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Hopping conductivity in heavily doped n-type GaAs layers in the quantum Hall effect regime
S. S. Murzin1,2 , M. Weiss1 , A. G. M. Jansen1 and K. Eberl3
1

arXiv:cond-mat/0106265v2 [cond-mat.mes-hall] 16 Jun 2001

Grenoble High Magnetic Field Laboratory, Max-Plank-Institut fur Festk¨rperforschung and Centre National de la Recherche ¨ o Scientifique, BP 166, F-38042, Grenoble Cedex 9, France 2 Institute of Solid State Physics RAS, 142432, Chernogolovka, Moscow District., Russia 3 Max-Plank-Institut fur Festk¨rperforschung, Postfach ¨ o 800 665 D-70569, Stuttgart, Germany

We investigate the magnetoresistance of epitaxially grown, heavily dop ed n-typ e GaAs layers with thickness (40-50 nm) larger than the electronic mean free path (23 nm). The temp erature dep endence of the dissipative resistance Rxx in the quantum Hall effect regime can b e well describ ed by a hopping law (Rxx exp {-(T0 /T )p }) with p 0.6. We discuss this result in terms of variable range hopping in a Coulomb gap together with a dep endence of the electron localization length on the energy in the gap. The value of the exp onent p 0.5 shows that electron-electron interactions have to b e taken into account in order to explain the occurrence of the quantum Hall effect in these samples, which have a three-dimensional single electron density of states.

For a two-dimensional electron system it is well known that the discrete electron spectrum in a high magnetic field leads to quantized Hall resistance (Quantum Hall Effect). However, Landau quantization is not a strict prerequisite for the QHE. According to gauge arguments1,2 it is sufficient that the dissipative conductance Gxx vanishes at the Fermi level, and that delocalized states exist below. The occurrence of the quantum Hall effect in not strictly two-dimensional systems has been considered by Khmelnitzkii3 in conjunction with the scaling theoretical treatment of the QHE4 . In our previous works5,6 , we observed the quantum Hall effect in a strongly disordered system, which consisted of a heavily Si-doped (n-type) GaAs layer between undoped GaAs. In this system a wide, smooth quantum well is formed by the impurity space charge potential that builds up at the layer interfaces. The electron gas is therefore confined inside the heavily doped GaAs layer, in the area of maximum disorder. The thickness d of the layers ranging from 50 up to 140 nm was larger than the electronic mean free path l of 15 - 30 nm. The density of states (DOS) of noninteracting electrons in these samples is therefore expected to be practically three-dimensional. As the very strong disorder broadening in the samples leads to a rather smooth density of states without the formation of gaps between Landau levels even at the highest magnetic fields ( 20 T), we have proposed a reduction of the diagonal conductance Gxx due to electron-electroninteraction effects in diffusive transport as a possible explanation for the observed quantization of Rxy in the investigated, strongly disordered systems. Evidence for this explanation comes from the temperature dependence in the quantum Hall minima of Gxx in samples with a thickness between 50 and 140 nm6 , which is logarithmic with temperature T , and thus resembles the temperature dependence that is caused by quantum corrections due to electron-electron interactions in disordered con-

ductors both in weak7 and in high magnetic fields8,9 . However, the logarithmic decrease of Gxx that was found in Ref.6 exceeds in amplitude the range, where the theory of quantum correction7­9 is applicable. Furthermore, in the thinnest sample (with a layer thickness d = 50 nm), which showed a fully developed quantization of the Hall effect, the conductance deviated from the logarithmic temperature dependence at the lowest temperatures for values of B where Gxx 0 and where the Hall conductance Gxy correspondingly shows a plateau at a value o f 2 e 2 / h. In the current work we have investigated an additional number of strongly disordered GaAs layers with smaller values of d (namely 40 and 50 nm), showing a fully developed QHE below 100 mK. The obtained hopping law for the temperature dependent dissipative resistance Rxx is discussed in terms of the opening of a Coulomb gap. The Hall conductance quantization in the aforementioned, quasi-three-dimensional systems with a "bare" (high temperature) conductance G0 x e2 /h can be x understood qualitatively in the following way. Usually, in systems with coherent diffusive transport the dissipative conductance Gxx decreases with temperature T due to quantum corrections. The weak localization (singleparticle) corrections are suppressed in a magnetic field B and reduce to10 Gxx (L ) = G0 x - x 2 e4 ln(L /L0 ) = 2 h2 G0 x x (1)

