Quantum fluctuations in n+ GaAs wires with invasive probes

SurfaceScience North-Holland

QUANTUM W. HANSEN

321

(1990) 321-325

FLUCTUATIONS *, T.P. SMITH

IN n+ GaAs WIRES WITH INVASIVE PROBES

III, D.P. DiVINCENZO,

K.Y LEE, Y.H. LEE and R. CHEUNG

**

IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA Received 11 July 1989; accepted for publication 14 September 1989

Conductance fluctuations in mesoscopic electronic systems depend strongly on the configuration of the measurement probes. We have found striking differences in the magnetoconductance fluctuations of n+ GaAs wires when measured in the longitudinal and Hall geometries. In our experiments, the elastic scattering length is smaller than the channel width, which is smaller than the phase coherence length which is comparable to the probe width. The voltage fluctuations and correlation fields of the conductance fluctuations are more than an order of magnitude larger in the longitudinal probe geometry than in the Hall geometry. These differences shed new light on mesoscopic transport phenomena and can be explained by extension of a recently developed multi-lead formalism.

In mesoscopic devices phase coherence of the electron wave leads to phenomema which have no analog in classical electron transport. For example, “magnetofingerprints” - reproducible fluctuations of the resistance measured as a function of an applied magnetic field - have been studied in metal wires [ 1,2] as well as in a variety of semiconductor structures: silicon inversion channels [ 3 1, doped semiconductor wires, [ 4-6 ] and quasi one-dimensional high mobility heterojunctions [ 7,8 1. Thus far the Hall voltage has been studied in detail only in high mobility samples, where the electrons motion past the part of the channel with the attached Hall probes is ballistic. Ballistic phenomena that have been studied previously include, for example, the quenching of the Hall effect [9,10], and the effect of the shape of the channel and of the junctions with the voltage probes on the quantization of the Hall voltage [lO,ll]. In this paper we examine the Hall effect in devices where the electron motion is diffusive. It might seem that making the conducting system dirtier (i.e., adding more impurity scattering) would inevitably make the physics more complicated and harder to inter-

* Permanent address: University of Hamburg, Institut fur Angewandte Physik, Fed. Rep. of Germany. ** Permanent address: University of Glasgow, Department of Electrical Engineering, Glasgow, UK.

pret. Fortunately, transport simplifies in many ways when electrons scatter repeatedly from impurities during passage through the sample - this is the diffusive limit, which is dominated by the physics of multiple wave interference. In several cases the physics of quantum diffusion has been clearly confirmed in observations of weak localization [ 12 1, conductance fluctuations [ 13 1, and Aharonov-Bohm phenomena [ 11. In this work we further illustrate the application of these simple ideas over new device structures, emphasizing their role in the fluctuations of the Hall conductivity. We have observed striking differences in magnitude and correlation field of the voltage fluctuations measured in Hall geometry as opposed to the fluctuations measured with probes on the same side of n+ doped GaAs wire where the transport is diffusive. Theory tells us that the important length scale for these differences is the phase coherence length Li - ballistic transport across the probes is not essential. The samples are fabricated using epitaxially grown GaAs on a semi-insulating substrate. The doping density is 1.4~ 1018 cm-3 in an 80 nm thick layer at the sample surface. The geometry of the devices is defined by electron-cyclotron-resonance reactive-ion etching through a NiCr mask defined by electronbeam lithography. We used two different Hall-bar geometries with the voltage probes 1.5 and 4.5 pm apart and with probe widths, W,, 0.5 and 1.0 pm,

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respectively. The geometrical width of the wires between the voltage probes is about 100 nm in our thinnest wire and 500 nm in the broadest ones. The widths of the conducting channels, w, are about 50 nm smaller than the geometrical widths due to depletion at the etched surfaces of the wires, and details of the fabrication process are described elsewhere [ 141. Figs. 1 and 2 show representative results for the longitudinal magnetoresistance and the Hall resistance measured on wires with channel widths of about 100 nm at T= 1.4 K. Before discussing the differences in the resistance fluctuations we estimate from these data the important length scales: the elastic mean free path, I,, the phase coherence length, L,, the thermal diffusion length, LT, the electron Fermi wavevector, kF, and the parameter k,l,, which determines the character of the electron motion. From the average Hall voltage we find that the carrier density in all the wires is essentially the same as the asgrown material (n= 1.4x 10i8/cmP3). From the zero-field resistance and channel width w, we determine mobilities of 2100 cm2/V*s for the sample in fig. 1 and 600 cm2/V*s for the device in fig. 2. The

