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Hall effect in hopping regime
Author’s Accepted Manuscript Hall effect in hopping regime A. Avdonin, P. Skupiński, K. Grasza
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S0921-4526(15)30356-2 http://dx.doi.org/10.1016/j.physb.2015.12.024 PHYSB309291
To appear in: Physica B: Physics of Condensed Matter Received date: 15 October 2015 Accepted date: 14 December 2015 Cite this article as: A. Avdonin, P. Skupiński and K. Grasza, Hall effect in hopping regime, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.12.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Hall effect in hopping regime A. Avdonina,∗, P. Skupi´ nskia , K. Graszaa,b a
Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ ow 32/46, 02-668 Warszawa, Poland b Institute of Electronic Materials Technology, ul. W´ olczy´ nska 133, 01-919 Warszawa, Poland
Abstract A simple description of the Hall effect in the hopping regime of conductivity in semiconductors is presented. Expressions for the Hall coefficient and Hall mobility are derived by considering averaged equilibrium electron transport in a single triangle of localization sites in a magnetic field. Dependence of the Hall coefficient is analyzed in a wide range of temperature and magnetic field values. Our theoretical result is applied to our experimental data on temperature dependence of Hall effect and Hall mobility in ZnO. Keywords: D. Hopping Transport, D. Hall Effect, D. Theory, A. ZnO PACS: 72.10.Bg, 72.20.Ee, 72.20.My 1. Introduction The dc Hall effect (HE) in the hopping regime of conductivity was addressed theoretically a number of times in the past [1–7], and various mathematical expressions for the Hall coefficient RH and Hall mobility μH were proposed. However, to our knowledge a well established description of HE in the hopping regime does not exist. Most theories of the hopping HE are based on calculation of three-site-jump probabilities [1] and subsequent construction of an equivalent resistor network in order to calculate the electron transport coefficients. Here we present a different and simpler derivation of the expression for the Hall coefficient and Hall mobility in case of hopping ∗
Corresponding author Email address: [email protected] (A. Avdonin)
Preprint submitted to Solid State Communications
December 14, 2015
conductance. We derive an expression for the transverse Hall-current based on the existing theory of two-site jump rate and then, from the balance of the Hall-current and transverse (compensating) electric-field-current, we derive the expressions for RH and μH . Our paper is organized as follows. We first calculate difference of jump probabilities and Hall current Izm in magnetic field (Section 2). In Section 3 we express Izm using the known formula of two-site transition rate. Then in Section 4 we derive the expression for RH and μH . Comparison with experiment is given in Section 5. 2. Tunneling probabilities The Hall effect in the hopping regime is not caused by the Lorentz force, because in the hopping conduction an electron can only follow a limited number of paths defined by the electron localization sites (LS). It is commonly considered that HE in the hopping regime is related to the self-interference effect of the electron wave function which propagates along different hopping paths in the magnetic field. Thus, to observe the interference, at least three localization centers should be taken into account. This mechanism was applied for the first time by Holstein [1], for description of hopping HE. An electron trapped on a LS can jump to any of its neighbors. We will consider only a fraction of jumps which contribute to the net current. Because hopping probability decreases exponentially with the distance [8], let us consider only two closest neighbors of any occupied LS, from the half-space in the direction of the net electron flow. In this manner for most occupied LSs we can construct a triangle of sites, which will contribute to HE. In order to simplify the calculation we are going to average over all possible triangles, and consider an equilateral triangle with the side length Δr equal to average distance between localization sites (Fig. 1a). In the initial state an electron is located on the site 1 (Fig. 1a). We will calculate the probability to find the electron on the sites 2 and 3 in magnetic field. The difference in the occupation probabilities for the sites 2 and 3 will produce over time a nonuniform charge distribution and the Hall voltage. We will use for the calculation the Feynman’s “probability amplitude” approach [9]. We denote by ψ12 , ψ13 and ψ32 the probability amplitudes of hopping between pairs of sites 1-2, 1-3 and 3-2, without any other localization sites around. Let us calculate first the probability of a jump to the right localization site in the considered triangle. The probability amplitude that 2
Figure 1: (a) Averaged triangle of localization sites. The occupied LS is marked by the black circle. Arrows show two possible hopping paths leading to localization site 2. Angle θ describes the direction of propagation of acoustic wave. (b) Direction of the applied electric field (current flow), direction of uniform magnetic field and the direction of rotation of the magnetic vector potential. (c) Coordinate system and schematic picture of the considered electric currents.
the electron will be detected on site 2 is Ψr = ψ12 + ψ13 ψ32 .
(1)
The hopping paths, which are taken into account in this formula, are shown in Fig. 1a by arrows. Then the probability of hopping to the right will be Pr = Ψr Ψ∗r .
(2)
We will represent probability amplitudes in the form ψij = fij eiϕij , where fij and ϕij are real values. The amplitude fij is a measurable value which is related to the hopping rate and ϕij is the additional phase gained during the transition from one site to another. Due to the symmetry of the problem there is f12 = f13 ≡fa . We denote these values as fa . There is f32 ≡fb < fa because the transition from site 3 to 2 is not facilitated by applied electric field. Thus ψij can be written as ψ12 = fa eiϕ12 , ψ13 = fa eiϕ13 , ψ32 = fb eiϕ32 .
(3)
After substitution of Eq. (3) into Eq. (1) we get Ψr = fa eiϕ12 [1 + fb ei(ϕ13 +ϕ32 −ϕ12 ) ]. 3
(4)
Phases ϕij can be written [9] as ϕij = ϕij0 + ϕijH , where ϕij0 is the phase gained in the absence of magnetic field, and the ϕijH is the phase change induced by magnetic field. The expression for ϕijH is [9] ϕijH
e =−
j Adr,
(5)
i
and the expression in the exponent in Eq. (4) can be written as ⎧ 3 ⎫ 2 2 ⎨ ⎬ e Adr + Adr − Adr . ϕ13 + ϕ32 − ϕ12 = β − ⎭ ⎩ 1
3
(6)
1
Here β stands for ϕ130 + ϕ320 − ϕ120 . The sum from the parentheses can be transformed into 3
2 Adr +
1
Adr + 3
1 Adr = 2
Adr = BS = Φ.