G0 x - x

m e4 ln(T1 /T ). 2 h2 G0 x x

0 Here L = Dxx is the distance an electron moves diffusively during the phase breaking time T -m , 0 Dxx is the "bare" high-temperature diffusion coefficient, 0 L 0 = d Dx x / D 0 zz

=

dG0 x /z x

z

(2)

1


is the electron displacement in the plane of the layer (perpendicular to the magnetic field) for the time of its diffusion across the layer (along the magnetic field), Dzz and zz are the diffusion coefficient and the conductivity in the direction parallel to the magnetic field, T1 is defined 0 from the equation d 2Dzz (T1 ). At low temperatures the phase breaks due to electron-electron interactions, leading to m = 1. The second order corrections in a magnetic field (Eq.1) are much smaller ( hG0 x /e2 x times) than the first-order corrections in zero field. Nevertheless, Gxx will eventually vanish, and in this case the Hall conductance Gxy should be quantized1,2 . Since Gxy tends to different quantum values for different bare Hall conductances G0 y , transitional values of the bare conducx tance G0 y should exist, for which Gxx tends to a finite x value and Gxy is not quantized. This approach, initially developed for spinless noninteracting electrons, can give a reasonable, qualitative explanation for the occurrence of the quantum Hall effect with even numbers of quantization i in the above mentioned, strongly disordered GaAs layers5,6 . Quantitative agreement with theory however does not exist because the quantum corrections (Eq.1) are small at real experimental conditions. To explain our results, we have proposed the inclusion of electron-electron interactions. In this case, the single-particle DOS and the conductance should decrease with decreasing temperature due to quantum corrections caused by interactions Gxx (LT ) = G0 x - x e2 2e2 ln(T2 /T ) = G0 x - ln(LT /L0 ) x h h (3)

T = 4 .2 K

0.6

1 0 .3 0 .0 8

(h/e )

R

2

xy

xx

,R

xy

0.4

R

R 0.2

xx

0.0 0 2 4 6 8 10 12

B(T)

FIG. 1. Magnetic field dep endence of the Hall (Rxy ) and transverse (Rxx ) resistance (p er square) for sample 40 in a magnetic field p erp endicular to the heavily dop ed GaAs layer at different temp eratures

that occur both in weak7 and in high magnetic fields8,9 . 0 Here LT (Dxx /kB T )1/2 , kB is the Boltzmann con0 stant, and T2 Dzz /kB d2 . 1 is the constant of interaction, which is of the order of unity and even somewhat larger in high magnetic fields (µB g B /kB T 1) than in zero field (µB is the Bohr magneton). For G0 x e2 /h these corrections are much larger than the x single-particle localization contributions (Eq.1). The interaction corrections (Eq.3) will lead to a vanishing of the dissipative conductance Gxx as a consequence of the opening of a Coulomb gap in the single particle DOS. Since also in this scenario Gxx will vanish at zero temperature, the Hall conductance should be quantized. The samples used were prepared by molecular-beam epitaxy: on a GaAs (100) substrate the following layers were successively grown: an undoped GaAs layer (0.1 µm), a periodic structure of 30 â GaAs/AlGaAs(10/10 nm), an undoped GaAs layer (0.5 µm), the heavily Sidoped GaAs layer with a nominal thickness of d = 40 (sample 40) and 50 nm (sample 50) and donor(Si) concentrations of 1.5 â 1017 cm-3 , and last a cap layer of 0.5 µm GaAs (undoped). Samples with Hall bar geometries of a width of 0.2 mm and a length of 1.4 mm were etched out of the wafers. A phase sensitive ac-technique was used for the magnetotransport measurements down 2

