- Email: [email protected]
The Coulomb gap and the transition to Mott hopping
surface science ELSEVIER
Surface Science 361/362 (1996) 945-948
The Coulomb gap and the transition to Mott hopping F . W . v a n K e u l s a, X.L. H u a, A.J. D a h m a , , , H . W . J i a n g b • Department of Physics, Case Western Reserve University, Cleveland, OH 44118, USA b Department of Physics, University of Cal~'ornia at Los Angeles, Los Angeles, CA 90024, USA Received 12 June 1995; accepted for publication 8 August 1995
Abstract Resistivity data are presented which confirm a Coulomb gap in the single particle density of states in two dimensional variable range hopping. For large hopping lengths the Coulomb interaction is screened bY a metallic gate. A universal crossover to Mort hopping is observed at low temperatures. Excellent agreement with theory is achieved with a single parameter fit.
The Coulomb gap in the density of states in electron systems is well established in three dimensions I-1]. Formulae for the resistivity in the presence [2] and absence of r31 a Coulomb gap are
giv
respectively, as
p~(T) = pc(T) exp (r~/~)--pc(T) exp ((Tt/T)I/2); Coulomb Gap (1) p=(T) = pc(T) exp (r~/~) = pc(T) exp ((To/T)1/3); Mott Hopping (2) where rm is the hopping length and ~ is the localiTation length. In two dimensions (2D), evidence for the Coulomb gap from resistance data has been unconvincing ['4]. Mott hopping is more commonly cited to explain the data. At high magnetic fields where data ['5,6] fit Eq. (1), an alternate theory [7], which does not involve a Coulomb gap, y/elds the same expression for the resistivity,
* Corresponding author. Fax: + 1 216 3684671; e-mail: [email protected].
and these data cannot unambiguously prove the existence of a Coulomb gap. Tunneling experiments suggest a depletion of states in the vicinity of the Fermi level [8]. In an attempt to answer the question of the existence of the Coulomb gap in 2D we have conducted experiments on gated samples, where the electron density can be varied across a predicted transition between Coulomb gap and Mott hopping behaviors. Screening of the 2D electrons by a metallic gate causes thi~ transition by bridging the Coulomb gap very close to the Fermi energy. The assumed form of the density of states g(E) is shown in Fig. 1. The distance d separating the 2D electron layer and the gate determines the height of the bridge. At very low temperatures the flat screened gs(Ev) governs the electron behavior producing a Mott hopping reTme. At higher temperatures the electrons probe the slopes of g(E), resulting in a Coulomb gap re,me. Aleiner and Shklovskii [9] have developed a theory for this crossover. They find that the hopping distance r~,, can be interpolated in the following universal
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved PII S0039-6028 (96) 00570-5
946
F W. mn Keuls et aZl Surface Science 361/362 (1996) 945-948
We studied two gated GaAs/Alo.3Gao.TAS heterostructures fabricated by molecular beam epitaxy. Samples A and B are from different wafers with d values of 320 and 90 nm; respectively. The donor potentials at the electron layer are enhanced by omitting the conventional undoped Alo.3Gao.7As spacer between the 2D electron layer and the doped layer. The low-temperature mobilities at Vg=0 of samples A and B are 1.1 and 2.3 m2/Vs, respectively. The electron densities at zero gate voltage were nA = 5.56 x 10 is and ne = 8.87 x 10 is
i
J °/ I I
~.(z,)
m-2.
E Fig. 1. The electronicdensityof statesg(E) with a soft Coulomb gap and screeningby a gate. The value of g(E) in the screened region is gs(Er)foce/eed. In the soft gap region, gc(E): 2&IE-ETI/ne4. forln
rJd:
[Cf(T/T*)/(T/T*)] ~/2~(~/d)(T~IT) ~/2 forT>> T*
-*(¢/d)(To/TY/3 forT<< T*,
(3)
where the characteristic temperature 7"* : e2:,/6keaa, e is the dielectric constant, andf(x) is a dimensionless function with the correct asymptotes given numerically in Ref. [9]. Several numerical coefficients enter the equations through the definitions of ~ in each r e , m e ~=Ce~/ekBT~
(Coulomb gap)
(4)
and ¢ = (fl/g,(EF)ka To) 1/2 = (fle2d/o~eke To)x/2 (Mort hopping).