-

I.8

-1.2

-0.6

0.0

0.6

1.2

1.6

B (Tesla)

Fig. I. The magnetoresistance (dotted curve) and Hall resistance for an n+ GaAs wire with a lithographic width of approximately 100 A. The inset shows the device geometry: W,=O.S pm, L= 1.5 pm.

in n+ GaAs w-es

SAMPLE 2 T=1.4K I

9y

0

1

2

3

1

I

4

5O

8 (T)

Fig. 2. The magnetoresistance (dotted curve) and Hall resistance for another n+ GaAs W,=l pm, L~4.5 pm. wire: width=lOOOA, WP=lfim,L=4.5pm.

mobility in macroscopic samples is about 200 cm2/ V-s. For the low-mobility sample the elastic mean free path, I, is 14 nm and k,l,= 5, indicating that we are in the weak-localization regime. Corresponding values for the high-mobility sample are I,=.50 nm, and kFfe= 18. Therefore, transport over distances comparable to the channel width and the voltage probe spacing is diffusive. The phase coherence length L, can be determined from the size of the conductance fluctuations or their correlation field. A further independent estimate can be made from a lit of the low-field (Bc0.2 T) magnetoresistance to weak-localization theory. The size of the conductance fluctuations is universal in wires with lengths less than the phase coherence length [ 15,16 1. The conductance fluctuations scale with the ratio L,/L if the probe distance L is larger than the phase coherence length [ 31: 6G= (LJL )3/2e2/h. In analyzing our results we assume that the transport is one-dimensional, i.e. the phase-coherence lengths is larger than the wire width. From the data in figs. I and 2 the phase-coherence lengths are Liz 0.5 and L, = 0.2 pm, respectively. The phase-coherence length is larger than the channel

W. Hansen et al./Quantum

width in all devices except those with 500 nm channel width, where Li- w. The conductance correlation functions [ 15 1,

F(W=([G(N-
(1)

(where ( ) denotes an average over magnetic field) derived from the data shown in fig. 2 are shown in fig. 3. The correlation field, A&, defined as the half width of the F(AB), is related to the phase coherence length by ABC=2.4(h/e)/wL,, where w is the channel width. From the correlation fields for R, we determine phase coherence lengths of L,= 1.24 and 0.66 pm for the samples in figs. 1 and 2, respectively. In all our devices the results of the two different approximations for Li are in reasonable agreement. Fits of the low-field magnetoresistance with a weak-localization theory (assuming one-dimensional transport in the wire and two-dimensional transport in the leads [ 121) yield values that are between those determined above, ranging between 0.4 and 0.6 pm. In all cases L> L,> w using these different approaches. In addition, the thermal diffusion length LT= (hD/kT)‘l’, is about twice as large as the L, in

Rxx

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-0.25

SAMPLE 2

0.00 A6 (T)

0.25

fluctuations

in n+ GaAs wires

323

all samples at T= 1.4 K. Therefore, we will assume the zero-temperature limit in the discussion below. The fluctuations in the Hall resistance R,, &Rx,,, differ from fluctuations in R, in two respects: the size of the fluctuations and their correlation fields. As shown in fig. 1, 6R,=O.O36R,. Here 6R, is evaluated after subtraction of a straight line corresponding to the average Hall resistance. The difference in the fluctuations is even more striking for the sample in fig. 2 where 6R,=O.O016R,. The reduction factor decreases with increasing W, and decreasing win samples with different geometries. The second striking difference is that the characteristic spacings of the fluctuations is much smaller in the Hall resistance. This observation is reflected in the correlation functions shown in fig. 3. The correlation field of R, is an order of magnitude larger than the correlation field of R,, and the difference in the correlation field decreases as the difference between the W, and the channel width w decreases. These experimental results can be understood in terms of recent developments in the theory of quantum transport [ 2,13,15,17]. In fact, several of our results illuminate some new features of quantum interference in small devices which have not been brought out in previous studies. The important theoretical idea [ 1,131 is that wave-interference phenomena are operative at small length scales, smaller than some phase-breaking length L,. Within each such phase coherent domain (illustrated in fig. 4)) elastic scattering of the electron wave (by impurities, etc.) will cause the resistance of this domain to fluctuate in accordance with the universal conductance fluctuation theory [ 15 1. This theory says that the conductance of a phase-coherent region will fluctuate by e2/h. The resistance of such a domain will therefore fluctuate by