to 80 mK. In the experiments the applied magnetic field of up to 15 T was directed perpendicular to the layers. Samples from the same wafer showed identical behavior. The electron densities per square as derived from the slope of the Hall resistance Rxy in weak magnetic fields (0.5 - 3 T) at T = 4.2 K are Ns = 4.5 and 5.1 â 1011 cm-2 . The "bare" mobilities µ0 are equal to 2500 and 2300 cm2 /Vs for sample 40 and 50 respectively, and the electron mean free path is about 23 nm for both samples. For the calculation of µ0 we took the value of the bare resistance R0 in the point of intersection of the curves Rxx (B ) for different temperatures at B = 3.4 T, taking into account that the classical resistance does not depend on field. In Fig.1 the magnetotransport data, namely the Hall (Rxy ) and transverse (Rxx , per square) resistance are plotted for sample 40 at temperatures below 4.2 K. The diagonal resistance Rxx decreases sharply at low magnetic fields due to the suppression of the weak localization corrections, and continues to decrease slightly between 0.5 and 4 T. It shows a deep minimum ranging from 6 to 11 T. The Hall resistance Rxy shows a linear increase up to 5 T, and then reveals a wide plateau from B = 6 T up to 11 T at the lowest temperatures with the value Rxy = h/2e2 (i.e. i = 2), in the same field range where Rxx shows a deep minimum. 2 2 The Hall conductance Gxy = Rxy /(Rxx + Rxy ) in the field range of B = 0.5 - 4 T does not depend on temperature. The diagonal conductance (per square) Gxx however shows a logarithmic temperature dependence with an only slightly field dependent coefficient, while the value of Gxx itself changes considerably. This behavior is in agreement with equation (3), giving an interaction constant 0.5. The magnetotransport data for sample 50 are similar to the data for sample 40. In our previous investigations of identical samples5,6,12


10

-1

chosen as a result of a fit of the experimental data to a hopping law Rxx = R0 exp {-(T0 /T ) }
p

(4)

10

-2

Rxx (h/e )

10.6 T 10
-3

in a range of temperature where Rxx (T ) < 0.1Rxx (4.2 K) 0.02h/e2. The fitting parameters R0 and T0 are listed in the table. Attempts to fit the data by an expression with a temperature dependent prefactor Rxx = T r exp {-(T0 /T )p } (5)

2

10

-4

9.8 8.7 8.8 1 2 3 4
-0.6 -0.6

10

-5

T
FIG. 2. of T -0.6 at larger minimum

(K )

The logarithm of the resistance Rxx as a function for sample 40 in the minimum (B = 8.8 T) and fields indicated by lines, and for sample 50 in the (B = 8.7 T)

with however a larger layer thickness, we found corrections to the conductivity due to electron-electron interactions. In a region of low magnetic field (B < 4 T) where G0 x e2 /h the magnetoresistance data can be quantix tatively described in terms of quantum corrections due to electron-electron-interaction effects12 . In high magnetic fields, even in samples with thicknesses d ranging up to 140 nm, quantization of the Hall conductance is observed. The mentioned samples show values of the bare conductance G0 x up to 2.6e2 /h5 . Even at these high fields the x different QHE minima in the transverse conductance Gxx of different samples show a universal logarithmic temperature dependence in a large range of a rescaled temperature T /Tsc , where Tsc exp(-3G0 x h/e2 )6 . Note x however, that the decrease of Gxx is not small and that a logarithmic temperature dependence is observed beyond the region of applicability of the theory of quantum corrections7. In the thinnest sample (d = 50 nm) investigated in Ref.6 , showing a well pronounced QH plateau, a deviation from the logarithmic behavior becomes visible at the lowest temperatures (T < 1 K). It is this range of temperature and layer thickness, that the present work is focused on. We therefore study the temperature dependence of the resistance Rxx of samples with a thickness d 50 nm, and therefore a rather low bare conductance G0 x of about e2 /h. These samples show a pronounced x plateau in Rxy and a strong T-dependence near the minimum of Rxx at low temperatures, as shown in Fig. 1. In Fig. 2 we plot the logarithm of the resistance Rxx as a function of T -0.6 in the minima of Rxx corresponding to the plateaus at Rxy = h/2e2 for samples 40 and 5013 , and additionally for sample 40 at somewhat larger B , but still not far from the minimum. The exponent p = 0.6 is 3