(5)
Calculations have found the various coefficients to be oc g0.1,]~=13.8 and C=6.2. We found good agreement with the universal curve (Eq. (3)) in an extensive study of two samples with different gate-to-electron separations. We found both Coulomb gap and screened Mott hopping behavior in their predicted regions of phase space. These equations were fitted over large ranges of the parameters: seven orders of magnitude for p,=, two orders of magnitude for T, two orders of magnitude for ~, and electron densities between 0.6 x 10 xs and 1.5 X 10 xs m -2.
The resistance was measured by two terminal methods. The temperature-independent resistance of the contacts was measured at Vg = 0 as a function of magnetic field, and this resistance is subtracted from measurements of the sample resistance. The contact resistance is small compared to the sample resistance, except in the conducting state where data are not presented. Chopped DC -measurements were used above about l0 s D, while faster 2 Hz lock in techniques were used at lower resistances. All data are taken in the ohmic r e , m e phase. The measuring current varied from 20 fA at the highest resistances to 15 nA at the lowest. To distinguish Coulomb gap from Mott hopping behavior, one must divide p,~ (T) by po(T). Fig. 2 illustrates some of our data at zero magnetic field
t ,
.
i 2'
'
,
.
,
,
,
,
,
.
'.:...^-oo"_o f'-:.°"'L.
;O,o,
.
,
,
,
.
....
,r--t/s (K--t/s) Fig. 2. The ratio 0~jJT°'s versus T - 1~. The n ~ b e f e d clm'es in increasing order correspond to densities of 0.61, 0.83, 0.94, 1.05, 1.27, 1.33,1.38, 1.44 and 1.49x10 ts m -2 for sample A. Solid lines
are fits to Eq. (1). Lettered curves correspond to densities of 0.70, 0.85, 0.95, 1.06, 1.13, 1.21and 1.31x10ts m -2 for sample B.
F. W. van Keuls et aL/Surface Science 361/362 (1996) 945-948
from both samples in this form. The lower electron density curves have a exponential dependence of T -1/2 while the higher densities are close to a T-v3 dependence. The frequently overlooked temperature dependence of the pre factor Po is critical in determining the exponential dependence. If resistance measurements consist of a set of data points of limited number and accuracy which vary over two orders of magnitude, then p ~ ~, T = exp ((T~/T) 1/3) with m in the range 0.5-1 will fit the data equally as well as p~,~ exp ((To/T)I/2). Calculations of the prefactor have proven difficult, leaving the temperature dependence of the prefactor an open question theoretically. To determine po(T), we fit p~(T) to AT ~ exp ((T/T) p) for all data from the insulating phase. The fitted values ofp fall into two categories. The larger values of p at higher temperatures and lower electron densities are set to 1/2. The average value of m is then determined to be m--0.8+0.2. Using this value of m, the two regimes coincide well with p = 1/2 and p - 1 / 3 . Fig. 3 shows the values of p obtained from setting m--0.8 for both samples. Fig. 3a reveals a wide Mott hopping region in data taken a fixed density wh/le the magnetic field was varied. Between 2 and 3 T the localization lengths are very long, and p,~ is not well described by the hopping formulae in thin cut of phase space. Fig. 3b shows the variation of/7 with density at zero magnetic field. The Coulomb gap region is clear at low densities, but the Mott hopping region is narrow since p drops further as one nears the conducting phase at large densities. All of the resistance measurements on both samples can be fit to the universal function for r~/d given in Eq. (3) by adjusting a single parameter, ~/. This parameter is the following combination of constants that enter the models of both temperature dependencies, t/= efl/ocC2. Fig. 4 shows data taken at 11 densities at zero magnetic field, plotted with a line denoting Eq. (3). The adjustment of r/ is necessary to cause data from all densities to bend at the same values of T/T*. This condition is equivalent to requiring that the values of ~ calculated in both re#rues for a given density be identical. For the data set shown in Fig. 4, we find the optimal value of ~ = 5.9 To, where the expected
947
I- .................... 0"4t "]~| II
0
•
0.2
1
2
4
3
6
5
(a) •
0.8
"
'
I
'
"
"
I
"
"
"
I
"
'
"
I
"
'
'
(b) '
~'"
!