(7)
1→3→2→1
In the last transformation we have used the Stokes’ theorem and relation ∇ × A = B. Since we assume a uniform magnetic field, the integral will result in the product BS, where B is the magnetic induction and S is the surface area outlined by the integration path 1 → 3 → 2 → 1. It is not obvious how this path looks like, but it is logical to assume that tunneling occurs along the shortest path. In such case S is the area of the triangle. Then, the expression in the parentheses equals to the magnetic flux through the triangle formed by three localization sites. Since the direction, followed by the integration path, coincides with the direction of rotation of the vector potential A (see Fig. 1a, b), the value of the integral is positive and hence ϕ13H + ϕ32H − ϕ12H = − e Φ < 0. After the substitution of Eqs. (7) → (6) → (4) → (2), we obtain the probability of electron jumping to the right
e 2 2 (8) Pr = fa 1 + fb + 2fb cos β − Φ . Since − e Φ < 0, the total phase in Eq. (8) will decrease and hence Pr will increase with the increasing magnetic field. 4
A similar consideration of the probability of electron jumping to the left (localization site 3) gives
e Pl = fa2 1 + fb2 + 2fb cos β + Φ . (9) This shows that the phase will increase and Pl will decrease with the increasing magnetic field. Thus, electron will tend to deviate more often to the right, which coincides with the direction of deviation that would be induced by the Lorentz force. In order to obtain normalized probabilities we introduce two new quantities Pr Pl Prn = , Pln = . (10) Pr + Pl Pr + Pl These are relative probabilities, indicationg the chance to find the electron on the right or left site, if the transition has already occurred. For the normalized values there is Prn + Pln = 1 and, in the absence of magnetic field, Prn (0) = Pln (0) = 12 , as one would expect in the symmetric system. The Hall voltage appears due to asymmetry between the probabilities of jumps to the right and left side of the triangle, and it is proportional to 2fb sin β sin e Φ Pr − Pl . = Prn − Pln = (11) Pr + Pl 1 + fb2 + 2fb cos β cos e Φ In the Appendix we show, that β = π/2, and hence this expression is simplified to 2fb sin e Φ . (12) Prn − Pln = 1 + fb2 Typical probability amplitude of the tunneling is small fb2 1, see the following section. Then the expression for the probability difference simplifies to e Φ . (13) Prn − Pln = 2fb sin This value changes periodically with the increasing magnetic field. Assuming a concentration of the localization centers to be 1019 cm−3 and magnetic field of B = 1 T, we estimate the phase change δφ = (e/)BS to be √ √ −2 2 3 about 2 × 10 . Here we have used S = Δr 4 , where Δr = (n/ 2)−1/3 and n is the concentration of localization sites. At such concentration the periodicity of the Hall voltage should be observable at B = 90 T, when δφ reaches 5
value of π/2. This critical field will decrease at smaller concentrations of localization sites. In small magnetic fields, when sin x ≈ x, the probability difference and Hall voltage will depend linearly on magnetic field e Prn − Pln = 2 fb BS.
(14)
The value of Prn − Pln indicates the net part of the tunneling events, contributing to a net current Izm along the z-axis, perpendicular to the applied current Ix (see Fig. 1c). Here index m shows that this is the current along the z-axis, induced by magnetic field. To calculate this horizontal current, the value Prn − Pln should be multiplied by the externally applied current along x, flowing thorough the considered triangle Izm = −Ix (Prn − Pln ).
(15)
Calculation of the current Ix and of the balancing current, opposite to Izm , along the z-axis will be given Section 4. 3. Probability amplitude In this section we will derive a formula for the probability amplitude fb . The latter has a straightforward relation to the tunneling probability √ 2 fb = Pb , and hence fb = Pb . A problem with the tunneling probability is that it depends on time. Thereby such probability is characterized by a probability increase rate, i.e. time derivative of probability γb = dPb /dt. For very short time intervals δt the tunneling probability can be calculated as Pb = γb δt. Thus for probability amplitude we obtain fb = γb δt. (16) There exists a reliable formula for the tunneling probability rate γb in case of phonon-assisted hopping [8, 10] 1 2Δr 0 γb = γb exp − , (17) a exp ΔE − 1 kB T
where γb0
E 2 ΔE = 1 5 4 πds
2e2 3κa
2
Δr a 6
2 2 −4 ΔEa 1+ . 2s
(18)
In this formula ΔE is the energy separation between initial and final states, Δr is the distance between localization sites, a = 2 κ/me2 is the Bohr’s radius of the localization, E1 is the acoustic deformation potential, d is the density of the crystal, s is speed of sound, κ = 4π 0 , where 0 is vacuum permittivity and is the relative permittivity of the crystal. It is not clear what time should be used in Eq. (16) and for the evaluation of the δt we propose the following considerations. The probability amplitude fb characterizes the electron transition between sites 3 and 2, perpendicular to the applied electric field. Before this transition occurs, however, the electron has to make a transition (1 → 3), and both these transitions occur during the time of transition (1 → 2). Thus we assume that average δt is equal to one half of the average two-site transition time Δt. This time may be evaluated using the uncertainty relation ΔEΔt > /2, where ΔE is the standard deviation of the energy in the initial state. In the above case it is the energy of absorbed phonon. Thus we use an estimated value δt =
. 4ΔE
For the probability amplitude (16) we then obtain γb fb = . 4ΔE
(19)
(20)
To estimate the value of fb we use the parameters for the silicon (d = 2.33 g/cm3 , s ≈ 8400 m/s, E1 = 9.6 eV [11], = 11.68). Assuming a concentration of localization sites (e.g. donors) of Nd = 1×1019 cm−3 , energy separation (Coulomb gap) of ΔE = 5 meV and temperature of T = 100 K, we obtain a = 3.3 nm, Δr = 5.2 nm, γb = 1.1 × 1012 s−1 and fb = 0.18. At smaller concentrations of localization sites and lower temperatures (below T = ΔE/kB T ) both γb and fb will decrease because of the exponential dependencies on Δr and T in Eq. (17). This proves that the condition fb2 1, which was assumed in Eq. (12), is correct for concentrations below 1019 cm−3 . 4. Hall coefficient and Hall mobility A macroscopic sample contains a large number of triangles. In a stationary state, the transverse flow of electrons caused by the imbalance of the 7
left/right jump probability is compensated by an opposite flow (see Ize in Fig. 1c). The opposite flow can be caused by a transverse electric field, produced by separated charges (drift flow) and by a flow caused by the nonuniform distribution of electrons in the transverse direction (diffusion flow). Similarly to the Hall effect of free electrons, we will assume (without demonstration) that the balancing flow in the bulk is caused by the transverse electric field, and the diffusion flow is essential only at the sample edges. The hopping rate of the electrons is given by Eq. (17). By considering only its exponential part, we write it as γb = C
exp
1 ΔE kB T
−1
.
