4 October 1999 Physics Letters A 261 Ž1999 108–113 www.elsevier.nlrlocaterphysleta Coulomb correlation effects in variable-range hopping thermopower Nguyen Van Lien a,1 , Dang Dinh Toi b a b Theoretical Department, Institute of Physics, P.O.Box 429 Bo Ho, Hanoi 10000, Viet Nam Physics Faculty, Hanoi National UniÕersity, 90 Nguyen Trai Str., Thanh-Xuan, Hanoi, Viet Nam Received May 1999; accepted 24 August 1999 Communicated by J Flouquet Abstract Expressions are presented for describing the variable-range hopping thermopower cross-overs from the Mott T Ž dy1.rŽ dq1.-behaviour to the temperature-independent behaviour as the temperature decreases for both two-dimensional Ž d s and three-dimensional Ž d s cases The cross-overs show a profound manifestation of the Coulomb correlation along with that observed in resistance cross-overs q 1999 Published by Elsevier Science B.V All rights reserved Introduction The variable-range hopping ŽVRH conception was first introduced by Mott w1,2x with his famous Ty1 rŽ dq1.-laws for the temperature dependence of resistivity: R Ž T s R exp Ž TMŽ d rT TMŽ d s b MŽ d r Ž k B G j d , 1r Ž dq1 , Ž where d s 2,3 is the dimensionality, j is the localization length, and b MŽ d are numerical coefficients The Mott optimizing argument in obtaining these laws consists of minimizing the exponent of the hopping probability, while the electron–electron interaction is assumed to be neglected, and consequently, the density of localized states is constant near the Fermi level, GŽ E ' G s constant Corresponding author Fax: q84-4-8349050; e-mail: nvlien@bohr.ac.vn Later, it was shown that w3x the Coulomb correlation between localized electron states leads to an appearance of a depressed gap in the density of states ŽDOS at the Fermi level, which, following Efros and Shklovskii ŽES w4,5x, has the form: d Gd Ž E s a d < E < dy1 , a d s Ž drp Ž kre , Ž where k is the dielectric constant and e the elementary charge The one-particle energy E is measured from the Fermi level The most observable manifestation of the Coulomb gap of Eq Ž2 is that the temperature dependence of VRH resistivity should behave as w4,5x: Žd R Ž T s R exp Ž TES rT Žd Žd T ES s b ES Ž e rk B kj , 1r2 ; Ž instead of the Mott laws of Eq Ž1 Experimentally, both the Mott laws of Eq Ž1 and the ES-laws of Eq Ž3 have been observed in a great number of measurements for various materials w5–8x 0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V All rights reserved PII: S - Ž 9 0 - Nguyen Van Lien, Dang Dinh Toi r Physics Letters A 261 (1999) 108–113 Moreover, for some materials the low-temperature measurements show a smooth cross-over from the Mott Ty1 rŽ dq1.-behaviour to the ES Ty1 r2-behaviour of VRH RŽT as temperature decreases Such a cross-over is considered as an evidence of the role of the Coulomb correlation at low temperatures To describe observed Mott–ES cross-overs, theoretically, there exist different arguments w9–12x Our theory w11,12x is based on using an ‘effective’ DOS of the form: < E < dy GdŽeff Ž E s a d Eddy1 dy1 , Ž Ed q < E < dy1 where Ed is a parameter, a d Eddy ' G The DOS of Eq Ž4 tend to the Mott constant DOS in the limit of < E < Ed and to the Coulomb gap DOS of Eq Ž2 in the opposite limit On the basis of the DOS Eq Ž4., using the standard Mott optimizing procedure, we obtained the following expressions for describing Mott–ES resistance cross-overs w11,12x: Fd Ž x y Ž dq1 rd Fd Ž x rdx Ž dF s Ž Q d pb MŽ d Edrk B TMŽ d rrj s 1rd Ž Edrk B T , Ž 1rd dr pb MŽ d Edrk B TMŽ d y1 rd = Fd x , Ž Ž Ž Ž hd s rrj q Ž Edrk B T x , Ž where x sgrEd with g being the optimum hopping energy, Q d s d Ž d y 1.Ž dy1.r2 d s 9r2 for d s and for d s 2, and where x t dy Fd Ž x s dt Ž dy1 1qt For a given value of d Ž2 or 3., by solving Eq Ž5 – Ž8 it is easy to obtain the exponent hd of the resistivity as a function of the temperature T : RŽT s R exphd ŽT The simple expressions of Eq Ž5 – Ž8., as was shown in Refs w11–14x, describe quite well the experimental Mott–ES resistance cross-overs observed in a y In xO y and a y Ni x Si 1yx films These cross-over expressions also predict the cross-over temperature H TcŽ d s d d Ž pb MŽ d d 2r2 1rd Ž Ž dq1 rd Ž d Edrk B TMŽ d TM , 109 In the limit of the Mott constant DOS, the expressions of Eq Ž5 – Ž8 give the Mott laws of Eq Ž1 with widely acceptable values of the coefficients: b MŽ2 s 27rp and b MŽ3 f 18.1 In the opposite limit of the ES Coulomb gap DOS they give ES laws of Ž2 Ž3 Eq Ž3 with b ES f and b ES f 7.27 A favourite of the cross-over expressions of Eq Ž5 – Ž8 is that in fitting these expressions to experimental data with a defined characteristic temperature TMŽ d deduced from the data, the temperature Td) s Edrk B is the only adjustable fitting parameter used Note also that the cross-over theory of Refs w11,12x has recently been generalized to describe VRH resistance cross-overs from the Mott Ty1 rŽ dq1.