--
t---t
0.2
density (IO ta m -m) Fig. 3. The exponent p. (a) p versus B. Open symbols: d = 9 0 nm, n = 1.13 x 10is m -2. Closed symbols: d = 320 nm, n = 1.33x10Is m -2. (b) p versus density for B = 0 . Open symbols: d = 90 nm, closed symbols: d--320 nm. Dotted fines are drawn at p-- 1/3 and 1/2.
value r/o contains the lattice dielectric constant, and the other constants given after Eq.(5). Analysis of data from both samples taken at a variety of magnetic fields up to 4 T show that ~ is consistent between samples and has tittle or no field dependence, with an average value of t/=(6.4+l.3)t/o. This analysis shows that the dielectric constant e is constant, while the localization length changes by two orders of magnitude. Our resisitivity measurements over wide ranges of resistance, temperature, density and localization length exhibit Coulomb gap behavior at small hopping lengths and Mott hopping behavior for large hops. The data agree with Aleiner and
948
F W. van I(Jsui~et aL/Surface Science 361/362 (1996) 945-948
convincing evidence for the existence of the Coulomb gap in 2D.
i 151
Acknowledgements
I •¢1 1 o
The authors are indebted to I.L. Aleiner, M.E. Raikh and B.I. Shklovskii for many stimulating conversations. This work was supported in part by NSF grants DMR 93 13786 and DMR 94 02647.
5
I
I
2
4
I 8
References Fig. 4. The ratio rffi/d versus ( ~ / T ) x/2. The solid line is given by Eq.(3) with q=5.gr/o. Symbols xepresent lldensities from 0.7 to 1.2 × 1015 m -2 for sample B.
Shklovskii's theory E9] of a universal screening transition between these two re~mes. We find a prefactor temperature dependence of approximately T °'s , and an adjustment parameter r/= 6.4r/o. We believe these data provide the most
[1] B3. Shklovskii and A.L. Efros, Ele~ronic Properties of Doped Semiconductors (Springer, Berlin, 1984). [2] A.L. Efros and B.I. Shklovskii, J. Phys. C 8 (1975) IA9. [3] N.F. Mott, J. Non Cryst. Solids 1 (1968) 1. [4] For a review, see M. Pollak, Philoa Mag B 65 (1992) 657. [5] G. Ebert et aL, Solid State Commun. 45 (1983) 625. [6] A. Briggs et aL, Phys. Rev. B 27 (1983) 6549. l'7] Y. Ono, L Phys. SOc. Jpn. 51 (1982) 237. [8] R.C. Ashoori et al., Phys. Rev. B 48 (1993) 4616. [9] LL. Aleiner and B3. Shklovskii, Phys. Rev. B 49 (1994) 13721.
Surface Science 361/362 (1996) 945-948
The Coulomb gap and the transition to Mott hopping F . W . v a n K e u l s a, X.L. H u a, A.J. D a h m a , , , H . W . J i a n g b • Department of Physics, Case Western Reserve University, Cleveland, OH 44118, USA b Department of Physics, University of Cal~'ornia at Los Angeles, Los Angeles, CA 90024, USA Received 12 June 1995; accepted for publication 8 August 1995
Abstract Resistivity data are presented which confirm a Coulomb gap in the single particle density of states in two dimensional variable range hopping. For large hopping lengths the Coulomb interaction is screened bY a metallic gate. A universal crossover to Mort hopping is observed at low temperatures. Excellent agreement with theory is achieved with a single parameter fit.
The Coulomb gap in the density of states in electron systems is well established in three dimensions I-1]. Formulae for the resistivity in the presence [2] and absence of r31 a Coulomb gap are
giv
respectively, as
p~(T) = pc(T) exp (r~/~)--pc(T) exp ((Tt/T)I/2); Coulomb Gap (1) p=(T) = pc(T) exp (r~/~) = pc(T) exp ((To/T)1/3); Mott Hopping (2) where rm is the hopping length and ~ is the localiTation length. In two dimensions (2D), evidence for the Coulomb gap from resistance data has been unconvincing ['4]. Mott hopping is more commonly cited to explain the data. At high magnetic fields where data ['5,6] fit Eq. (1), an alternate theory [7], which does not involve a Coulomb gap, y/elds the same expression for the resistivity,
* Corresponding author. Fax: + 1 216 3684671; e-mail: [email protected].
