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Variable range hopping in the Coulomb gap and gate screening in two dimensions
Physics Letters A 349 (2006) 404–410 www.elsevier.com/locate/pla
Variable range hopping in the Coulomb gap and gate screening in two dimensions V. Duc Nguyen a , V. Lien Nguyen a,∗ , D. Toi Dang b a Theoretical Department, Institute of Physics, VAST, P.O. Box 429 Bo Ho, Hanoi 10000, Vietnam b Physics Faculty, School of Natural Sciences, HSU, 90 Nguyen Trai, Hanoi, Vietnam
Received 8 August 2005; accepted 21 October 2005 Available online 15 November 2005 Communicated by J. Flouquet
Abstract We simulate the single particle ground state and variable range hopping (VRH) in the two-dimensional model of doped semiconductors. The obtained results show a universality of the linear Coulomb gap in the density of states at the Fermi energy as well as the Efros–Shklovskii (ES) T −1/2 -law for the temperature dependence of VRH resistivity. The screening due to the metallic gate smears the Coulomb gap, and therefore, may produce the crossover from ES to Mott VRH as the temperature decreases. The full range of temperature of such a crossover seems however to be very large, up to three orders of magnitude. 2005 Elsevier B.V. All rights reserved. PACS: 72.20.Ee; 72.80.Ng; 73.40.Gk
1. Introduction
ates a soft gap in the DOS at the Fermi energy (Coulomb gap):
The variable range hopping (VRH) is a main electric conduction mechanism in systems with strongly localized electronic states at low temperatures. It is a problem of long time discussion. Almost four decades ago Mott has suggested the famous law for the temperature dependence of VRH resistivity [1]: ρ(T ) = ρ0 exp (TM /T )1/d+1 ,
TM = βM /kB g0 ξ d ,
(1)
where ρ0 is a pre-exponential factor, depending weakly on temperature, d is the dimensionality, g0 is the density of states (DOS) at the Fermi level, ξ is the localization length and βM is a numerical coefficient, depending on d. The law (1) was obtained, assuming that the DOS near the Fermi level is a constant, g(E) ≡ g0 . In 1975 Efros and Shklovskii (ES) [2] pointed out that the long-range Coulomb interaction between localized states cre* Corresponding author.
E-mail address: [email protected] (V. Lien Nguyen). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.094
d g(E) = (d/π) κ/e2 |E|d−1 ,
(2)
where κ is the dielectric constant, e is the elementary charge and the dimensionality d = 2, 3. The energy E in this expression is counted from the Fermi energy. Extending Mott’s argument of calculating ρ(T ) to the case of DOS with the Coulomb gap (2), ES obtained the well-known T −1/2 -law: ρ(T ) = ρ0 exp (TC /T )1/2 ,
TC = βC e2 /kB κξ.
(3)
The coefficient βC has been evaluated using a simple percolation approximation: βC ≈ 2.7 [3] and ≈ 6.2 [4] for d = 3 and 2, respectively. The former number 2.7 is in very good agreement with the simulation result βC (d = 3) ≈ 2.6 obtained by solving numerically the percolation problem in the so-called doped semiconductor model [3]. The latter number 6.2 is close to the value βC (d = 2) ≈ 5.8 reported by Tsigankov and Efros [5], simulating VRH in the square lattice model. Experimentally, though the ES-law was observed in a great number of experiments for various bulk materials (see references in [6]) as well
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
as two-dimensional (2D) systems [7], there exists however a large discrepancy in the value of βC for the 2D-case [8]. The fact that the T −1/2 -VRH law of Eq. (3) is recently observed in a variety of 2D-systems with great interest such as the insulating side of (assumed) metal-insulating transition (for a review, see [9]), the 2D doped array of self-assembled germanium quantum dots on the silicon substrate [10], and CdSe-nanocrystal films [11], attracts a renewed attention on this VRH-law, and particularly, on the value of the coefficient βC (2D). Although the value of 6.2 derived in Ref. [4], as mentioned above, is well approved by the simulation data for the lattice model, in order to check a universality of laws (2) and (3), in this Letter we will simulate both the DOS and VRH, including the coefficient βC (2D), in another fundamental disordered model, the model of doped semiconductors, as considered in Refs. [3,12,13] for the three-dimensional (3D) case. Besides, as is generally suggested [10,14], in the gated 2D systems, the metallic gate screening may make the DOS finite at the Fermi energy, and therefore may cause a crossover from ES to Mott VRH as the temperature reduces. We will exclusively examine these effects, using Monte Carlo simulation and Efros’s selfconsistent equation method. For these aims, at first, the single particle ground state (GS) in the lightly doped-semiconductor 2D-model is simulated, using the GS-simulation method suggested in Ref. [12] for the 3D-model. Using the GS obtained we calculate the single particle DOS and check a universality of the Coulomb gap (2). Then, at the same GS we solve directly the percolation problem in computer and thereby calculate the temperature dependence of the exponent of VRH resistivity [13]: ln ρ(T )/ρ0 = ηC (T ), (4) where ηC is the percolation threshold. The simulation results seem to fit well the ES-law of Eq. (3) and give βC (2D) ≈ 6.5 ± 1.0, close to the value 6.2 calculated in [4] and to the value 5.8 reported in [5] for the square lattice model. Thereby, we demonstrate a universality of the ES-law of Eq. (3) with βC ≈ 6. For the case of gate screened electron–electron interaction potential, both the GS-simulation and Efros’s selfconsistent equation [15] are in good agreement showing an expected smear of the Coulomb gap. Moreover, numerical calculations of the exponent of VRH-resistivity, ηC (T ), using the obtained self-consistent DOS and the percolation approximation suggested in Refs. [3,4], really show a gate induced ES to Mott VRH crossover. However, the range of temperature of such a crossover may be very large, up to three orders of magnitude. 2. Model and simulation We consider the 2D n-type doped semiconductor model with N donors and NA = CN acceptors (the degree of compensation C < 1), arranged at random in a square of side L = (N/ND )1/2 . Thus, the donor concentration in the model is always equal to ND regardless of the number N . At zero temperature all acceptors, possessing excess electrons each by one, become negatively charged; an equal number (NA ) of donors, donating elec-
405
trons to acceptors, are charged positively; the rest, N (1 − C), of donors remains neutral. Thus, though the system is totally neutral, there is a random potential, created by charged impurities, which acts on donor states. Experimentally, the model suggested can be used to describe the impurity band and hopping conduction in, for example, the sodium-doped silicon MOSFET measured by Timp et al. [16] or n-GaAs MESFET measured by Savchenko et al. [17]. Without screening the total electrostatic energy of the system reads N e2 (1 − ni )(1 − nj ) H= κ |ri − rj | i>j N,N NA A (1 − ni ) 1 + , − (5) |ri − rl | |rl − rk | i,l
l>k
ri (rl )
is the coordinate of donor i (acceptor l), ni is the where occupation number: ni = 1 for a neutral donor and ni = 0 for an ionized one. Correspondingly, the one-electron energy at donor i is N N A (1 − nj ) 1 e2 i = (6) − . κ |ri − rl | |ri − rj | l
j
To simulate a single particle GS we minimize the total energy (5) with respect to all possible single electron transfers from a neutral to an ionized donor. The transfer of an electron from i to j is allowed only if it decreases H, i.e., if j − i − e2 /κrij > 0 (hereafter rij ≡ |ri − rj |). At the same time, as a rule, in GS the electrons occupy energy levels gradually from down, i.e., for any neutral donor i and ionized donor j there always has the relation j > i . The GS is the state, where both conditions stated are simultaneously satisfied. At the GS all the energies [i ] are known and therefore one can calculate the DOS and determine the Fermi energy EF . Since the simulation is restricted to only single electron transfers the DOS obtained is the single particle one. The program of the GS simulation as well as of the DOS calculation is similar to that discussed in very detail in Ref. [12]. In Fig. 1 we show the DOS for the sample of N = 25600. Throughout this work the degree of compensation C is chosen to be 0.5. The data in Fig. 1 was averaged over 200 realizations of random impurity coordinates and the statistical error is too small to be shown. We have checked and believed that at this sample size the finite size effect for the DOS is unimportant. There exists a clear soft gap in the DOS at the Fermi energy EF . The simulation DOS in the close vicinity of EF is also shown in a larger scale in the inset of this figure (solid curve) in comparison with the linear DOS of Eq. (2) (dashed straight line). Obviously, two results are practically coincident in the energy range of |ε − EF | 0.2. At higher energies, however, they gradually diverge from each other. The situation recognized is similar to that observed in the 3D-case (see Fig. 4 in Ref. [12]), where the discrepancy between the simulation DOS and the parabolic gap (2) has been related to a strong fluctuation of one-particle energies, which smears the simulation impurity
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V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
Fig. 1. The simulation DOS for the long-range Coulomb interaction case (N = 25600, C = 0.5, averaging over 200 random realizations). Inset: the simulation DOS (solid curve) is compared with the linear Coulomb gap of Eq. (2) (dashed line).