-behaviours to the soft gap Tyn -behaviours with any n from 1rŽ d q to 1, including the ES value of n s 1r2 as a special case w13,14x Thus, the role of the Coulomb correlation effects in temperature dependence of VRH resistivity is well understood both experimentally and theoretically, while much less is known about the role of this correlation in other transport properties The thermopower, as was originally noted by Mott w2x, is sensitive to the material parameters and is expected to provide a good test of the principle ideas of the transport theory of disordered systems On the other hand, the thermopower is easier to measure than other thermal transport coefficients most commonly studied In this work we present expressions for describing Mott–ES cross-overs in temperature dependences of VRH thermopower for both two-dimensional Ž2D and three dimensional Ž3D cases The obtained expressions show a profound manifestation of the Coulomb correlation along with that observed in VRH resistances Besides, they are simple and easy to be used in comparison with experiments Cross-over expressions Based on the percolation method the VRH thermoelectric power Žthermopower could be found as w15–18x: Ž which is in good agreement with measured values S s WreT , Ž 10 Nguyen Van Lien, Dang Dinh Toi r Physics Letters A 261 (1999) 108–113 110 where the transport energy W is given by Ws HEG Ž E p Ž E dE , G E p E dE Ž Ž H Ž 11 simply, though longishly To the terms linear in the small parameter gd g the VRH thermopowers obtained from Eq Ž11 – Ž13 are the following: For the 2D case Ž S Ž d s ' S : S s g Ž k B re Ž E2rk B T and where H H ž =u h y j , Ž 14 P2 s 3600 Ž C q ln2 Ž C q < E < q < EX < q < E y EX < y k BT / y Ž 120C q 3600C q 12300C q 13200C q4620 ln Ž C q q 75C y 76C Ž 12 Here u is the Heaviside step function, h is the percolation threshold which defines the exponent of the VRH resistivity, which is hd Ž d s 2,3 formulated in the previous section The expressions Eqs Ž10 and Ž11 show that, as for the VRH resistivity described above, the thermopower S is entirely determined by the form of the density GŽ E of localized states close to the Fermi level Qualitatively, Burn and Chaikin w19x suggested that for the Mott constant DOS the thermopowers depend on temperature as S ŽMott A T Ž dy1.rŽ dq1., while for the ES electron–electron correlation DOS of Eq Ž2 the VRH thermopowers are temperature-independent Hence, the temperature dependence of VRH thermopower seems to be much more sensitive to the electron–electron correlation than that of VRH resistivity To derive the expressions of VRH thermopower in a large range of temperature, covering both the high-limit of Mott constant DOS regime and the low-limit of ES electron–electron correlation Coulomb gap regime, we start from the ‘effective’ asymmetric DOS of the form: Gd Ž E s GdŽeff Ž E Ž q gd E , Q2 Ž T where p Ž E s dr dEX G Ž EX 2r P2 Ž T Ž 13 where the symmetric part GdŽeff Ž E is just the DOS of Eq Ž4 This symmetric part of the DOS does not give any contribution to the VRH thermopower w18x The asymmetric correction that responds to the VRH thermopower is assumed to be small, i.e it is assumed in Eq Ž13 that gd g< 1, where g is the optimum hopping energy Using the suggested DOS Gd Ž E of Eq Ž13 the expressions of Eqs Ž11 and Ž12 could be evaluated q 995C q 3640C q 7290C q 4620C, Q2 s 1800 Ž C q ln2 Ž C q y Ž 2400C q9000C q 10800C q 4200 ln Ž C q q 450C q 3200C q 6900C q 4200C ln Ž x q C Ž1yr q 1800C dr dx CŽ1yr yxq1 0 H H For the 3D case Ž S Ž d s ' S : S s g Ž k B re Ž E3rk B T P3 Ž T Q3 Ž T , where P3 s Ž 210C y 7000C y 6930C ln Ž C q q Ž y140C q 7350C y 6930 arctgC q Ž 75C y 1799C q 4620C q 6930C q 6300C Ž I1r2 y I2 y I3 y I4 , Q3 s 4200C Ž q C ln Ž q C q 4200 Ž y C arctgC q 630C y 2800C y 4200C q 12600C Ž I2 q I3 , and where 2 I1 s H0 dr ln Ž D I2 s H0 drarctg D ; I3 s H0 drH0 I4 s 1 q ; D D H0 drH0 arctg Ž D y x x2q1 dx ; Ž D y x ln Ž x q Ž Dyx q1 with D ' C Ž1 y r dx Ž 15 Nguyen Van Lien, Dang Dinh Toi r Physics Letters A 261 (1999) 108–113 In all the expressions