(21)
In the presence of the transverse electric (Hall) field, electrons hopping to the right (3→2 in Fig. 1a) will “see” a higher potential barrier ΔEr = ΔE+ΔEel , and electrons hopping to the left will “see” a lower potential barrier ΔEl = ΔE − ΔEel . Here ΔEel is the additional energy acquired in the presence of electric Hall field. Thus Hall field will also introduce a difference between hopping rates to the right γbr and to the left γbl side. The probability of electron jumping to the right Pre under the action of transverse electric field can be calculated as Pre = γbr /(γbr + γbl ). The probability of left jump will be Ple = γbl /(γbl + γbr ). The flow in the horizontal direction is proportional to Pre − Ple . After some manipulation we obtain 2ΔEel 1 − exp kB T γbl − γbr . = (22) Pre − Ple = γbl + γbr 1 + exp 2ΔEel kB T
Because the transverse electric field is small, there is ΔEel ΔE and ΔEel kT . Thus Eq. (22) can be further simplified by expanding the exponential function Pre − Ple = −
eΔzEz ΔEel =− . kB T kB T
(23)
Here Ez is the intensity of electric Hall field. The value Pre − Ple indicates the net part of all tunneling events occuring in the direction of electric field Ez . The electric current through triangle under application of field Ez can be calculated by multiplying Pre − Ple by 8
the number of jumps per second γb b(1 − b) and by the electric charge, for every possible path 3ΔrEz b(1 − b). (24) Ize = e2 γb 2kB T Here b is the occupation probability, b(1−b) is the probability that the initial state is occupied and the final state is empty, index ze indicates that this is the current produced by the transverse electric field along z-axis. If we apply similar considerations to the current along the x-axis produced by the external electric field Ex , we obtain a formula similar to Eq. (24) √ 3ΔrEx 2 b(1 − b). (25) Ix = e γ b kB T Now we can calculate the electric filed Ez and the Hall coefficient RH . To calculate Ez we use the condition that current does not flow along the z-axis in the stationary state Izm + Ize = 0. (26) By substituting Eqs. (15), (24) and (25) into Eq. (26) we obtain 4e Ez = √ fb BSEx . 3
(27)
√
In this equation we can further substitute S = 43 (Δr)2 and Ex = jx /σ, where jx is the current density and σ is the conductance. Now we can calculate the Hall coefficient Ez e fb (Δr)2 . (28) RH = = Bjx σ The Hall mobility in the hopping regime is e μH = RH σ = fb (Δr)2 . (29) The conductance σ = j/E is obtained from Eq. (25) by dividing the current by √ √ the cross section area of a single triangle in the yz plane. It is 2 2Δr / 3 for tightly packed placement of localization sites. Thus we obtain 3e2 γb b(1 − b) . σ= √ 2kB T Δr
(30)
Until now we have assumed that the triangle of sites is perpendicular to magnetic field and the external electric field is in the plane of triangle. In 9
a random system the orientation of the triangle will vary with respect to vectors B and E. To account for this feature one needs to substitute Δr by Δr/2 in Eqs. (28), (29), (24) and (25), as a simple averaging shows. By substituting Eqs. (30), (20) into Eq. (28) and taking into account √ that (Δr)3 = 2/n, where n is the concentration of localization centers, we obtain kB T 1 √ RH = . (31) 6e nb(1 − b) ΔEγb We can now briefly analyze the temperature dependence of Hall coefficient. If we neglect the temperature dependence of b and ΔE in Eq. (31), √ the temperature dependence of RH stems from the T / γb, where γb is given by Eq. (17). In the low-temperature range, when exp(ΔE/kB T ) 1, we obtain ΔE RH ∼ T exp . (32) 2kB T At very low temperatures RH increases with an activation energy of ΔE/2. At higher temperatures this function has a minimum at T ≈ ΔE/(2kB ). In the high-temperature range, √ where exp(ΔE/kB T ) ≈ 1, RH increases with the temperature as RH ∼ T . 5. Comparison with experiment We have applied the above theory to our experimental data on ZnO. The information about the studied samples was reported elsewhere [12]. Our measurements of the Hall effect and electrical resistivity were carried out in Van der Pauw configuration on transparent as-grown and annealed monocrystalline ZnO samples with typical dimensions of 4 × 4 × 0.5 mm3 , in magnetic field of 1.5 T, and under applied current of 1–5 mA. An example of temperature dependence of the Hall coefficient of the asgrown n-type ZnO is shown in Fig. 2a (open circles). The appearance of the minimum in ln RH vs 1/T dependence is attributed to the transition from free-carrier transport to hopping transport mechanism [8]. A standard test for hopping transport is a linear dependence of ln σ versus 1 1/n plot (see Fig. 2b). According the model of Mott n = 4 and according T the model of Shklovskii-Efros n = 2 [8]. In our case the linear dependence at low temperatures has a much larger range for n = 2, suggesting that the model of Shklovskii-Efros [8], which assumes an existence of the Coulomb gap, describes our data better. 10
Figure 2: (a) Comparison of the experimental and theoretical temperature dependencies of Hall coefficient. Dashed and dotted lines show “free” and “hopping” components of RH . 1/4 1/2 (b) Logarithm of electrical conductivity vs (1/T ) and (1/T ) , for comparison with the models of Mott and Shklovskii-Efros, respectively. (c) Comparison of experimental and theoretical temperature dependence of Hall mobility.
11
In order to fit the experimental data, the transport of free carriers has to be also taken into account. For the fitting we have used the formula for RH , assuming a “two-band” transport Rf σf2 + Rh σh2 RH = , (σf + σh )2
(33)
where indexes f and h indicate free-carrier or hopping transport coefficients, respectively, σh and Rh are given by Eqs. (30) and (31) respectively, and Rf = 1/(nf e), σf = enf μf . Here μf and nf are the mobility and concentration of free electrons, given by (Nd − Na ) NC − kEdT nf = e B . (34) 2 Here NC is the density of states at the bottom of conduction band, Ed is activation energy of donors and Nd and Na are concentrations of donors and compensating acceptors: Nd ≡ n in Eq. (31), Na = b0 Nd , where b0 is the compensation coefficient. Occupation probability b in Eqs. (30) and (31) was calculated as b = (Nd − Na − nf )/Nd . During the fitting procedure, five coefficients were used as fitting parameters, ΔE, Nd , b0 , Ed and μf . The temperature dependence of μf was not taken into account. Among these, only the first three are used to describe hopping transport, the latter two are required for description of free carriers transport. The results of fitting procedure are shown in Fig. 2a. Here solid line shows the combined Hall coefficient calculated using Eq. (33), dotted and dashed lines show the RH for hopping and free electron transport, calculated separately. The obtained values of fitting parameters are ΔE = 4.6 meV, Nd = 6.6 × 1018 cm−3 , b0 = 0.2, Ed = 18.3 meV, μf = 130 cm2 V−1 s−1 . The Hall mobility for hopping transport was calculated using Eq. (29) with the same set of parameters’ values. The plot of the calculated and measured temperature dependence of Hall mobility is shown in Fig. 2c. The experimental values of RH and μH are obtained from two independent measurements. Since μH was calculated using the same parameters as RH , without additional fitting, the correspondence between experiment and theory is reasonably good.