band. Besides, as the first approximation, the law (2) is expected to be valid only at the energies near the Fermi level. Now, using the set of donor coordinates [ri ] and energies [i ], already known at the GS, we can solve the standard percolation problem with the criteria [13]: ηij = 2rij /ξ + Eij /kB T ηC , where Eij =
|Ei − Ej | − e2 /κrij , max(|Ei |, |Ej |),
(7)
if Ei Ej < 0, if Ei Ej > 0.
(8)
The energy Ei as mentioned above is the one-electron energy measured from the Fermi level, Ei = i − EF . The percolation threshold ηC (T ), obtained at a given temperature, gives the exponent of VRH-resistivity (see Eq. (4)). In practical simulations Eq. (7) is reasonably rewritten in term of the dimensionless distance rij∗ and energy Eij∗ : ∗ , rij∗ + t −1 Eij∗ ηC
(9)
where t ≡ (2κ/e2 ND ξ )kB T is the dimensionless temperature ∗ is the effective percolation threshold: and ηC 1/2 ∗ ηC (10) = ξ ND /2 ηC . The simulation units of length and energy are here chosen to −1/2 1/2 be ND and e2 ND /κ, respectively. The percolation problem, formulated in the form of Eq. (9), is very convenient for a quantitative examination of VRH-laws, (1) or (3). For example, using Eqs. (4) and (10), the ES-law (3) takes the simple form ∗ = βC /2 t −1/2 , ηC (11) which provides an easy way for determining the coefficient βC . Following the standard procedure of solving percolation problems [13,18], for a given dimensionless temperature t , the ∗ (t) can be obtained by (i) calculating percolation threshold ηC
∗ (t, N ) versus N −1/2 for various t (from top): Fig. 2. Finite size effect: ηC −1 t = 40, 30, 20, 15, 10, 5, 2 and 1. The number of random realizations used ∗ (t, N ) decreases from 3200 to for calculating ensemble-averaged quantities ηC 100 as N increases from 800 to 25600, respectively. The fitting dashed straight ∗ (t, ∞) ≡ η∗ (t) in the limit of N → ∞. lines give ηC C
∗ (N, t) for finite ensemble-averaged percolation thresholds ηC ∗ (N, t) as systems of different N and then (ii) extrapolating ηC ∗ ∗ N → ∞ to get ηC (t) ≡ ηC (∞, t) for an infinite system. The ∗ (N, t) as well as the procedure of exprogram of simulating ηC ∗ (t) trapolation of these ensemble-averaged quantities to gain ηC were described in very detail in Ref. [18]. ∗ (N, t) are plotted In Fig. 2 ensemble-averaged quantities ηC versus the inverse of square-system side N −1/2 for different N , from 800 to 25600, and at various temperatures t, from 1 down to 0.025. The number of realizations of random impurity arrangements used for calculating these averaging quantities, depends on N as given in the figure. Apparently, all the simulation points, belonging to each value of temperature t , ∗ (N, t) = η∗ (∞, t) × follow well a linear scaling relation, ηC C ∗ ∗ (∞, t) in the limit of −1/2 (1 + const N ), which gives ηC (t) = ηC N → ∞ as indicated by dashed straight lines in the figure. The ∗ (N, t), which is larger for smaller samstatistical error of ηC ∗ (t), will be totally ples and which leads to an uncertainty of ηC counted in Fig. 3. ∗ (t), deduced from Fig. 2, Using the percolation thresholds ηC ∗ (t) as a function of t −1/2 . For we plot in Fig. 3 the quantities ηC each temperature point the error bar is an over-estimation of ∗ (N, t) and the error due to the both the statistical error of ηC data extrapolation in Fig. 2. Importantly, in the whole range of temperature under study the simulation points can be well fitted with a straight line which describes the ES T −1/2 -law of Eq. (11) for 2D VRH (see the dashed straight line as an example). From the slope of reasonably fitting straight lines we obtain βC (2D) = 6.5 ± 1.0, close to the value 5.8 reported in Ref. [5] for the square lattice model as well as the value 6.2
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
∗ (t) is plotted versus t −1/2 . Fitting Fig. 3. The exponent of VRH-resistivity ηC straight lines describe the ES-law of Eq. (3) (see the dashed straight line as an example). The slope of fitting straight lines gives possible values of (βC /2)1/2 (see Eq. (11)).
