 11 is 4 % larger for the +y side than for the −y side. Therefore, there is an unaccounted excess of electrons; so this is not quite the actual shape of the charge-redistribution function. However, it is the potential and current distributions that are of primary importance to the transport properties—not the charge-redistribution. The charge was certainly conserved in the experiment of Fontein et al. [8], and yet their measured potential distributions are symmetrical. This fact demonstrates that the slight charge asymmetry does not significantly affect the potential and current distributions.

We could conserve the charge by adjusting the origin slightly to the right until the area under the curves are equal for ±y in Fig. 11, and then self-consistently recalculating the potentials with the new coordinates. This would greatly complicate the calculations however, and with all the approximations that have been made in this paper, and with the excellent agreement with experiment [8], it seems unnecessary. It may be a consequence of this charge nonconservation problem that the value of ymin is inside the confining potential for the case in Table 1 when ISD = 225μA.

5.4 Sample-Width Dependence of the Critical Current

Balaban, Meirav, and Shtrikman [10] have found that the critical current for breakdown of the quantum Hall effect, Icr, scales logarithmically with the sample width w for all Landau levels. We verify this dependence by: (a) using the result in Sec. 3.3 that Ic(rhs) = − Ic(lhs) = 0.81 μA for the i = 2 plateau at 12.3 T when ISD = 0 μA; (b) calculating the value of ymax from Eqs. (2) and (5) for each value of w at ISD = 0 μA; (c) calculating the value of G from Eq. (17) for each value of w; (d) defining Icr as the applied current ISD sufficient to excite the lowest, M = 1, QUILLS transitions [11,25-28]; (e) assuming the value of Ec(ymax) is the same for all values of ISD that excite M = 1 QUILLS transitions (The value used is Ec(ymax) = 5.96×105 V/m, obtained from Table 1 at ISD = 215 μA, w = 400 μm, and Et(ymax) = 1.1×106 V/m. This is equivalent to fixing the value of Ic(rhs) to be 2.30 μA for each value of Icr); (f) calculating the value of ymax from Eq. (2) for each value of w; (g) adjusting the value of Ir so that Et(ymax) = 1.1×106 V/m in Eqs. (2), (17), (20), and (23); and (h) adjusting the value of ymin to give the correct Hall voltage for each current by using Eqs. (2), (15), (17), (21), and (22).

The results of Icr versus w are plotted in Fig. 12. The shape of the curve is identical to the experimental data of Balaban et al. [10]. The scaling is very different, however because their critical currents are about two orders of magnitude smaller than ours. We note that the experiment of Haug, von Klitzing, and Plog [29] tends to agree with the experimental curve shapes of Balaban et al. [10], but the experiment of Kawaji, Hirakawa, and Nagata [30] found a linear, rather than a logoarithmic, dependence of Icr with w. Perhaps this difference is due to nonuniformities in the values of the charge-depletion width along the sides of the samples, e.g., we have observed different values of Icr along the lengths of some of our samples. If we assume that the value of Vm remains constant along a sample edge, allow to vary by changing the average ionized donor density ND, and assume the ratio Ec(ymax)/Et(ymax) remains constant, then we find that the critical current required to excite M = 1 QUILLS transitions with Et(ymax) = 1.1×106 V/m decreases when Δ decreases and ND increases, i.e., the steeper the confining potential, the smaller the critical current.

Fig. 12

The critical current Icr versus sample width w. Refer to Sec. 5.4 for details of the calculation.

6. Conclusions

We have calculated potential and current distributions across the width of a GaAs/AlGaAs heterostructure sample for applied currents between 0 μA and 225 μA, using: (a) a quadratic confining potential Vc(y) arising from charge-depletion regions along the sides of the sample; (b) parameters for that potential obtained from a localization experiment [9]; (c) a logarithmic charge-redistribution potential Vr(y) of the 2DEG; and (d) a maximum electric field Et(ymax) calculated from breakdown measurements and a QUILLS model [11]. Our predictions are in excellent agreement with experiments [8,10].

Referring to Table 1, the confining potential component Ec(ymax) of the electric field at ymax contributes 88 %, 54 %, and 57 % to Et(ymax) at 25 μA, 215 μA, and 225 μA, respectively. The maximum current density Jt(ymax) is 34 A/m, 85 A/m, and 325 A/m, respectively at these three currents. A significant amount of current is distributed within the sample interior. For example, Ir is 99 %, 99 %, and 83 % of ISD, respectively at these three currents. We predict the current to be much farther from the sides of the sample than in other models, e.g., no current flows within 60, 55, and 13 magnetic lengths of the sample side for these currents. It would require a lateral resolution of about 0.1 μm, rather than the 70 μm resolution of Fontein et al. [8], to verify this result.