12
6. Discussion The above theory does not take into account percolation effects and variable range hopping effects. It assumes that localization centers are equally spaced in a tight packing fashion and the energy separation between the occupied and empty states is the same. The last two assumptions are the main reasons of the discrepancy of the temperature dependence of RH and μH at low temperatures. In the present state Aof the theory all transport parameters depend on the temperature as exp ± T1 in the low-temperature range. Consideration of the effects of variable range hopping, as it was done by Mott or Shklovskii A2 1/n and Efros, will result in the dependence of exp ± T and will improve the agreement with the experiment. On the other hand, the obtained fitting parameters are quite reasonable. For example the concentration of donors of Nd = 6.6 × 1018 cm−3 , at the compensation level of b0 = 0.2, is in agreement with the concentration of free electrons of 2.7 × 1018 cm−3 , measured at room temperature. The obtained mobility of free electrons μf = 130 cm2 V−1 s−1 is also in agreement with the measured value of about 100 cm2 V−1 s−1 (Fig. 2c). Finally, we note that although it is usually assumed that hopping is important only at low temperatures, the presence of the hopping transport channel may influence the interpretation of the Hall effect even at room temperature. Figure 2a shows that neglecting the hopping conductivity at high temperatures will result in an underestimation of the donor activation energy Ed and an overestimation of the free electron concentration. It also will lead to an underestimated value of the Hall mobility and will distort the shape of the μH (T ) curve. 7. Summary The presented expressions of Hall coefficient RH and Hall mobility μH for hopping conduction require only three parameters: magnitude of the Coulomb gap Ed , concentration of localization sites n and occupation probability b. These expressions have been used to describe the temperature dependence of RH and μH in ZnO mono-crystal. The same set of parameters has provided a correct order of magnitudes of RH and μH , obtained from two independent measurements. The agreement between theory and experiment can be further improved by considering the variable range hopping. The 13
calculated non-linearity of Hall effect in the hopping regime is of the form eΦ RH ∼ sin . Acknowledgments This work was supported by the National Science Centre in Poland, under grant no.: UMO-2011/01/D/ST7/02657. Appendix A. Calculation of the phase β According to perturbation theory, the probability amplitude of a transition induced by a periodical perturbation is given by [13] ψij (t) =
exp[2πi(ν − νij )t] − 1 1 (j|F |i) , i 2πi(ν − νij )
(A.1)
where νij = ΔEij /(2π) depends on the energy separation between initial and final state, ν is the frequency of the perturbation. By taking into account that ν = νij , we can write ψij (t) =
1 (j|F |i)t. i
(A.2)
The (j|F |i) in the model of deformation potential has the following form [8, 10] iE1 I (j|F |i) = ΔE where
qNq 2V0 sd
2 e2 I= 3 κa
12
Δr a
qa 2 −2 1+ eiqΔr − 1 , 2
Δr exp − . a
(A.3)
(A.4)
Here q is the wave vector of the phonon, V0 is the volume of the crystal, Nq is the number of phonons with wave vector q, and the other symbols are the same as in Eq. (18). After substitution of (A.3) into (A.2) one can see that the phase of the iqΔr probability amplitude is determined by e − 1 factor. Thus qΔrij π ϕij0 = arg eiqΔrij − 1 = + . 2 2 14
(A.5)
From Fig. 1a, qΔr13 = qΔr cos(120◦ + θ), qΔr32 = qΔr cos(θ), qΔr12 = qΔr cos(60◦ + θ). By substituting these relations into (A.5) we obtain β = ϕ130 + ϕ320 − ϕ120 = π/2,
(A.6)
for any direction of q vector of the acoustic wave. References [1] T. Holstein, Hall effect in impurity conduction, Phys. Rev. 124 (1961) 1329–1347. doi:10.1103/PhysRev.124.1329. [2] L. Friedman, M. Pollak, Hall mobility due to hopping-type conduction in disordered systems, Phil. Mag. B 38 (1978) 173–189. doi:10.1080/13642817808245674. [3] P. Butcher, Calculation of hopping transport coefficients, Phil. Mag. B 42 (1980) 799–824. doi:10.1080/01418638008222328. [4] L. Friedman, M. Pollak, The Hall effect due to hopping conduction in the localized states of amorphous semiconductors, Journal de Physique Colloques 42 (1981) C4–87–C4–90. doi:10.1051/jphyscol:1981414. [5] M. Gruenewald, H. Mueller, P. Thomas, D. Wuertz, The hopping Hall mobility – a percolation approach, Solid State Comm. 38 (1981) 1011– 1014. doi:10.1016/0038-1098(81)90006-5. [6] H. B¨ottger, V. Bryksin, Hopping conduction in solids, Akademie-Verlag, Berlin, 1985. [7] Y. Gal’perin, E. German, V. Karpov, Hall effect under hopping conduction conditions, Sov. Phys. JETP 72 (1991) 193–200. [8] B. Shklovskii, A. Efros, Electronic properties of doped semiconductors, Springer-Verlag, Berlin, 1984. [9] R. Feynman, R. Leighton, M. Sands, Feynman lectures on physics, Addison-Wesley, US, 1964. [10] A. Miller, E. Abrahams, Impurity conduction at low concentrations, Phys. Rev. 120 (1960) 745–755. doi:10.1103/PhysRev.120.745. 15
[11] J.-S. Lim, X. Yang, T. Nishida, S. Thompson, Measurement of conduction band deformation potential constants using gate direct tunneling current in n-type metal oxide semiconductor field effect transistors under mechanical stress, Appl. Phys. Lett. 89 (2006) 073509. doi:10.1063/1.2245373. [12] K. Grasza, P. Skupi´ nski, A. Mycielski, E. L usakowska, V. Domukhovski, J. Cryst. Growth 310 (2008) 1823. doi:10.1016/j.jcrysgro.2007.11.128. [13] N. Mott, I. Sneddon, Wave Mechanics and its Applications, Clarendon Press, Oxford, 1948.