calculated analytically by one of us [4]. Such an agreement of βC -values calculated by different methods for different models gives an argument to believe a universality of the law (3) with βC ≈ 6 for the single particle VRH in 2D systems with the Coulomb electron–electron interaction. In closing this section we would like to note that the value t = 1 of the simulation dimensionless temperature t 1/2 corresponds to an actual temperature of kB T = (e2 ND /κ) × 1/2 (ξ ND /2). Then, for instance, for n-GaAs MESFET measured in Ref. [14], the range of simulation temperatures in Fig. 3 corresponds to experimental temperatures of about 40 K to 1 K. 3. Gate screening Both the Mott and ES VRH laws have been observed in a great number of experiments. One also observed crossovers between two VRH regimes in the same sample as the temperature varies (see references in [7]). While the Mott to ES crossover as the temperature decreases is believed associating with the change in the form of DOS [19], the inverse crossover, ES to Mott, as the temperature decreases is still in discussion. Experimentally, the latter crossover has been observed by Van Keuls et al. [14] in the gated GaAs/Alx Ga1−x As heterostructure and by Yakimov et al. [10] in the 2D doped array of self-assembled germanium quantum dots. In both works the crossover was qualitatively assigned to the gate induced screening effect [20]. In the presence of a metallic gate the long range Coulomb interaction energy e2 /κr is replaced by the screening one of the form: V (r) = e2 /κ 1/r − 1/ r 2 + 4s 2 , (12) where s is the distance from the 2D layer to the metallic gate and plays the role of the screening length. For a given s, the screening energy (12) turns into the standard long-range Coulomb energy if the distance r is small enough, r 2s, and becomes vanished if r is large, r 2s. Since it is the longrange Coulomb interaction that makes the DOS vanished at the
407
Fig. 4. The simulation DOS in the case of gate-screened interaction energy of Eq. (12) for some values of the screening length s: (2) s = 2 and (3) s = 1. The un-screened DOS of Fig. 1 (curve (1)) and the gate-screening DOS calculated from Efros’s self-consistent equation (15) for the case of s = 1 (dashed line) are also included for a comparison.
Fermi energy, leading to the Coulomb gap (2), then, it is natural to expect that the gate screened interaction energy (12), vanishing at large distances ( 2s), should cause a smear of the Coulomb gap, making the DOS finite at EF . We have performed the GS-simulation, using the same program as mentioned in the preceding section, but everywhere the long-range Coulomb potential was correspondingly replaced by the screening one of Eq. (12). In Fig. 4 we show the simulation DOS obtained for two values of the screening length, s = 2 (curve 2) and 1 (curve 3). The DOS taken from Fig. 1 for the case of purely Coulomb potential (curve 1) and the gate-screened DOS calculated from the Efros’s self-consistent equation (see below) for the case of s = 1 (dashed line) are also included for a comparison. Obviously, the gate-screening causes a smear of the Coulomb gap, giving rise to a finite DOS at the Fermi energy. And, as expected, the smaller the length s, the stronger the screening effect becomes. Reminding that in −1/2 the present simulation s is measured in the units of ND , the sample of s = 1 should be then regarded as the very strong screening case. In practice, the distance from the 2D-layer to the gate is often greater than the average distance between −1/2 donors, ND (see, for example, [10,14]). Thus, Fig. 4 demonstrates how the DOS changes as the screening length varies (between ∞ for the un-screening case of curve (1) and 1 for the strongly screening case of curve (3)). Such a gate-induced change in the shape of DOS near EF should be manifested in the temperature-dependent behavior of VRH. Physically, the only characteristic length in the VRH problem is the typical hopping distance, which depends on the temperature as RH = (ξ/3)(TM /T )1/3 for Mott-VRH or RH = (ξ/4)(TC /T )1/2 for ES-VRH. The ratio between RH and the screening length s should be then discussed as the parameter, driving the gate screening effect in the VRH problem. One can merely suggest a crossover in the ρ(T )-behavior as the following. At relatively high temperatures, when RH 2s, the
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V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
is screened interaction energy of Eq. (12), the rE -dependence √ determined from the relation E = (e2 /κ)[1/r − 1/ r 2 + 4s 2 ]. Introducing dimensionless variables R, E, and G: g = κ/e2 s G, r = sR, = e2 /κs E, (14) we can rewrite Eq. (13) in the form of self-consistent equations [21]: −1
G
π (E) = exp −F (E) 2
∞
2 dE RE exp F (E ) ,
E
F (E) =
π 2
∞
2 dE RE G(E ) − G(E − E)
(15)
E
with Fig. 5. The gate induced ES to Mott VRH crossover: ln ηC is plotted versus ln T ∗ for two cases: s = ξ/2 (higher curve) and s = ξ (lower curve). The dashed straight lines describe the limiting regimes of Mott T −1/3 -VRH and ES T −1/2 -VRH.
energy (12) is practically Coulombic, the corresponding part (at relatively high energies) of DOS follows the linear form of Eq. (2), and therefore the VRH should obey the ES T −1/2 -law of Eq. (3). With decreasing temperature the typical hopping distance increases and at temperature low enough, when RH 2s, the interaction energy (12) becomes neglectfully small, i.e. the Coulomb interaction is strongly screened, the DOS is somewhat constant and remains finite around the Fermi energy, and consequently the Mott T −1/3 -law of Eq. (1) should be observed. In principle, using the GS with the DOS as shown in Fig. 4, one can solve the percolation problem in the same way as discussed in the preceding section and can search for the crossover of interest. We have done such calculations for some values of s. However, it seems that the crossover proceeds too slowly (see Fig. 5) and we were unable to reach the low-temperature limit of the fully screened Mott VRH even in the largest simulation sample and in the strongly screening regime. The problem is that the lower the temperature, the larger the quantity ηC (t) becomes, and therefore, the larger simulation sample should be used. Thus, it is practically impossible to fully observe the gateinduced Mott-to-ES crossover in the way of solving directly the percolation problem in our computer. In order to see the whole range of this crossover we describe below an alternative way of calculating ηC (t), based on the percolation approximation suggested in Refs. [3,4], but with the DOS deduced from the Efros’s self-consistent equation. In the 2D case, Efros’s equation for the DOS g() has the form [15]:
∞ π 2 dE g(E)r+E , g() = g∞ exp − 2
2 + 4. E = 1/RE − 1/ RE In the lowest approximation, setting F(E) to be zero, the integral (15) can be easily calculated that gives 2 2 π G−1 (E) = (16) + 8 / RE +4 . RE − RE 2 For the most interesting limit of energies close to the Fermi level, E → 0, corresponding to RE → ∞, this DOS can be further approximated as