of P2 , Q2 , P3 and Q3 presented above the temperature is just being in the quantity C defined as C ' hd Ž k B TrEd Since the exponent hd also depends on the temperature, the resistance expressions Ž5 – Ž8 should be included as the first part in the full expressions for VRH thermopowers To calculate VRH thermopowers at a given temperature one has first to solve Eq Ž5 – Ž8 in getting hd , and afterward, to put the obtained value of hd into Eq Ž14 Žor Eq Ž15 for further calculating Sd The factor gd , measuring an asymmetricalness of DOS, should be considered as a material parameter, which might even be negative w18x We would also note that, consistently, the resistance exponent hd in thermopower expressions Ž14., Ž15 should be calculated using the same DOS of Eq Ž13., including the asymmetric part ; gd g Such an inclusion, however, will lead to thermopower corrections, which are ; Žgd g and which are therefore assumed to be negligible small Discussion In the high-energy limit, when the DOS GdŽeff Ž E of Eq Ž4 tends to the Mott constant one, and therefore when the resistance expressions Ž5 – Ž8 give hd s ŽTMŽ d rT 1rŽ dq1 of the Mott law of Eq Ž1., the expressions Ž14., Ž15 give for 2D and 3D Mott VRH thermopowers the well-known expressions w15–17x, respectively, as: S Ž Mott s 16 g TMŽ2.2r3 T 1r3 k B2 re , Ž 16 S Ž Mott s 425 g 3TMŽ3.1r2 T 1r2 k B2 re Ž 17 111 tained by Zvyagin w18x and is very close to those obtained by Pollak and Friedman w16x, and by Overhof and Thomas w17x, the numerical coefficients in other expressions of Eqs Ž16 and Ž18., and Ž19 are, to our knowledge, new Certainly, the values of the coefficients in all the expressions of Eq Ž16 – Ž19 for the limit cases should be independent of the chosen model of DOS Thus, the obtained expressions of Eq Ž5 – Ž8 and Eqs Ž14 and Ž15 really describe the smooth VRH thermopower cross-overs from the Mott T Ž dy1.rŽ dq1.-behaviours of Eq Ž16 or Eq Ž17 to the temperature-independent behaviours of Eq Ž18 or Eq Ž19., respectively, as the temperature decreases The cross-over temperature TcŽ d of Eq Ž9 should keep having the same sense for the thermopower cross-overs It seems from Eq Ž16 – Ž19 that the VRH thermopower cross-overs are more sensitive to the temperature than the VRH resistance cross-overs above mentioned As an illustration, a solution of the cross-over expressions Ž5 – Ž8 and Eq Ž14 Žfor 2D or Eq Ž15 Žfor 3D is presented in Fig together with the limit expressions of Eqs Ž16 and Ž18 or Eqs Ž17 and Ž19., respectively The thermopowers are here measured in units of S ' Ž k B re gd Ed , and the temperature in units of Ed The values of the parameter Ž Edrk B TMŽ d are arbitrarily chosen for this figure as E2rk B TMŽ2 s 2.10y2 and E3rk B TMŽ3 s 10y3 Note In the opposite limit, when the DOS of Eq Ž4 takes the forms of the ES Coulomb gap of Eq Ž2., and therefore when the resistance expressions of Eq Žd Ž5 – Ž8 give hd s ŽT ES rT 1r2 of the ES law of Eq Ž3., we receive from Eqs Ž14 and Ž15 the 2D and 3D Coulomb gap VRH thermopowers, respectively, as follows: Ž2 S Ž ES s 43 98 g T ES k B re , Ž 18 87 159 Ž 19 S Ž ES s Ž3 g 3TES k B re We would like here to note that while the number 5r42 in Eq Ž17 exactly coincided with that ob- Fig The numerical solutions of cross-over expressions Žsolid lines are presented in together with the high limits of Eqs Ž16 and Ž17 Ždots and the low limits of Eqs Ž18 and Ž19 Ždashed lines.; S ' Ž k B r e gd Ed ; the parameters used: E2 r k B TMŽ2 s 2.10y2 Žfor 2D., E3 r k B TMŽ3 s10y3 Žfor 3D 112 Nguyen Van Lien, Dang Dinh Toi r Physics Letters A 261 (1999) 108–113 that in practical comparisons of these cross-over expressions with experiments since the characteristic temperatures TMŽ d could always be deducted from data; the only adjustable parameter used in fitting is Ed Fig shows a non-monotonous behaviour of the VRH thermopower with a slight minimum at some temperature between two limits for both 2D and 3D cases We assume that such a minimum may be resulted from a concurrence of two effects: the first is related to the hopping energy, which increases as the temperature increases; the second is related to the relative role of the asymmetrical part in the DOS, which is more essential at low temperatures Experimentally, one might expect that the thermopower laws of Eqs Ž16 and Ž17 should be observed along with the Mott conduction laws of Eq Ž1 However, there are very few data on VRH thermopowers could be found in the literature An early