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PII: DOI: Reference:
S0921-4526(15)30356-2 http://dx.doi.org/10.1016/j.physb.2015.12.024 PHYSB309291
To appear in: Physica B: Physics of Condensed Matter Received date: 15 October 2015 Accepted date: 14 December 2015 Cite this article as: A. Avdonin, P. Skupiński and K. Grasza, Hall effect in hopping regime, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.12.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Hall effect in hopping regime A. Avdonina,∗, P. Skupi´ nskia , K. Graszaa,b a
Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ ow 32/46, 02-668 Warszawa, Poland b Institute of Electronic Materials Technology, ul. W´ olczy´ nska 133, 01-919 Warszawa, Poland
Abstract A simple description of the Hall effect in the hopping regime of conductivity in semiconductors is presented. Expressions for the Hall coefficient and Hall mobility are derived by considering averaged equilibrium electron transport in a single triangle of localization sites in a magnetic field. Dependence of the Hall coefficient is analyzed in a wide range of temperature and magnetic field values. Our theoretical result is applied to our experimental data on temperature dependence of Hall effect and Hall mobility in ZnO. Keywords: D. Hopping Transport, D. Hall Effect, D. Theory, A. ZnO PACS: 72.10.Bg, 72.20.Ee, 72.20.My 1. Introduction The dc Hall effect (HE) in the hopping regime of conductivity was addressed theoretically a number of times in the past [1–7], and various mathematical expressions for the Hall coefficient RH and Hall mobility μH were proposed. However, to our knowledge a well established description of HE in the hopping regime does not exist. Most theories of the hopping HE are based on calculation of three-site-jump probabilities [1] and subsequent construction of an equivalent resistor network in order to calculate the electron transport coefficients. Here we present a different and simpler derivation of the expression for the Hall coefficient and Hall mobility in case of hopping ∗
Corresponding author Email address: [email protected] (A. Avdonin)
Preprint submitted to Solid State Communications
December 14, 2015
conductance. We derive an expression for the transverse Hall-current based on the existing theory of two-site jump rate and then, from the balance of the Hall-current and transverse (compensating) electric-field-current, we derive the expressions for RH and μH . Our paper is organized as follows. We first calculate difference of jump probabilities and Hall current Izm in magnetic field (Section 2). In Section 3 we express Izm using the known formula of two-site transition rate. Then in Section 4 we derive the expression for RH and μH . Comparison with experiment is given in Section 5. 2. Tunneling probabilities The Hall effect in the hopping regime is not caused by the Lorentz force, because in the hopping conduction an electron can only follow a limited number of paths defined by the electron localization sites (LS). It is commonly considered that HE in the hopping regime is related to the self-interference effect of the electron wave function which propagates along different hopping paths in the magnetic field. Thus, to observe the interference, at least three localization centers should be taken into account. This mechanism was applied for the first time by Holstein [1], for description of hopping HE. An electron trapped on a LS can jump to any of its neighbors. We will consider only a fraction of jumps which contribute to the net current. Because hopping probability decreases exponentially with the distance [8], let us consider only two closest neighbors of any occupied LS, from the half-space in the direction of the net electron flow. In this manner for most occupied LSs we can construct a triangle of sites, which will contribute to HE. In order to simplify the calculation we are going to average over all possible triangles, and consider an equilateral triangle with the side length Δr equal to average distance between localization sites (Fig. 1a). In the initial state an electron is located on the site 1 (Fig. 1a). We will calculate the probability to find the electron on the sites 2 and 3 in magnetic field. The difference in the occupation probabilities for the sites 2 and 3 will produce over time a nonuniform charge distribution and the Hall voltage. We will use for the calculation the Feynman’s “probability amplitude” approach [9]. We denote by ψ12 , ψ13 and ψ32 the probability amplitudes of hopping between pairs of sites 1-2, 1-3 and 3-2, without any other localization sites around. Let us calculate first the probability of a jump to the right localization site in the considered triangle. The probability amplitude that 2
Figure 1: (a) Averaged triangle of localization sites. The occupied LS is marked by the black circle. Arrows show two possible hopping paths leading to localization site 2. Angle θ describes the direction of propagation of acoustic wave. (b) Direction of the applied electric field (current flow), direction of uniform magnetic field and the direction of rotation of the magnetic vector potential. (c) Coordinate system and schematic picture of the considered electric currents.
the electron will be detected on site 2 is Ψr = ψ12 + ψ13 ψ32 .
(1)
The hopping paths, which are taken into account in this formula, are shown in Fig. 1a by arrows. Then the probability of hopping to the right will be Pr = Ψr Ψ∗r .
(2)
We will represent probability amplitudes in the form ψij = fij eiϕij , where fij and ϕij are real values. The amplitude fij is a measurable value which is related to the hopping rate and ϕij is the additional phase gained during the transition from one site to another. Due to the symmetry of the problem there is f12 = f13 ≡fa . We denote these values as fa . There is f32 ≡fb < fa because the transition from site 3 to 2 is not facilitated by applied electric field. Thus ψij can be written as ψ12 = fa eiϕ12 , ψ13 = fa eiϕ13 , ψ32 = fb eiϕ32 .
(3)
After substitution of Eq. (3) into Eq. (1) we get Ψr = fa eiϕ12 [1 + fb ei(ϕ13 +ϕ32 −ϕ12 ) ]. 3
(4)
Phases ϕij can be written [9] as ϕij = ϕij0 + ϕijH , where ϕij0 is the phase gained in the absence of magnetic field, and the ϕijH is the phase change induced by magnetic field. The expression for ϕijH is [9] ϕijH
e =−
j Adr,
(5)
i
and the expression in the exponent in Eq. (4) can be written as ⎧ 3 ⎫ 2 2 ⎨ ⎬ e Adr + Adr − Adr . ϕ13 + ϕ32 − ϕ12 = β − ⎭ ⎩ 1
3
(6)
1
Here β stands for ϕ130 + ϕ320 − ϕ120 . The sum from the parentheses can be transformed into 3
2 Adr +
1
Adr + 3
1 Adr = 2
Adr = BS = Φ.