and a fixed p different from 0.6 resulted in a less optimal fit. Moreover, the resulting fitting parameters are unphysical. For instance, for the case of p = 0.5 the fit gives r = 0.65, = 17.1 and T0 = 25.5 K. For this situation, the prefactor T r in equation (5) at T = 1K corresponds to a conductance Gxx = 2 Rxx /Rxy = Rxx /(0.5h/e2)2 70e2 /h while Gxx = 2 0.95e /h only at T = 10 K. The large prefactor in the conductance is compensated by a small exponential facp 1/2 tor exp {-(T 0 /T ) } = exp {-25.5 } 6.4 â 10-3 , while Gxx (10K)/Gxx (1 K) has a value of about 3 only. The small difference between Gxx (1 K) and Gxx (10K) would be the result of a compensation of the two, which is not realistic. Thus we conclude, that the temperature dependence in the i = 2 minimum in Rxx is rather described by a hopping law according to Eq.(4) with a hopping exponent p near 0.6. Without the existence of a Coulomb gap the Mott theory of variable range hopping14 predicts the temperature dependence of Rxx to follow equation (4) with p = 1/3. According to the theory from Efros and Sklovskii15,16 , p is equal to 1/2 in the presence of a Coulomb gap around the Fermi energy EF (both in zero magnetic field and in the QHE regime). This theory was developed for situations where the localization length does not depend on the energy = |E - EF | in the gap. In the case of Anderson localization the localization length should depend on the energy near the Coulomb gap. In the single-particle approach, at G0 y (B ) = ie2 /h x with even i, the localization length sp of an electron at the Fermi level equals sp L0 exp 0.5 2 G0 x
x 22

h /e

4

(6)

estimated from the equation Gxx (sp ) = 0 with Gxx taken from equation (1). According to the scaling theoretical treatment of the QHE, the localization length sp generally depends both on G0 x and G0 y . It diverges at x x G0 y (B ) = (i + 1/2)e2 /h. x However, electron-electron interactions should result in a decrease of the localization length in the Coulomb gap. A lower limit of this decrease can be estimated from the equation Gxx (0 ) = 0 with Gxx taken from equation (3) 0 L0 exp G0 x h x 2e2 . (7)


Outside the gap interaction is not important, and the localization length is equal or larger than the one given by expression (6) with G0 x = G0 x (E ) for the energy x x 0 E . For typical values of Gxx e2 /h and 1, 0 is much smaller than sp . As it will be shown below, such an energy dependence of should result in p > 1/2 in Eq.(4). The single-particle density of states should be unaffected by an energy dependence of the localization length, unless the distance between electrons is much larger than the localization length, i.e. g () ()2 || 1. It should still be linear: g () = || with = 22 / e4 ( is the dielectric constant of the lattice). Let us suppose that in some range of energy = ||s . Then by analogy with the Mott-law derivation14,16 we obtain Rxx Gxx exp {-(T 0 /T ) where T0 = S kB C e2
1/(s+1) (s+1)/(s+2)

Sample 40 40 40 50

B (T) 8.8 9.8 10.6 8.7

T0 (K ) 6.0 4.3 2.04 4.5

R0 (h/e2 ) 2.41 1.55 0.66 1.6

h (µm) 0.63 0.95 2.3 0.9

T (µm) 0.24 0.4 1.2 0.37

TABLE I. Values of the magnetic fields B , the constant T0 and the prefactor R0 , the localization length h of the electrons giving the main contribution to the conductivity, at T = 0.1 K and the localization length T of the electrons with energy /kB = 0.1 K.

},

(8)

,
(s+2)/(s+1)

S = (s + 1)-

(s+1)/(s+2)

+ (s + 1)1/

(s+2)

. (9)

The coefficient depends on the magnetic field. For s = 0 equations (8) and (9) reduce to the results from Efros and Shklovskii. For s 1 one finds activated behavior and p = (s + 1)/(s + 2) = 0.6 is obtained for s = 1/2. The main contribution to the conductivity is given by hopping electrons with an energy of h = C kB e2 (s + 1) T
1/(s+2)

field the power of T in equation (4) could be larger than 1/2 in some range of temperature. In summary, in low magnetic fields (but still larger than 0.5 T) where Gxx > 3e2 /h, the temperature dependence of the diagonal conductance Gxx of heavily doped n-type GaAs layers with thicknesses (d = 40 Â 140 nm) larger than the mean free path of the electrons (l = 23 nm) is well described by the theory of quantum corrections due to electron-electron interactions. In high magnetic fields where Gxx < 3e2 /h the temperature dependence of the conductance in the minima of Gxx,min is still close to logarithmic down to 0.25e2 /h, although the theory of quantum corrections is no more applicable. In the region of Gxx < 0.25e2 /h the dissipative conductance shows an exponential decrease with a power p 0.6, indicating the presence of a Coulomb gap. The data display the relevance of electron-electron interactions for the quantum Hall effect in these systems which have a 3D single-particle spectrum. We have pointed out, that a dependence of the localization length on energy could result in an exponent p > 1/2 both in zero and nonzero magnetic field.