3 1 1/3 1 + (E/2) . G(E) ≈ (17) 2π 2 Thus, to the lowest approximation the Efros’s self-consistent equation (13) gives the DOS at the Fermi level G(0) ≈ 1/2π ≈ 0.16 (in units of (e2 s/κ)−1 ). For a large range of energy, Eqs. (15) can be self-consistently solved. The DOS obtained for the case of s = 1 is presented by the dashed curve in Fig. 4, which seems to be in good agreement with the simulation DOS for the same screening length (curve (3)) in a narrow range of energy, close to the Fermi level, | − EF | 0.1. At higher energies two curves, simulation and self-consistent, diverge from each other. Due to the fact that the law (2) is nothing but the solution (to the first approximation) of the self-consistent equation (13) with the Coulomb interaction potential, this discrepancy of two curves in Fig. 4 and that discussed in the inset of Fig. 1 should have the same root. Next, to calculate the resistivity exponent ηC ≡ ln(ρ(T )/ρo ) we use the approximation based on the percolation statement that the average number of connections to a site at the percolation threshold is an invariant quantity, depending only on the dimensionality [3,13]. For 2D systems this quantity is equal to [4] ∞
(13)
0
where g∞ is the DOS at large energies (where the electron– electron interaction can be neglected) and the functional form of rE ≡ r(E) depends on the interaction. In the case of gate
πR 2 F(R, Ω)Θ(ηc − Ω/T ∗ − 2R/ξ ∗ ), 4 0 (18) where T ∗ and ξ ∗ are the dimensionless temperature and localization length, which for convenience in numerical calculations are defined as kB T = kB T ∗ (e2 /κs) and ξ = ξ ∗ s, respectively,
α=
dΩ
d 2R
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
Θ is the step function, and the distribution function 1 F(R, Ω) = 2
∞
dEi dEj G(Ei )G(Ej )δ Eij (R) − Ω
−∞
(19) is defined so that 2πF(R, Ω)R dR dΩ is the probability to find a donor pair in an area unit with the pair length within the range of [R, R + dR] and the energy Eij within the range of [Ω, Ω + dΩ]. Here the energy Eij is the dimensionless form of (8) with the Coulomb potential replaced by the screening potential (12): |Ei − Ej | − V (R), if Ei Ej < 0, Eij = max(|Ei |, |Ej |), if Ei Ej > 0, V = 1/R − 1/ R 2 + 4. Eq. (18) with F (R, Ω) from (19) can be transformed into the form T ∗ ηC Ω π2 ∗ ∗ 4 dΩ G(Ω)(T ∗ ηC − Ω)4 G(E) dE (ξ /2T ) 4 0 ξ ∗ ηC /2
+2
0 V (R)+T ∗ ηC −2RT ∗ /ξ ∗
dR R 3 0
W
abrupt that it may not be related to the gate. Since the screening energy (9) is a continuous function of r, and the hopping distance, in turn, continuously depends on the temperature, the gate-induced crossover should be then smoothly proceeded. As an alternative argument to understand the ES-to-Mott crossover observed, particularly, in the 2D doped array of selfassembled germanium quantum dots [10] we would like to mention the recent idea by Zhang and Shklovskii [23]. Following these authors an uncontrolled or intentional doping of an insulator around dots by impurities can lead to random charging of dots and therefore to a finite bare DOS at the Fermi energy. Then the Coulomb interaction between electrons in distant dots creates a soft Coulomb gap at the Fermi energy that leads to the ES VRH. Zhang and Shklovskii also show an oscillation of the DOS as the impurity concentration varies that may lead to a periodic transition between the Mott and ES VRH regime. Thus, one can assume that the metallic gate induces a change in the impurity concentration around dots, which causes a corresponding change in the form of DOS near the Fermi energy, and consequently, a crossover between two VRH regimes. 4. Conclusion
dW V (R)
dE G(E)G(W − E)
×
409
= α.
(20)
0
The quantity α is basically assumed independent of temperature. In the limit of an infinite temperature we simply have the “r-percolation” problem with α = BC2 /16 ≈ 1.2656, where BC = 4.5 [13]. Using this α-value and the DOS G(E) calculated self-consistently from (15) we can solve Eq. (20) numerically to get the percolation threshold ηC as a function of the dimensionless temperature T ∗ . The obtained results are shown in Fig. 5 for two cases, ξ ∗ = 1 and 2, in a very large range of T ∗ . There really exists a crossover between two VRH regimes, from ES (T ∗ )−1/2 -VRH (dashed line on the relatively high temperature side) to Mott (T ∗ )−1/3 -VRH (dashed line on the lower temperature side). However, it is also clear from the figure that even for the strongly screening case of ξ ∗ = 2, i.e. of small s, s = ξ/2, the range of temperature of the crossover is still very large, about three orders of magnitude. Actually, this can be understood, considering the gate screened DOS in Fig. 4. To maintain the true Mott VRH, there has to exist a finite interval of energy with a flat DOS around the Fermi level. In Fig. 4 such an energy interval is still not visible even in the strongly screening case of s = 1, though the DOS really remains finite at the Fermi level. Experimentally, Van Keuls et al. [14], on the one hand, claimed that their data fits well the crossover curve suggested in Ref. [20], and on the other hand, stated that in the samples measured [14] the correlation hopping process plays an important role. Though the correlation does not qualitatively change the VRH laws in two limiting regimes [5,14,22], it may quantitatively affect the crossover between them. As for the crossover observed in Ref. [10], the change between two regimes is so
We have simulated the ground state and the VRH in the 2D model of lightly doped semiconductors. The simulation shows a clear soft gap in the single particle DOS near the Fermi level EF . In the close vicinity of EF the simulation DOS fits well the linear law of the Coulomb gap of Eq. (2). For the VRH, our simulation results strongly support a universality of the ES T −1/2 -law of Eq. (3) for the single particle VRH with the coefficient βC ≈ 6. It should be here noted that though the many particle effect (correlation hopping) does not alter the qualitative behavior of this law, it may lower the hopping energy [18], leading to much smaller values of βC as observed in Refs. [14,17]. We have also studied the gate induced screening effect on the DOS and the VRH. Using the standard metallic gate screening potential, the DOS has been calculated by simulating the GS and by solving Efros’s self-consistent equation. The screening effect is found important making the DOS finite at the Fermi level. Further, using the self-consistent DOS and the percolation method, the exponent of VRH resistivity has been calculated in a large range of temperature for some values of the screening length. The crossover from ES to Mott VRH regime as the temperature decreases was recognized. However, the range of temperature of such a crossover is very large, about three orders of magnitude. It is worthy to check if many particle processes can accelerate the crossover, making it easy to be fully observable. A possible explanation of this crossover can be also found in Ref. [23]. Acknowledgements One of authors (V.L.N.) thanks Prof. Boris Shklovskii (Minneapolis) for very helpful discussion. He also thanks Profs. Woods Halley and Boris Shklovskii for kind hospitality at University of Minnesota, where parts of this work have been done.
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This work was supported by the Natural Science Council of Vietnam and VAST research grant on simulation of nanodevices. References [1] N.F. Mott, J. Non-Cryst. Solids 1 (1968) 1. [2] A.L. Efros, B.I. Shklovskii, J. Phys. C: Solid State Phys. 8 (1975) L49. [3] A.L. Efros, V. Lien Nguyen, B.I. Shklovskii, Solid State Commun. 32 (1979) 851. [4] V. Lien Nguyen, Sov. Phys. Semicond. 18 (1984) 207. [5] D.N. Tsigankov, A.L. Efros, Phys. Rev. Lett. 88 (2002) 176602. [6] V. Lien Nguyen, R. Rosenbaum, J. Phys.: Condens. Matter 10 (1998) 6083. [7] V. Lien Nguyen, R. Rosenbaum, Phys. Rev. B 56 (1997) 14980. [8] A.I. Yakimov, A.V. Dvurechenskii, A.V. Nenashev, A.I. Nikiforov, Phys. Rev. B 68 (2003) 205310. [9] E. Abrahams, S.V. Kravchenko, M.P. Sarachick, Rev. Mod. Phys. 73 (2001) 251. [10] A.I. Yakimov, A.V. Dvurechenskii, V.V. Kirienko, Yu.I. Yakovlev, A.I. Nikiforov, C.J. Adkins, Phys. Rev. B 61 (2000) 10868. [11] D. Yu, C. Wang, B.L. Wehrenberg, P.G. Guyot-Sionnest, cond-mat/ 0403597.