(7)
1→3→2→1
In the last transformation we have used the Stokes’ theorem and relation ∇ × A = B. Since we assume a uniform magnetic field, the integral will result in the product BS, where B is the magnetic induction and S is the surface area outlined by the integration path 1 → 3 → 2 → 1. It is not obvious how this path looks like, but it is logical to assume that tunneling occurs along the shortest path. In such case S is the area of the triangle. Then, the expression in the parentheses equals to the magnetic flux through the triangle formed by three localization sites. Since the direction, followed by the integration path, coincides with the direction of rotation of the vector potential A (see Fig. 1a, b), the value of the integral is positive and hence ϕ13H + ϕ32H − ϕ12H = − e Φ < 0. After the substitution of Eqs. (7) → (6) → (4) → (2), we obtain the probability of electron jumping to the right
e 2 2 (8) Pr = fa 1 + fb + 2fb cos β − Φ . Since − e Φ < 0, the total phase in Eq. (8) will decrease and hence Pr will increase with the increasing magnetic field. 4
A similar consideration of the probability of electron jumping to the left (localization site 3) gives
e Pl = fa2 1 + fb2 + 2fb cos β + Φ . (9) This shows that the phase will increase and Pl will decrease with the increasing magnetic field. Thus, electron will tend to deviate more often to the right, which coincides with the direction of deviation that would be induced by the Lorentz force. In order to obtain normalized probabilities we introduce two new quantities Pr Pl Prn = , Pln = . (10) Pr + Pl Pr + Pl These are relative probabilities, indicationg the chance to find the electron on the right or left site, if the transition has already occurred. For the normalized values there is Prn + Pln = 1 and, in the absence of magnetic field, Prn (0) = Pln (0) = 12 , as one would expect in the symmetric system. The Hall voltage appears due to asymmetry between the probabilities of jumps to the right and left side of the triangle, and it is proportional to 2fb sin β sin e Φ Pr − Pl . = Prn − Pln = (11) Pr + Pl 1 + fb2 + 2fb cos β cos e Φ In the Appendix we show, that β = π/2, and hence this expression is simplified to 2fb sin e Φ . (12) Prn − Pln = 1 + fb2 Typical probability amplitude of the tunneling is small fb2 1, see the following section. Then the expression for the probability difference simplifies to e Φ . (13) Prn − Pln = 2fb sin This value changes periodically with the increasing magnetic field. Assuming a concentration of the localization centers to be 1019 cm−3 and magnetic field of B = 1 T, we estimate the phase change δφ = (e/)BS to be √ √ −2 2 3 about 2 × 10 . Here we have used S = Δr 4 , where Δr = (n/ 2)−1/3 and n is the concentration of localization sites. At such concentration the periodicity of the Hall voltage should be observable at B = 90 T, when δφ reaches 5
value of π/2. This critical field will decrease at smaller concentrations of localization sites. In small magnetic fields, when sin x ≈ x, the probability difference and Hall voltage will depend linearly on magnetic field e Prn − Pln = 2 fb BS.
(14)
The value of Prn − Pln indicates the net part of the tunneling events, contributing to a net current Izm along the z-axis, perpendicular to the applied current Ix (see Fig. 1c). Here index m shows that this is the current along the z-axis, induced by magnetic field. To calculate this horizontal current, the value Prn − Pln should be multiplied by the externally applied current along x, flowing thorough the considered triangle Izm = −Ix (Prn − Pln ).
(15)
Calculation of the current Ix and of the balancing current, opposite to Izm , along the z-axis will be given Section 4. 3. Probability amplitude In this section we will derive a formula for the probability amplitude fb . The latter has a straightforward relation to the tunneling probability √ 2 fb = Pb , and hence fb = Pb . A problem with the tunneling probability is that it depends on time. Thereby such probability is characterized by a probability increase rate, i.e. time derivative of probability γb = dPb /dt. For very short time intervals δt the tunneling probability can be calculated as Pb = γb δt. Thus for probability amplitude we obtain fb = γb δt. (16) There exists a reliable formula for the tunneling probability rate γb in case of phonon-assisted hopping [8, 10] 1 2Δr 0 γb = γb exp − , (17) a exp ΔE − 1 kB T
where γb0
E 2 ΔE = 1 5 4 πds
2e2 3κa
2
Δr a 6
2 2 −4 ΔEa 1+ . 2s
(18)
In this formula ΔE is the energy separation between initial and final states, Δr is the distance between localization sites, a = 2 κ/me2 is the Bohr’s radius of the localization, E1 is the acoustic deformation potential, d is the density of the crystal, s is speed of sound, κ = 4π 0 , where 0 is vacuum permittivity and is the relative permittivity of the crystal. It is not clear what time should be used in Eq. (16) and for the evaluation of the δt we propose the following considerations. The probability amplitude fb characterizes the electron transition between sites 3 and 2, perpendicular to the applied electric field. Before this transition occurs, however, the electron has to make a transition (1 → 3), and both these transitions occur during the time of transition (1 → 2). Thus we assume that average δt is equal to one half of the average two-site transition time Δt. This time may be evaluated using the uncertainty relation ΔEΔt > /2, where ΔE is the standard deviation of the energy in the initial state. In the above case it is the energy of absorbed phonon. Thus we use an estimated value δt =
. 4ΔE
For the probability amplitude (16) we then obtain γb fb = . 4ΔE
(19)
(20)
To estimate the value of fb we use the parameters for the silicon (d = 2.33 g/cm3 , s ≈ 8400 m/s, E1 = 9.6 eV [11], = 11.68). Assuming a concentration of localization sites (e.g. donors) of Nd = 1×1019 cm−3 , energy separation (Coulomb gap) of ΔE = 5 meV and temperature of T = 100 K, we obtain a = 3.3 nm, Δr = 5.2 nm, γb = 1.1 × 1012 s−1 and fb = 0.18. At smaller concentrations of localization sites and lower temperatures (below T = ΔE/kB T ) both γb and fb will decrease because of the exponential dependencies on Δr and T in Eq. (17). This proves that the condition fb2 1, which was assumed in Eq. (12), is correct for concentrations below 1019 cm−3 . 4. Hall coefficient and Hall mobility A macroscopic sample contains a large number of triangles. In a stationary state, the transverse flow of electrons caused by the imbalance of the 7
left/right jump probability is compensated by an opposite flow (see Ize in Fig. 1c). The opposite flow can be caused by a transverse electric field, produced by separated charges (drift flow) and by a flow caused by the nonuniform distribution of electrons in the transverse direction (diffusion flow). Similarly to the Hall effect of free electrons, we will assume (without demonstration) that the balancing flow in the bulk is caused by the transverse electric field, and the diffusion flow is essential only at the sample edges. The hopping rate of the electrons is given by Eq. (17). By considering only its exponential part, we write it as γb = C
exp
1 ΔE kB T
−1
.