The corresponding localization length h = |h |s = C kB e2 (s + 1) T
s/(s+2)

,

(10)
1 2 3

of the electrons giving the main contribution to the conductivity for T = 0.1 K and s = 1/2 is listed in the table. The numerical coefficient C is taken to be 1.55 as defined from the equation for T0 from the Efros-Shklovskii theory (T0 = 6.2e2 / )17 . Since h T 1/5 can not be smaller than 0 it should become constant at the lowest temperatures and the temperature dependence should reduce to the EfrosShklovskii law. In our experimental conditions h approaches 0 < 200 nm at the very small temperature of T < 3 â 10-4 K at B = 8.8 T and T < 5 â 10-7 K at B = 10.6 T for sample 40. A dependence of the localization length on energy could probably also account for hopping exponents p > 1/2, observed in zero-field experiments18­20 . An energydependence as described above is also indicated by numerical simulations21,22 . Therefore, also in zero magnetic 4

4

5

6

7

8

9

R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). H. Aoki and T. Ando, Phys. Rev. Lett. 57, 3093 (1986). D.E. Khmel'nitski, Pis'ma Zh. Eksp. Teor. Fiz. 38, 454 i (1983) [JETP Lett. 38, 552 (1983)]. H. Levine S. B. Libby and A. M. Pruisken, Phys. Rev. Lett. 51, 1915 (1983). S. S. Murzin, A. G. M. Jansen, and P. v. d. Linden, Phys. Rev. Lett. 80, 2681 (1998);.S. S. Murzin, I. Claus, and A. G. M. Jansen, Pis'ma Zh. Eksp. Teor. Fiz. 68, 305 (1998) [JETP Lett. 68 (1998)]. S. S. Murzin, I. Claus, A. G. M. Jansen et. al., Phys. Rev. B 59, 7330 (1999). B. L. Al'tshuler and A. G. Aronov, in Electron-Electron Interaction in Disordered Systems, edited by A. L. Efros and M. Pollak, North-Holland, Amsterdam, 1987. A. Houghton, J. R. Senna and S. C. Ying, Phys. Rev. B 25, 2196, (1982); 25, 6468 (1982). S. M. Girvin, M. Johnson and P. A. Lee, Phys. Rev. B 26,


10

11

12

13

14 15 16

17 18

19

20

21

22

23 24 25

1651 (1982). The numerical coefficient in Eq.(1) differs from an earlier presented one23,12 . Therefore, we have recalculated it using the equation for the -function derived by S. Hikami24 and K. B. Efetov25 , taking into account that the equation is written for a spinless electron system. S. Kivelson, D.Lee, and S. Zang, Phys. Rev. B 46, 2223 (1992). S. S. Murzin, Pis'ma Zh. Eksp. Teor. Fiz. 67, 201 (1998) [JETP Lett. 67, 216 (1998)]. The very first measurement of sample 506 showed an activated temp erature dep endence b elow 0.3 K. The results of subsequent measurements, presented here and taken with sp ecial care to avoid heating effects, are identical for two samples from the same wafer with d = 50 nm. N. F. Mott, J. Non-Cryst. Solids 1, 1 (1968). A. L. Efros, B. I. Shklovskii, J. Phys. C 8, L49 (1975). B. I. Shklovskii, A. L. Efros, Electronic Properties of Doped Semiconductors, Springer Heidelb erg, Berlin, 1984. V. L. Nguen, Sov. Phys. Semicond.18, 207 (1984) W. F. Van Keuks, X. L. Hu, H. W. Jiang, and A. J. Dacm Phys. Rev. B 56, 1161 (1997). M. E. Gershenson, Yu. B. Khavin, D. Reuter et al, Phys. Rev. Lett. 85, 1718 (2000) N. Markvi´, C. Christiasen, D. E. Grupp and et al, Phys. c Rev. B 62, 2195 (2000). F. Epp erlein, M. Schreib er and T. Vo jta, Phys. Rev. B 56, 5890 (1997). Gun Sang Jeon, Seongho Wu, H.-W. Lee and M. Y. Choi, Phys. Rev. B 59, 3033 (1999). Bodo Huckestein, Phys. Rev. Lett. 84, 3141 (2000). S. Hikami, Phys. Rev. B 24, 2671 (1981) K. B. Efetov, Adv. Phys., 32, 53 (1983)

5