[12] A.L. Efros, V. Lien Nguyen, B.I. Shklovskii, J. Phys. C: Solid State Phys. 12 (1979) 1859. [13] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer-Verlag, Heidelberg, 1984. [14] F.W. Van Keuls, X.L. Hu, H.W. Jiang, A.J. Dahm, Phys. Rev. B 56 (1997) 1161. [15] A.L. Efros, J. Phys. C: Solid State Phys. 9 (1976) 2021. [16] G. Timp, A.B. Fowler, A. Hartstein, B.N. Butcher, Phys. Rev. B 33 (1986) 1499; G. Timp, A.B. Fowler, A. Hartstein, B.N. Butcher, Phys. Rev. B 34 (1986) 8771. [17] A.K. Savchenko, S.S. Safonov, S.H. Roshko, D.A. Bagrets, O.N. Jouravlev, Y.V. Nazarov, E.H. Linfield, D.A. Ritchie, Phys. Status Solidi B 241 (2004) 26. [18] V. Lien Nguyen, B.I. Shklovskii, A.L. Efros, Sov. Phys. Semicond. 13 (1979) 1281. [19] V. Lien Nguyen, Phys. Lett. A 207 (1995) 379. [20] I.L. Aleiner, B.I. Shklovskii, Phys. Rev. B 49 (1994) 13721. [21] V. Lien Nguyen, M.E. Raikh, A.L. Efros, Sov. Phys. Solid State 28 (1986) 1129. [22] A.M. Somoza, M. Ortuno, M. Pollak, Phys. Status Solidi C 1 (2004) 42. [23] J. Zhang, B.I. Shklovskii, Phys. Rev. B 70 (2004) 115317.
Variable range hopping in the Coulomb gap and gate screening in two dimensions V. Duc Nguyen a , V. Lien Nguyen a,∗ , D. Toi Dang b a Theoretical Department, Institute of Physics, VAST, P.O. Box 429 Bo Ho, Hanoi 10000, Vietnam b Physics Faculty, School of Natural Sciences, HSU, 90 Nguyen Trai, Hanoi, Vietnam
Received 8 August 2005; accepted 21 October 2005 Available online 15 November 2005 Communicated by J. Flouquet
Abstract We simulate the single particle ground state and variable range hopping (VRH) in the two-dimensional model of doped semiconductors. The obtained results show a universality of the linear Coulomb gap in the density of states at the Fermi energy as well as the Efros–Shklovskii (ES) T −1/2 -law for the temperature dependence of VRH resistivity. The screening due to the metallic gate smears the Coulomb gap, and therefore, may produce the crossover from ES to Mott VRH as the temperature decreases. The full range of temperature of such a crossover seems however to be very large, up to three orders of magnitude. 2005 Elsevier B.V. All rights reserved. PACS: 72.20.Ee; 72.80.Ng; 73.40.Gk
1. Introduction
ates a soft gap in the DOS at the Fermi energy (Coulomb gap):
The variable range hopping (VRH) is a main electric conduction mechanism in systems with strongly localized electronic states at low temperatures. It is a problem of long time discussion. Almost four decades ago Mott has suggested the famous law for the temperature dependence of VRH resistivity [1]: ρ(T ) = ρ0 exp (TM /T )1/d+1 ,
TM = βM /kB g0 ξ d ,
(1)
where ρ0 is a pre-exponential factor, depending weakly on temperature, d is the dimensionality, g0 is the density of states (DOS) at the Fermi level, ξ is the localization length and βM is a numerical coefficient, depending on d. The law (1) was obtained, assuming that the DOS near the Fermi level is a constant, g(E) ≡ g0 . In 1975 Efros and Shklovskii (ES) [2] pointed out that the long-range Coulomb interaction between localized states cre* Corresponding author.
E-mail address: [email protected] (V. Lien Nguyen). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.094
d g(E) = (d/π) κ/e2 |E|d−1 ,
(2)
where κ is the dielectric constant, e is the elementary charge and the dimensionality d = 2, 3. The energy E in this expression is counted from the Fermi energy. Extending Mott’s argument of calculating ρ(T ) to the case of DOS with the Coulomb gap (2), ES obtained the well-known T −1/2 -law: ρ(T ) = ρ0 exp (TC /T )1/2 ,
TC = βC e2 /kB κξ.
(3)
The coefficient βC has been evaluated using a simple percolation approximation: βC ≈ 2.7 [3] and ≈ 6.2 [4] for d = 3 and 2, respectively. The former number 2.7 is in very good agreement with the simulation result βC (d = 3) ≈ 2.6 obtained by solving numerically the percolation problem in the so-called doped semiconductor model [3]. The latter number 6.2 is close to the value βC (d = 2) ≈ 5.8 reported by Tsigankov and Efros [5], simulating VRH in the square lattice model. Experimentally, though the ES-law was observed in a great number of experiments for various bulk materials (see references in [6]) as well
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
as two-dimensional (2D) systems [7], there exists however a large discrepancy in the value of βC for the 2D-case [8]. The fact that the T −1/2 -VRH law of Eq. (3) is recently observed in a variety of 2D-systems with great interest such as the insulating side of (assumed) metal-insulating transition (for a review, see [9]), the 2D doped array of self-assembled germanium quantum dots on the silicon substrate [10], and CdSe-nanocrystal films [11], attracts a renewed attention on this VRH-law, and particularly, on the value of the coefficient βC (2D). Although the value of 6.2 derived in Ref. [4], as mentioned above, is well approved by the simulation data for the lattice model, in order to check a universality of laws (2) and (3), in this Letter we will simulate both the DOS and VRH, including the coefficient βC (2D), in another fundamental disordered model, the model of doped semiconductors, as considered in Refs. [3,12,13] for the three-dimensional (3D) case. Besides, as is generally suggested [10,14], in the gated 2D systems, the metallic gate screening may make the DOS finite at the Fermi energy, and therefore may cause a crossover from ES to Mott VRH as the temperature reduces. We will exclusively examine these effects, using Monte Carlo simulation and Efros’s selfconsistent equation method. For these aims, at first, the single particle ground state (GS) in the lightly doped-semiconductor 2D-model is simulated, using the GS-simulation method suggested in Ref. [12] for the 3D-model. Using the GS obtained we calculate the single particle DOS and check a universality of the Coulomb gap (2). Then, at the same GS we solve directly the percolation problem in computer and thereby calculate the temperature dependence of the exponent of VRH resistivity [13]: ln ρ(T )/ρ0 = ηC (T ), (4) where ηC is the percolation threshold. The simulation results seem to fit well the ES-law of Eq. (3) and give βC (2D) ≈ 6.5 ± 1.0, close to the value 6.2 calculated in [4] and to the value 5.8 reported in [5] for the square lattice model. Thereby, we demonstrate a universality of the ES-law of Eq. (3) with βC ≈ 6. For the case of gate screened electron–electron interaction potential, both the GS-simulation and Efros’s selfconsistent equation [15] are in good agreement showing an expected smear of the Coulomb gap. Moreover, numerical calculations of the exponent of VRH-resistivity, ηC (T ), using the obtained self-consistent DOS and the percolation approximation suggested in Refs. [3,4], really show a gate induced ES to Mott VRH crossover. However, the range of temperature of such a crossover may be very large, up to three orders of magnitude. 2. Model and simulation We consider the 2D n-type doped semiconductor model with N donors and NA = CN acceptors (the degree of compensation C < 1), arranged at random in a square of side L = (N/ND )1/2 . Thus, the donor concentration in the model is always equal to ND regardless of the number N . At zero temperature all acceptors, possessing excess electrons each by one, become negatively charged; an equal number (NA ) of donors, donating elec-