(21)
In the presence of the transverse electric (Hall) field, electrons hopping to the right (3→2 in Fig. 1a) will “see” a higher potential barrier ΔEr = ΔE+ΔEel , and electrons hopping to the left will “see” a lower potential barrier ΔEl = ΔE − ΔEel . Here ΔEel is the additional energy acquired in the presence of electric Hall field. Thus Hall field will also introduce a difference between hopping rates to the right γbr and to the left γbl side. The probability of electron jumping to the right Pre under the action of transverse electric field can be calculated as Pre = γbr /(γbr + γbl ). The probability of left jump will be Ple = γbl /(γbl + γbr ). The flow in the horizontal direction is proportional to Pre − Ple . After some manipulation we obtain 2ΔEel 1 − exp kB T γbl − γbr . = (22) Pre − Ple = γbl + γbr 1 + exp 2ΔEel kB T
Because the transverse electric field is small, there is ΔEel ΔE and ΔEel kT . Thus Eq. (22) can be further simplified by expanding the exponential function Pre − Ple = −
eΔzEz ΔEel =− . kB T kB T
(23)
Here Ez is the intensity of electric Hall field. The value Pre − Ple indicates the net part of all tunneling events occuring in the direction of electric field Ez . The electric current through triangle under application of field Ez can be calculated by multiplying Pre − Ple by 8
the number of jumps per second γb b(1 − b) and by the electric charge, for every possible path 3ΔrEz b(1 − b). (24) Ize = e2 γb 2kB T Here b is the occupation probability, b(1−b) is the probability that the initial state is occupied and the final state is empty, index ze indicates that this is the current produced by the transverse electric field along z-axis. If we apply similar considerations to the current along the x-axis produced by the external electric field Ex , we obtain a formula similar to Eq. (24) √ 3ΔrEx 2 b(1 − b). (25) Ix = e γ b kB T Now we can calculate the electric filed Ez and the Hall coefficient RH . To calculate Ez we use the condition that current does not flow along the z-axis in the stationary state Izm + Ize = 0. (26) By substituting Eqs. (15), (24) and (25) into Eq. (26) we obtain 4e Ez = √ fb BSEx . 3
(27)
√
In this equation we can further substitute S = 43 (Δr)2 and Ex = jx /σ, where jx is the current density and σ is the conductance. Now we can calculate the Hall coefficient Ez e fb (Δr)2 . (28) RH = = Bjx σ The Hall mobility in the hopping regime is e μH = RH σ = fb (Δr)2 . (29) The conductance σ = j/E is obtained from Eq. (25) by dividing the current by √ √ the cross section area of a single triangle in the yz plane. It is 2 2Δr / 3 for tightly packed placement of localization sites. Thus we obtain 3e2 γb b(1 − b) . σ= √ 2kB T Δr
(30)
Until now we have assumed that the triangle of sites is perpendicular to magnetic field and the external electric field is in the plane of triangle. In 9
a random system the orientation of the triangle will vary with respect to vectors B and E. To account for this feature one needs to substitute Δr by Δr/2 in Eqs. (28), (29), (24) and (25), as a simple averaging shows. By substituting Eqs. (30), (20) into Eq. (28) and taking into account √ that (Δr)3 = 2/n, where n is the concentration of localization centers, we obtain kB T 1 √ RH = . (31) 6e nb(1 − b) ΔEγb We can now briefly analyze the temperature dependence of Hall coefficient. If we neglect the temperature dependence of b and ΔE in Eq. (31), √ the temperature dependence of RH stems from the T / γb, where γb is given by Eq. (17). In the low-temperature range, when exp(ΔE/kB T ) 1, we obtain ΔE RH ∼ T exp . (32) 2kB T At very low temperatures RH increases with an activation energy of ΔE/2. At higher temperatures this function has a minimum at T ≈ ΔE/(2kB ). In the high-temperature range, √ where exp(ΔE/kB T ) ≈ 1, RH increases with the temperature as RH ∼ T . 5. Comparison with experiment We have applied the above theory to our experimental data on ZnO. The information about the studied samples was reported elsewhere [12]. Our measurements of the Hall effect and electrical resistivity were carried out in Van der Pauw configuration on transparent as-grown and annealed monocrystalline ZnO samples with typical dimensions of 4 × 4 × 0.5 mm3 , in magnetic field of 1.5 T, and under applied current of 1–5 mA. An example of temperature dependence of the Hall coefficient of the asgrown n-type ZnO is shown in Fig. 2a (open circles). The appearance of the minimum in ln RH vs 1/T dependence is attributed to the transition from free-carrier transport to hopping transport mechanism [8]. A standard test for hopping transport is a linear dependence of ln σ versus 1 1/n plot (see Fig. 2b). According the model of Mott n = 4 and according T the model of Shklovskii-Efros n = 2 [8]. In our case the linear dependence at low temperatures has a much larger range for n = 2, suggesting that the model of Shklovskii-Efros [8], which assumes an existence of the Coulomb gap, describes our data better. 10
Figure 2: (a) Comparison of the experimental and theoretical temperature dependencies of Hall coefficient. Dashed and dotted lines show “free” and “hopping” components of RH . 1/4 1/2 (b) Logarithm of electrical conductivity vs (1/T ) and (1/T ) , for comparison with the models of Mott and Shklovskii-Efros, respectively. (c) Comparison of experimental and theoretical temperature dependence of Hall mobility.
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In order to fit the experimental data, the transport of free carriers has to be also taken into account. For the fitting we have used the formula for RH , assuming a “two-band” transport Rf σf2 + Rh σh2 RH = , (σf + σh )2
(33)
where indexes f and h indicate free-carrier or hopping transport coefficients, respectively, σh and Rh are given by Eqs. (30) and (31) respectively, and Rf = 1/(nf e), σf = enf μf . Here μf and nf are the mobility and concentration of free electrons, given by (Nd − Na ) NC − kEdT nf = e B . (34) 2 Here NC is the density of states at the bottom of conduction band, Ed is activation energy of donors and Nd and Na are concentrations of donors and compensating acceptors: Nd ≡ n in Eq. (31), Na = b0 Nd , where b0 is the compensation coefficient. Occupation probability b in Eqs. (30) and (31) was calculated as b = (Nd − Na − nf )/Nd . During the fitting procedure, five coefficients were used as fitting parameters, ΔE, Nd , b0 , Ed and μf . The temperature dependence of μf was not taken into account. Among these, only the first three are used to describe hopping transport, the latter two are required for description of free carriers transport. The results of fitting procedure are shown in Fig. 2a. Here solid line shows the combined Hall coefficient calculated using Eq. (33), dotted and dashed lines show the RH for hopping and free electron transport, calculated separately. The obtained values of fitting parameters are ΔE = 4.6 meV, Nd = 6.6 × 1018 cm−3 , b0 = 0.2, Ed = 18.3 meV, μf = 130 cm2 V−1 s−1 . The Hall mobility for hopping transport was calculated using Eq. (29) with the same set of parameters’ values. The plot of the calculated and measured temperature dependence of Hall mobility is shown in Fig. 2c. The experimental values of RH and μH are obtained from two independent measurements. Since μH was calculated using the same parameters as RH , without additional fitting, the correspondence between experiment and theory is reasonably good.