0.50

Fig. 3. The correlation function for 6R, and 6R, for the data shown in fig. 2. The half width of the correlation function (shown by arrows) for 6R, is about IO times smaller than the half width for 6R,.

Fig. 4. A schematic diagram of the device geometry and the important length scales for diffusive transport.

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where tin is the sheet resistance. To understand the magnitude of the measured resistance fluct.uations, we must consider the actual device geometry. In the longitudinal resistance geometry, since the distance between the voltage probes L is much greater than L,, we are effectively probing n= L/L, phase coherent regions connected in series. The magnitude of the resistance fluctuations can be estimated by applying the following principle on long length scales [ 13 1: the fluctuations should add just like the fluctuations of classical resistors connected in series: SR,

= JNGR,

.

(3)

In the Hall geometry, the voltage probes are within one phase-breaking length of one another; therefore, in this case, the resistance fluctuations of one phasecoherent region are probed directly (see fig. 4) _ However, the phase coherent volume under the Hall probes (A, in fig. 4) has a different aspect ratio than the ones inside the channel (A,), because IV’,> w. Thus, eq. (2) must be modified:

(4) A more careful calculation based on the diffusive electron transport theory [2,15 1 provides a detailed quantitative prediction for the reduction of the Hall voltage fluctuations with probe width. The reduction factor is

(5) This result is obtained in the approximation that the electron transport is purely diffusive (that is, the elastic scattering length is smaller than the sample dimensions), and also that L, is greater than w or IVp. In this approximation, the probe width has no effect on the magnitude of 6R,. Using eq. ( 5 ), we predict that 6R,/6R,=O.O9 for the geometry of fig. 1, and GR,/FR, = 0.02 for the geometry of fig. 2. In both

cases the theoretical value is an order of magnitude larger than the experimental result, but the trend of the data and the substantial suppression of fluctuations in the Hall resistance relative to the longitudinal resistance is correctly represently by the theory. We turn to an explanation of the observed correlation fields A& in the magnetoresistance and the Hall resistance. As noted earlier, we observe that A&tx > A&?‘. The basic principle that we apply here is that A& is the magnetic field such that the magnetic flux through one coherence volume is equal to one flux quantum ( aO= h/e) [ 15 1. Again, the difference in the Hall resistance fluctuations and the magnetoresistance fluctuations arise from the difference in aspect ratio of the coherence volume near the voltage probes. As fig. 4 illustrates, the area A, under the Wall probes is greater than the area of the coherence volume along the wire A2 by roughly the ratio of the probe width to the sample width. This predicts

(6) Eq. (6) gives a ratio of 9 for the device of fig. 1 (compared with 9 experimentally), and 19 for the device of fig. 2 (compared with 12 experimentaIly ). So, the theory provides a satisfactory qualitative explanation for the wide differences between the correlation fields. The results presented here serve to reinforce in a dramatic way the idea that electrical measurements in a quantum system are highly invasive, that is, the nature and configuration of the measurement probe have strong effects on the result of the measurement [ 17 1. The Hall resistance is not a unique quantity: as our results illustrate, quantum features of R, (e.g., conductance fl uctuations~ are profoundly effected by how the voltage measurement is performed. It is satisfying that we can account for these effects with the simple physical picture of fig. 4 [ 13 1, with quantum interference phenomena operative at length scales shorter than Z+ and classical electrical ideas applied on longer scales. The GaAs material was grown by W.A. Wang at Columbia University. We are very grateful to D.P. Kern, P. Santhanam, S. Washburn, and L.L. Chang

W. Hansen et al./Quantumfluctuations

for helpful discussions. This work was supported part by the OffIce of Naval Research.

in

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