405
trons to acceptors, are charged positively; the rest, N (1 − C), of donors remains neutral. Thus, though the system is totally neutral, there is a random potential, created by charged impurities, which acts on donor states. Experimentally, the model suggested can be used to describe the impurity band and hopping conduction in, for example, the sodium-doped silicon MOSFET measured by Timp et al. [16] or n-GaAs MESFET measured by Savchenko et al. [17]. Without screening the total electrostatic energy of the system reads N e2 (1 − ni )(1 − nj ) H= κ |ri − rj | i>j N,N NA A (1 − ni ) 1 + , − (5) |ri − rl | |rl − rk | i,l
l>k
ri (rl )
is the coordinate of donor i (acceptor l), ni is the where occupation number: ni = 1 for a neutral donor and ni = 0 for an ionized one. Correspondingly, the one-electron energy at donor i is N N A (1 − nj ) 1 e2 i = (6) − . κ |ri − rl | |ri − rj | l
j
To simulate a single particle GS we minimize the total energy (5) with respect to all possible single electron transfers from a neutral to an ionized donor. The transfer of an electron from i to j is allowed only if it decreases H, i.e., if j − i − e2 /κrij > 0 (hereafter rij ≡ |ri − rj |). At the same time, as a rule, in GS the electrons occupy energy levels gradually from down, i.e., for any neutral donor i and ionized donor j there always has the relation j > i . The GS is the state, where both conditions stated are simultaneously satisfied. At the GS all the energies [i ] are known and therefore one can calculate the DOS and determine the Fermi energy EF . Since the simulation is restricted to only single electron transfers the DOS obtained is the single particle one. The program of the GS simulation as well as of the DOS calculation is similar to that discussed in very detail in Ref. [12]. In Fig. 1 we show the DOS for the sample of N = 25600. Throughout this work the degree of compensation C is chosen to be 0.5. The data in Fig. 1 was averaged over 200 realizations of random impurity coordinates and the statistical error is too small to be shown. We have checked and believed that at this sample size the finite size effect for the DOS is unimportant. There exists a clear soft gap in the DOS at the Fermi energy EF . The simulation DOS in the close vicinity of EF is also shown in a larger scale in the inset of this figure (solid curve) in comparison with the linear DOS of Eq. (2) (dashed straight line). Obviously, two results are practically coincident in the energy range of |ε − EF | 0.2. At higher energies, however, they gradually diverge from each other. The situation recognized is similar to that observed in the 3D-case (see Fig. 4 in Ref. [12]), where the discrepancy between the simulation DOS and the parabolic gap (2) has been related to a strong fluctuation of one-particle energies, which smears the simulation impurity
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V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
Fig. 1. The simulation DOS for the long-range Coulomb interaction case (N = 25600, C = 0.5, averaging over 200 random realizations). Inset: the simulation DOS (solid curve) is compared with the linear Coulomb gap of Eq. (2) (dashed line).
band. Besides, as the first approximation, the law (2) is expected to be valid only at the energies near the Fermi level. Now, using the set of donor coordinates [ri ] and energies [i ], already known at the GS, we can solve the standard percolation problem with the criteria [13]: ηij = 2rij /ξ + Eij /kB T ηC , where Eij =
|Ei − Ej | − e2 /κrij , max(|Ei |, |Ej |),
(7)
if Ei Ej < 0, if Ei Ej > 0.
(8)
The energy Ei as mentioned above is the one-electron energy measured from the Fermi level, Ei = i − EF . The percolation threshold ηC (T ), obtained at a given temperature, gives the exponent of VRH-resistivity (see Eq. (4)). In practical simulations Eq. (7) is reasonably rewritten in term of the dimensionless distance rij∗ and energy Eij∗ : ∗ , rij∗ + t −1 Eij∗ ηC
(9)
where t ≡ (2κ/e2 ND ξ )kB T is the dimensionless temperature ∗ is the effective percolation threshold: and ηC 1/2 ∗ ηC (10) = ξ ND /2 ηC . The simulation units of length and energy are here chosen to −1/2 1/2 be ND and e2 ND /κ, respectively. The percolation problem, formulated in the form of Eq. (9), is very convenient for a quantitative examination of VRH-laws, (1) or (3). For example, using Eqs. (4) and (10), the ES-law (3) takes the simple form ∗ = βC /2 t −1/2 , ηC (11) which provides an easy way for determining the coefficient βC . Following the standard procedure of solving percolation problems [13,18], for a given dimensionless temperature t , the ∗ (t) can be obtained by (i) calculating percolation threshold ηC
∗ (t, N ) versus N −1/2 for various t (from top): Fig. 2. Finite size effect: ηC −1 t = 40, 30, 20, 15, 10, 5, 2 and 1. The number of random realizations used ∗ (t, N ) decreases from 3200 to for calculating ensemble-averaged quantities ηC 100 as N increases from 800 to 25600, respectively. The fitting dashed straight ∗ (t, ∞) ≡ η∗ (t) in the limit of N → ∞. lines give ηC C
∗ (N, t) for finite ensemble-averaged percolation thresholds ηC ∗ (N, t) as systems of different N and then (ii) extrapolating ηC ∗ ∗ N → ∞ to get ηC (t) ≡ ηC (∞, t) for an infinite system. The ∗ (N, t) as well as the procedure of exprogram of simulating ηC ∗ (t) trapolation of these ensemble-averaged quantities to gain ηC were described in very detail in Ref. [18]. ∗ (N, t) are plotted In Fig. 2 ensemble-averaged quantities ηC versus the inverse of square-system side N −1/2 for different N , from 800 to 25600, and at various temperatures t, from 1 down to 0.025. The number of realizations of random impurity arrangements used for calculating these averaging quantities, depends on N as given in the figure. Apparently, all the simulation points, belonging to each value of temperature t , ∗ (N, t) = η∗ (∞, t) × follow well a linear scaling relation, ηC C ∗ ∗ (∞, t) in the limit of −1/2 (1 + const N ), which gives ηC (t) = ηC N → ∞ as indicated by dashed straight lines in the figure. The ∗ (N, t), which is larger for smaller samstatistical error of ηC ∗ (t), will be totally ples and which leads to an uncertainty of ηC counted in Fig. 3. ∗ (t), deduced from Fig. 2, Using the percolation thresholds ηC ∗ (t) as a function of t −1/2 . For we plot in Fig. 3 the quantities ηC each temperature point the error bar is an over-estimation of ∗ (N, t) and the error due to the both the statistical error of ηC data extrapolation in Fig. 2. Importantly, in the whole range of temperature under study the simulation points can be well fitted with a straight line which describes the ES T −1/2 -law of Eq. (11) for 2D VRH (see the dashed straight line as an example). From the slope of reasonably fitting straight lines we obtain βC (2D) = 6.5 ± 1.0, close to the value 5.8 reported in Ref. [5] for the square lattice model as well as the value 6.2
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
∗ (t) is plotted versus t −1/2 . Fitting Fig. 3. The exponent of VRH-resistivity ηC straight lines describe the ES-law of Eq. (3) (see the dashed straight line as an example). The slope of fitting straight lines gives possible values of (βC /2)1/2 (see Eq. (11)).