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6. Discussion The above theory does not take into account percolation effects and variable range hopping effects. It assumes that localization centers are equally spaced in a tight packing fashion and the energy separation between the occupied and empty states is the same. The last two assumptions are the main reasons of the discrepancy of the temperature dependence of RH and μH at low temperatures. In the present state Aof the theory all transport parameters depend on the temperature as exp ± T1 in the low-temperature range. Consideration of the effects of variable range hopping, as it was done by Mott or Shklovskii A2 1/n and Efros, will result in the dependence of exp ± T and will improve the agreement with the experiment. On the other hand, the obtained fitting parameters are quite reasonable. For example the concentration of donors of Nd = 6.6 × 1018 cm−3 , at the compensation level of b0 = 0.2, is in agreement with the concentration of free electrons of 2.7 × 1018 cm−3 , measured at room temperature. The obtained mobility of free electrons μf = 130 cm2 V−1 s−1 is also in agreement with the measured value of about 100 cm2 V−1 s−1 (Fig. 2c). Finally, we note that although it is usually assumed that hopping is important only at low temperatures, the presence of the hopping transport channel may influence the interpretation of the Hall effect even at room temperature. Figure 2a shows that neglecting the hopping conductivity at high temperatures will result in an underestimation of the donor activation energy Ed and an overestimation of the free electron concentration. It also will lead to an underestimated value of the Hall mobility and will distort the shape of the μH (T ) curve. 7. Summary The presented expressions of Hall coefficient RH and Hall mobility μH for hopping conduction require only three parameters: magnitude of the Coulomb gap Ed , concentration of localization sites n and occupation probability b. These expressions have been used to describe the temperature dependence of RH and μH in ZnO mono-crystal. The same set of parameters has provided a correct order of magnitudes of RH and μH , obtained from two independent measurements. The agreement between theory and experiment can be further improved by considering the variable range hopping. The 13
calculated non-linearity of Hall effect in the hopping regime is of the form eΦ RH ∼ sin . Acknowledgments This work was supported by the National Science Centre in Poland, under grant no.: UMO-2011/01/D/ST7/02657. Appendix A. Calculation of the phase β According to perturbation theory, the probability amplitude of a transition induced by a periodical perturbation is given by [13] ψij (t) =
exp[2πi(ν − νij )t] − 1 1 (j|F |i) , i 2πi(ν − νij )
(A.1)
where νij = ΔEij /(2π) depends on the energy separation between initial and final state, ν is the frequency of the perturbation. By taking into account that ν = νij , we can write ψij (t) =
1 (j|F |i)t. i
(A.2)
The (j|F |i) in the model of deformation potential has the following form [8, 10] iE1 I (j|F |i) = ΔE where
qNq 2V0 sd
2 e2 I= 3 κa
12
Δr a
qa 2 −2 1+ eiqΔr − 1 , 2
Δr exp − . a
(A.3)
(A.4)
Here q is the wave vector of the phonon, V0 is the volume of the crystal, Nq is the number of phonons with wave vector q, and the other symbols are the same as in Eq. (18). After substitution of (A.3) into (A.2) one can see that the phase of the iqΔr probability amplitude is determined by e − 1 factor. Thus qΔrij π ϕij0 = arg eiqΔrij − 1 = + . 2 2 14
(A.5)
From Fig. 1a, qΔr13 = qΔr cos(120◦ + θ), qΔr32 = qΔr cos(θ), qΔr12 = qΔr cos(60◦ + θ). By substituting these relations into (A.5) we obtain β = ϕ130 + ϕ320 − ϕ120 = π/2,
(A.6)
for any direction of q vector of the acoustic wave. References [1] T. Holstein, Hall effect in impurity conduction, Phys. Rev. 124 (1961) 1329–1347. doi:10.1103/PhysRev.124.1329. [2] L. Friedman, M. Pollak, Hall mobility due to hopping-type conduction in disordered systems, Phil. Mag. B 38 (1978) 173–189. doi:10.1080/13642817808245674. [3] P. Butcher, Calculation of hopping transport coefficients, Phil. Mag. B 42 (1980) 799–824. doi:10.1080/01418638008222328. [4] L. Friedman, M. Pollak, The Hall effect due to hopping conduction in the localized states of amorphous semiconductors, Journal de Physique Colloques 42 (1981) C4–87–C4–90. doi:10.1051/jphyscol:1981414. [5] M. Gruenewald, H. Mueller, P. Thomas, D. Wuertz, The hopping Hall mobility – a percolation approach, Solid State Comm. 38 (1981) 1011– 1014. doi:10.1016/0038-1098(81)90006-5. [6] H. B¨ottger, V. Bryksin, Hopping conduction in solids, Akademie-Verlag, Berlin, 1985. [7] Y. Gal’perin, E. German, V. Karpov, Hall effect under hopping conduction conditions, Sov. Phys. JETP 72 (1991) 193–200. [8] B. Shklovskii, A. Efros, Electronic properties of doped semiconductors, Springer-Verlag, Berlin, 1984. [9] R. Feynman, R. Leighton, M. Sands, Feynman lectures on physics, Addison-Wesley, US, 1964. [10] A. Miller, E. Abrahams, Impurity conduction at low concentrations, Phys. Rev. 120 (1960) 745–755. doi:10.1103/PhysRev.120.745. 15
[11] J.-S. Lim, X. Yang, T. Nishida, S. Thompson, Measurement of conduction band deformation potential constants using gate direct tunneling current in n-type metal oxide semiconductor field effect transistors under mechanical stress, Appl. Phys. Lett. 89 (2006) 073509. doi:10.1063/1.2245373. [12] K. Grasza, P. Skupi´ nski, A. Mycielski, E. L usakowska, V. Domukhovski, J. Cryst. Growth 310 (2008) 1823. doi:10.1016/j.jcrysgro.2007.11.128. [13] N. Mott, I. Sneddon, Wave Mechanics and its Applications, Clarendon Press, Oxford, 1948.
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