and these data cannot unambiguously prove the existence of a Coulomb gap. Tunneling experiments suggest a depletion of states in the vicinity of the Fermi level [8]. In an attempt to answer the question of the existence of the Coulomb gap in 2D we have conducted experiments on gated samples, where the electron density can be varied across a predicted transition between Coulomb gap and Mott hopping behaviors. Screening of the 2D electrons by a metallic gate causes thi~ transition by bridging the Coulomb gap very close to the Fermi energy. The assumed form of the density of states g(E) is shown in Fig. 1. The distance d separating the 2D electron layer and the gate determines the height of the bridge. At very low temperatures the flat screened gs(Ev) governs the electron behavior producing a Mott hopping reTme. At higher temperatures the electrons probe the slopes of g(E), resulting in a Coulomb gap re,me. Aleiner and Shklovskii [9] have developed a theory for this crossover. They find that the hopping distance r~,, can be interpolated in the following universal
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved PII S0039-6028 (96) 00570-5
946
F W. mn Keuls et aZl Surface Science 361/362 (1996) 945-948
We studied two gated GaAs/Alo.3Gao.TAS heterostructures fabricated by molecular beam epitaxy. Samples A and B are from different wafers with d values of 320 and 90 nm; respectively. The donor potentials at the electron layer are enhanced by omitting the conventional undoped Alo.3Gao.7As spacer between the 2D electron layer and the doped layer. The low-temperature mobilities at Vg=0 of samples A and B are 1.1 and 2.3 m2/Vs, respectively. The electron densities at zero gate voltage were nA = 5.56 x 10 is and ne = 8.87 x 10 is
i
J °/ I I
~.(z,)
m-2.
E Fig. 1. The electronicdensityof statesg(E) with a soft Coulomb gap and screeningby a gate. The value of g(E) in the screened region is gs(Er)foce/eed. In the soft gap region, gc(E): 2&IE-ETI/ne4. forln
rJd:
[Cf(T/T*)/(T/T*)] ~/2~(~/d)(T~IT) ~/2 forT>> T*
-*(¢/d)(To/TY/3 forT<< T*,
(3)
where the characteristic temperature 7"* : e2:,/6keaa, e is the dielectric constant, andf(x) is a dimensionless function with the correct asymptotes given numerically in Ref. [9]. Several numerical coefficients enter the equations through the definitions of ~ in each r e , m e ~=Ce~/ekBT~
(Coulomb gap)
(4)
and ¢ = (fl/g,(EF)ka To) 1/2 = (fle2d/o~eke To)x/2 (Mort hopping).
(5)
Calculations have found the various coefficients to be oc g0.1,]~=13.8 and C=6.2. We found good agreement with the universal curve (Eq. (3)) in an extensive study of two samples with different gate-to-electron separations. We found both Coulomb gap and screened Mott hopping behavior in their predicted regions of phase space. These equations were fitted over large ranges of the parameters: seven orders of magnitude for p,=, two orders of magnitude for T, two orders of magnitude for ~, and electron densities between 0.6 x 10 xs and 1.5 X 10 xs m -2.
The resistance was measured by two terminal methods. The temperature-independent resistance of the contacts was measured at Vg = 0 as a function of magnetic field, and this resistance is subtracted from measurements of the sample resistance. The contact resistance is small compared to the sample resistance, except in the conducting state where data are not presented. Chopped DC -measurements were used above about l0 s D, while faster 2 Hz lock in techniques were used at lower resistances. All data are taken in the ohmic r e , m e phase. The measuring current varied from 20 fA at the highest resistances to 15 nA at the lowest. To distinguish Coulomb gap from Mott hopping behavior, one must divide p,~ (T) by po(T). Fig. 2 illustrates some of our data at zero magnetic field
t ,
.
i 2'
'
,
.
,
,
,
,
,
.
'.:...^-oo"_o f'-:.°"'L.
;O,o,
.
,
,
,
.
....
,r--t/s (K--t/s) Fig. 2. The ratio 0~jJT°'s versus T - 1~. The n ~ b e f e d clm'es in increasing order correspond to densities of 0.61, 0.83, 0.94, 1.05, 1.27, 1.33,1.38, 1.44 and 1.49x10 ts m -2 for sample A. Solid lines
are fits to Eq. (1). Lettered curves correspond to densities of 0.70, 0.85, 0.95, 1.06, 1.13, 1.21and 1.31x10ts m -2 for sample B.