calculated analytically by one of us [4]. Such an agreement of βC -values calculated by different methods for different models gives an argument to believe a universality of the law (3) with βC ≈ 6 for the single particle VRH in 2D systems with the Coulomb electron–electron interaction. In closing this section we would like to note that the value t = 1 of the simulation dimensionless temperature t 1/2 corresponds to an actual temperature of kB T = (e2 ND /κ) × 1/2 (ξ ND /2). Then, for instance, for n-GaAs MESFET measured in Ref. [14], the range of simulation temperatures in Fig. 3 corresponds to experimental temperatures of about 40 K to 1 K. 3. Gate screening Both the Mott and ES VRH laws have been observed in a great number of experiments. One also observed crossovers between two VRH regimes in the same sample as the temperature varies (see references in [7]). While the Mott to ES crossover as the temperature decreases is believed associating with the change in the form of DOS [19], the inverse crossover, ES to Mott, as the temperature decreases is still in discussion. Experimentally, the latter crossover has been observed by Van Keuls et al. [14] in the gated GaAs/Alx Ga1−x As heterostructure and by Yakimov et al. [10] in the 2D doped array of self-assembled germanium quantum dots. In both works the crossover was qualitatively assigned to the gate induced screening effect [20]. In the presence of a metallic gate the long range Coulomb interaction energy e2 /κr is replaced by the screening one of the form: V (r) = e2 /κ 1/r − 1/ r 2 + 4s 2 , (12) where s is the distance from the 2D layer to the metallic gate and plays the role of the screening length. For a given s, the screening energy (12) turns into the standard long-range Coulomb energy if the distance r is small enough, r 2s, and becomes vanished if r is large, r 2s. Since it is the longrange Coulomb interaction that makes the DOS vanished at the
407
Fig. 4. The simulation DOS in the case of gate-screened interaction energy of Eq. (12) for some values of the screening length s: (2) s = 2 and (3) s = 1. The un-screened DOS of Fig. 1 (curve (1)) and the gate-screening DOS calculated from Efros’s self-consistent equation (15) for the case of s = 1 (dashed line) are also included for a comparison.
Fermi energy, leading to the Coulomb gap (2), then, it is natural to expect that the gate screened interaction energy (12), vanishing at large distances ( 2s), should cause a smear of the Coulomb gap, making the DOS finite at EF . We have performed the GS-simulation, using the same program as mentioned in the preceding section, but everywhere the long-range Coulomb potential was correspondingly replaced by the screening one of Eq. (12). In Fig. 4 we show the simulation DOS obtained for two values of the screening length, s = 2 (curve 2) and 1 (curve 3). The DOS taken from Fig. 1 for the case of purely Coulomb potential (curve 1) and the gate-screened DOS calculated from the Efros’s self-consistent equation (see below) for the case of s = 1 (dashed line) are also included for a comparison. Obviously, the gate-screening causes a smear of the Coulomb gap, giving rise to a finite DOS at the Fermi energy. And, as expected, the smaller the length s, the stronger the screening effect becomes. Reminding that in −1/2 the present simulation s is measured in the units of ND , the sample of s = 1 should be then regarded as the very strong screening case. In practice, the distance from the 2D-layer to the gate is often greater than the average distance between −1/2 donors, ND (see, for example, [10,14]). Thus, Fig. 4 demonstrates how the DOS changes as the screening length varies (between ∞ for the un-screening case of curve (1) and 1 for the strongly screening case of curve (3)). Such a gate-induced change in the shape of DOS near EF should be manifested in the temperature-dependent behavior of VRH. Physically, the only characteristic length in the VRH problem is the typical hopping distance, which depends on the temperature as RH = (ξ/3)(TM /T )1/3 for Mott-VRH or RH = (ξ/4)(TC /T )1/2 for ES-VRH. The ratio between RH and the screening length s should be then discussed as the parameter, driving the gate screening effect in the VRH problem. One can merely suggest a crossover in the ρ(T )-behavior as the following. At relatively high temperatures, when RH 2s, the
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V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
is screened interaction energy of Eq. (12), the rE -dependence √ determined from the relation E = (e2 /κ)[1/r − 1/ r 2 + 4s 2 ]. Introducing dimensionless variables R, E, and G: g = κ/e2 s G, r = sR, = e2 /κs E, (14) we can rewrite Eq. (13) in the form of self-consistent equations [21]: −1
G
π (E) = exp −F (E) 2
∞
2 dE RE exp F (E ) ,
E
F (E) =
π 2
∞
2 dE RE G(E ) − G(E − E)
(15)
E
with Fig. 5. The gate induced ES to Mott VRH crossover: ln ηC is plotted versus ln T ∗ for two cases: s = ξ/2 (higher curve) and s = ξ (lower curve). The dashed straight lines describe the limiting regimes of Mott T −1/3 -VRH and ES T −1/2 -VRH.