F. W. van Keuls et aL/Surface Science 361/362 (1996) 945-948
from both samples in this form. The lower electron density curves have a exponential dependence of T -1/2 while the higher densities are close to a T-v3 dependence. The frequently overlooked temperature dependence of the pre factor Po is critical in determining the exponential dependence. If resistance measurements consist of a set of data points of limited number and accuracy which vary over two orders of magnitude, then p ~ ~, T = exp ((T~/T) 1/3) with m in the range 0.5-1 will fit the data equally as well as p~,~ exp ((To/T)I/2). Calculations of the prefactor have proven difficult, leaving the temperature dependence of the prefactor an open question theoretically. To determine po(T), we fit p~(T) to AT ~ exp ((T/T) p) for all data from the insulating phase. The fitted values ofp fall into two categories. The larger values of p at higher temperatures and lower electron densities are set to 1/2. The average value of m is then determined to be m--0.8+0.2. Using this value of m, the two regimes coincide well with p = 1/2 and p - 1 / 3 . Fig. 3 shows the values of p obtained from setting m--0.8 for both samples. Fig. 3a reveals a wide Mott hopping region in data taken a fixed density wh/le the magnetic field was varied. Between 2 and 3 T the localization lengths are very long, and p,~ is not well described by the hopping formulae in thin cut of phase space. Fig. 3b shows the variation of/7 with density at zero magnetic field. The Coulomb gap region is clear at low densities, but the Mott hopping region is narrow since p drops further as one nears the conducting phase at large densities. All of the resistance measurements on both samples can be fit to the universal function for r~/d given in Eq. (3) by adjusting a single parameter, ~/. This parameter is the following combination of constants that enter the models of both temperature dependencies, t/= efl/ocC2. Fig. 4 shows data taken at 11 densities at zero magnetic field, plotted with a line denoting Eq. (3). The adjustment of r/ is necessary to cause data from all densities to bend at the same values of T/T*. This condition is equivalent to requiring that the values of ~ calculated in both re#rues for a given density be identical. For the data set shown in Fig. 4, we find the optimal value of ~ = 5.9 To, where the expected
947
I- .................... 0"4t "]~| II
0
•
0.2
1
2
4
3
6
5
(a) •
0.8
"
'
I
'
"
"
I
"
"
"
I
"
'
"
I
"
'
'
(b) '
~'"
!
--
t---t
0.2
density (IO ta m -m) Fig. 3. The exponent p. (a) p versus B. Open symbols: d = 9 0 nm, n = 1.13 x 10is m -2. Closed symbols: d = 320 nm, n = 1.33x10Is m -2. (b) p versus density for B = 0 . Open symbols: d = 90 nm, closed symbols: d--320 nm. Dotted fines are drawn at p-- 1/3 and 1/2.
value r/o contains the lattice dielectric constant, and the other constants given after Eq.(5). Analysis of data from both samples taken at a variety of magnetic fields up to 4 T show that ~ is consistent between samples and has tittle or no field dependence, with an average value of t/=(6.4+l.3)t/o. This analysis shows that the dielectric constant e is constant, while the localization length changes by two orders of magnitude. Our resisitivity measurements over wide ranges of resistance, temperature, density and localization length exhibit Coulomb gap behavior at small hopping lengths and Mott hopping behavior for large hops. The data agree with Aleiner and
948
F W. van I(Jsui~et aL/Surface Science 361/362 (1996) 945-948
convincing evidence for the existence of the Coulomb gap in 2D.
i 151
Acknowledgements
I •¢1 1 o
The authors are indebted to I.L. Aleiner, M.E. Raikh and B.I. Shklovskii for many stimulating conversations. This work was supported in part by NSF grants DMR 93 13786 and DMR 94 02647.
5
I
I
2
4
I 8
References Fig. 4. The ratio rffi/d versus ( ~ / T ) x/2. The solid line is given by Eq.(3) with q=5.gr/o. Symbols xepresent lldensities from 0.7 to 1.2 × 1015 m -2 for sample B.
Shklovskii's theory E9] of a universal screening transition between these two re~mes. We find a prefactor temperature dependence of approximately T °'s , and an adjustment parameter r/= 6.4r/o. We believe these data provide the most
[1] B3. Shklovskii and A.L. Efros, Ele~ronic Properties of Doped Semiconductors (Springer, Berlin, 1984). [2] A.L. Efros and B.I. Shklovskii, J. Phys. C 8 (1975) IA9. [3] N.F. Mott, J. Non Cryst. Solids 1 (1968) 1. [4] For a review, see M. Pollak, Philoa Mag B 65 (1992) 657. [5] G. Ebert et aL, Solid State Commun. 45 (1983) 625. [6] A. Briggs et aL, Phys. Rev. B 27 (1983) 6549. l'7] Y. Ono, L Phys. SOc. Jpn. 51 (1982) 237. [8] R.C. Ashoori et al., Phys. Rev. B 48 (1993) 4616. [9] LL. Aleiner and B3. Shklovskii, Phys. Rev. B 49 (1994) 13721.