energy (12) is practically Coulombic, the corresponding part (at relatively high energies) of DOS follows the linear form of Eq. (2), and therefore the VRH should obey the ES T −1/2 -law of Eq. (3). With decreasing temperature the typical hopping distance increases and at temperature low enough, when RH 2s, the interaction energy (12) becomes neglectfully small, i.e. the Coulomb interaction is strongly screened, the DOS is somewhat constant and remains finite around the Fermi energy, and consequently the Mott T −1/3 -law of Eq. (1) should be observed. In principle, using the GS with the DOS as shown in Fig. 4, one can solve the percolation problem in the same way as discussed in the preceding section and can search for the crossover of interest. We have done such calculations for some values of s. However, it seems that the crossover proceeds too slowly (see Fig. 5) and we were unable to reach the low-temperature limit of the fully screened Mott VRH even in the largest simulation sample and in the strongly screening regime. The problem is that the lower the temperature, the larger the quantity ηC (t) becomes, and therefore, the larger simulation sample should be used. Thus, it is practically impossible to fully observe the gateinduced Mott-to-ES crossover in the way of solving directly the percolation problem in our computer. In order to see the whole range of this crossover we describe below an alternative way of calculating ηC (t), based on the percolation approximation suggested in Refs. [3,4], but with the DOS deduced from the Efros’s self-consistent equation. In the 2D case, Efros’s equation for the DOS g() has the form [15]:
∞ π 2 dE g(E)r+E , g() = g∞ exp − 2
2 + 4. E = 1/RE − 1/ RE In the lowest approximation, setting F(E) to be zero, the integral (15) can be easily calculated that gives 2 2 π G−1 (E) = (16) + 8 / RE +4 . RE − RE 2 For the most interesting limit of energies close to the Fermi level, E → 0, corresponding to RE → ∞, this DOS can be further approximated as
3 1 1/3 1 + (E/2) . G(E) ≈ (17) 2π 2 Thus, to the lowest approximation the Efros’s self-consistent equation (13) gives the DOS at the Fermi level G(0) ≈ 1/2π ≈ 0.16 (in units of (e2 s/κ)−1 ). For a large range of energy, Eqs. (15) can be self-consistently solved. The DOS obtained for the case of s = 1 is presented by the dashed curve in Fig. 4, which seems to be in good agreement with the simulation DOS for the same screening length (curve (3)) in a narrow range of energy, close to the Fermi level, | − EF | 0.1. At higher energies two curves, simulation and self-consistent, diverge from each other. Due to the fact that the law (2) is nothing but the solution (to the first approximation) of the self-consistent equation (13) with the Coulomb interaction potential, this discrepancy of two curves in Fig. 4 and that discussed in the inset of Fig. 1 should have the same root. Next, to calculate the resistivity exponent ηC ≡ ln(ρ(T )/ρo ) we use the approximation based on the percolation statement that the average number of connections to a site at the percolation threshold is an invariant quantity, depending only on the dimensionality [3,13]. For 2D systems this quantity is equal to [4] ∞
(13)
0
where g∞ is the DOS at large energies (where the electron– electron interaction can be neglected) and the functional form of rE ≡ r(E) depends on the interaction. In the case of gate
πR 2 F(R, Ω)Θ(ηc − Ω/T ∗ − 2R/ξ ∗ ), 4 0 (18) where T ∗ and ξ ∗ are the dimensionless temperature and localization length, which for convenience in numerical calculations are defined as kB T = kB T ∗ (e2 /κs) and ξ = ξ ∗ s, respectively,
α=
dΩ
d 2R
V. Duc Nguyen et al. / Physics Letters A 349 (2006) 404–410
Θ is the step function, and the distribution function 1 F(R, Ω) = 2
∞
dEi dEj G(Ei )G(Ej )δ Eij (R) − Ω
−∞
(19) is defined so that 2πF(R, Ω)R dR dΩ is the probability to find a donor pair in an area unit with the pair length within the range of [R, R + dR] and the energy Eij within the range of [Ω, Ω + dΩ]. Here the energy Eij is the dimensionless form of (8) with the Coulomb potential replaced by the screening potential (12): |Ei − Ej | − V (R), if Ei Ej < 0, Eij = max(|Ei |, |Ej |), if Ei Ej > 0, V = 1/R − 1/ R 2 + 4. Eq. (18) with F (R, Ω) from (19) can be transformed into the form T ∗ ηC Ω π2 ∗ ∗ 4 dΩ G(Ω)(T ∗ ηC − Ω)4 G(E) dE (ξ /2T ) 4 0 ξ ∗ ηC /2
+2
0 V (R)+T ∗ ηC −2RT ∗ /ξ ∗
dR R 3 0
W
abrupt that it may not be related to the gate. Since the screening energy (9) is a continuous function of r, and the hopping distance, in turn, continuously depends on the temperature, the gate-induced crossover should be then smoothly proceeded. As an alternative argument to understand the ES-to-Mott crossover observed, particularly, in the 2D doped array of selfassembled germanium quantum dots [10] we would like to mention the recent idea by Zhang and Shklovskii [23]. Following these authors an uncontrolled or intentional doping of an insulator around dots by impurities can lead to random charging of dots and therefore to a finite bare DOS at the Fermi energy. Then the Coulomb interaction between electrons in distant dots creates a soft Coulomb gap at the Fermi energy that leads to the ES VRH. Zhang and Shklovskii also show an oscillation of the DOS as the impurity concentration varies that may lead to a periodic transition between the Mott and ES VRH regime. Thus, one can assume that the metallic gate induces a change in the impurity concentration around dots, which causes a corresponding change in the form of DOS near the Fermi energy, and consequently, a crossover between two VRH regimes. 4. Conclusion
dW V (R)
dE G(E)G(W − E)
×
409
= α.
(20)
0
The quantity α is basically assumed independent of temperature. In the limit of an infinite temperature we simply have the “r-percolation” problem with α = BC2 /16 ≈ 1.2656, where BC = 4.5 [13]. Using this α-value and the DOS G(E) calculated self-consistently from (15) we can solve Eq. (20) numerically to get the percolation threshold ηC as a function of the dimensionless temperature T ∗ . The obtained results are shown in Fig. 5 for two cases, ξ ∗ = 1 and 2, in a very large range of T ∗ . There really exists a crossover between two VRH regimes, from ES (T ∗ )−1/2 -VRH (dashed line on the relatively high temperature side) to Mott (T ∗ )−1/3 -VRH (dashed line on the lower temperature side). However, it is also clear from the figure that even for the strongly screening case of ξ ∗ = 2, i.e. of small s, s = ξ/2, the range of temperature of the crossover is still very large, about three orders of magnitude. Actually, this can be understood, considering the gate screened DOS in Fig. 4. To maintain the true Mott VRH, there has to exist a finite interval of energy with a flat DOS around the Fermi level. In Fig. 4 such an energy interval is still not visible even in the strongly screening case of s = 1, though the DOS really remains finite at the Fermi level. Experimentally, Van Keuls et al. [14], on the one hand, claimed that their data fits well the crossover curve suggested in Ref. [20], and on the other hand, stated that in the samples measured [14] the correlation hopping process plays an important role. Though the correlation does not qualitatively change the VRH laws in two limiting regimes [5,14,22], it may quantitatively affect the crossover between them. As for the crossover observed in Ref. [10], the change between two regimes is so
We have simulated the ground state and the VRH in the 2D model of lightly doped semiconductors. The simulation shows a clear soft gap in the single particle DOS near the Fermi level EF . In the close vicinity of EF the simulation DOS fits well the linear law of the Coulomb gap of Eq. (2). For the VRH, our simulation results strongly support a universality of the ES T −1/2 -law of Eq. (3) for the single particle VRH with the coefficient βC ≈ 6. It should be here noted that though the many particle effect (correlation hopping) does not alter the qualitative behavior of this law, it may lower the hopping energy [18], leading to much smaller values of βC as observed in Refs. [14,17]. We have also studied the gate induced screening effect on the DOS and the VRH. Using the standard metallic gate screening potential, the DOS has been calculated by simulating the GS and by solving Efros’s self-consistent equation. The screening effect is found important making the DOS finite at the Fermi level. Further, using the self-consistent DOS and the percolation method, the exponent of VRH resistivity has been calculated in a large range of temperature for some values of the screening length. The crossover from ES to Mott VRH regime as the temperature decreases was recognized. However, the range of temperature of such a crossover is very large, about three orders of magnitude. It is worthy to check if many particle processes can accelerate the crossover, making it easy to be fully observable. A possible explanation of this crossover can be also found in Ref. [23]. Acknowledgements One of authors (V.L.N.) thanks Prof. Boris Shklovskii (Minneapolis) for very helpful discussion. He also thanks Profs. Woods Halley and Boris Shklovskii for kind hospitality at University of Minnesota, where parts of this work have been done.
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