observation of the law S f T 1r2 in the fluorine-substituted magnetite Fe O4yx Fx had been reported by Graener et al.w20x Recently, measuring various transport characters of sintered semiconducting compositions FeŽNb1y xWx O4 in a large temperature range, Schmidbauer w21x parallelly observed the Mott Ty1 r4-law of Eq Ž1 and the thermopower T 1r2-law of Eq Ž17 with a negative sign of the factor g at not very low temperatures up to ; 300 K A positive VRH thermopower with a similar behavior was recognized in the transmutation-doped Ge:Ga at T F K by Andreev et al w22x The only 2D data we know are those reported by Buhannic et al w23x for the parallel thermopower in the layered intercalation compounds Fe x ZrSe with x s 0.09–0.2 In the temperature range of the Mott 2D VRH Ty1 r3-law of Eq Ž1., the VRH thermopower roughly follows the 2D VRH thermopower T 1r3-law of Eq Ž16 There are a number of reports on temperature-independent-like behaviours of VRH thermopowers at a low temperature w24x, but we not find any data available to compare with the expressions of Eqs Ž18 and Ž19 quantitatively The difficulties in observing low temperature VRH thermopowers might be due to: Ži the magnitude of thermopowers is often so small Ž< S < F 20 VKy1 that could even not dominate measurement errors; Žii VRH thermopower is very sensitive to the conditions in preparing measurement samples Žvacuum level, impurity content, deposition rate, substrate temperature , they might uncontrollably affect the form of the DOS and the position of the Fermi level.; Žiii a possible compensation between the thermopower related to the asymmetry of the DOS studied here and the Hubbard correlation contribution associated with the features of the electron distribution function, when two of these parts of thermopowers are opposite on sign w22x Regarding all these difficulties, we assume that the cross-over curves in the figure qualitatively describe the data for Fe x ZrSe presented in Fig Žfor 2D and the data for a–Ge films presented in Fig Žfor 3D in Ref w18x, including an existence of a shallow minimum Conclusion We have presented the expressions for describing the VRH thermopower cross-overs from the wellknown Mott T Ž dy1.rŽ dq1.-behaviours to the temperature-independent behaviours as the temperature decreases The expressions are obtained by using the ‘effective’ DOS of Eq Ž4., which tends to the Mott constant DOS in the high energy limit and tends to the ES Coulomb gap DOS in the opposite limit This form of DOS, on the one hand, is no other than the solution to the first approximation of Efros’s selfconsistent equations w25x Žwhile the zero approximation gives the Coulomb gap and, on the other hand, was previously suggested for describing the Mott–ES VRH resistance cross-overs w11,12x The obtained cross-over expressions are simple and show a profound manifestation of the Coulomb correlation They could also be extended for the whole class of the Mott to any soft gap regime cross-overs by a way similar to that for VRH resistance cross-overs w13,14x Note again that while the limit expressions of Eq Ž16 – Ž19 are well defined, independent of the chosen model of DOS, the energy Ed Žmeasure of the gap width should be used as an adjustable parameter in fitting theoretical cross-over curves to experimental data Thus, the Mott–ES VRH thermopower cross-overs could be described by the same way of percolation methods using the same ‘effective’ DOS of Eq Ž4 Nguyen Van Lien, Dang Dinh Toi r Physics Letters A 261 (1999) 108–113 and with the same fitting parameter Ed as for corresponding VRH resistance cross-overs We hope that the present work will stimulate further investigations of electron–electron correlation effects on the thermopower as well as other VRH transport characters which even might promise important technological applications w26x The interesting behaviours of the Coulomb gap, analyzed in recent works w27,28x should be manifested in the VRH transport properties To find possible relations between different VRH transport characters as was done in Ref w29x may also be interesting since, as is stated by Polyakov and Shklovskii w30x, the resistivity of the Ty1 r2-behaviour observed at the resistivity minimum in the quantum Hall effect should be considered as an effect of the 2D Coulomb gap Acknowledgements This work is partly supported by the collaboration fund from the Solid State Group of Lund 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Semiconductors ŽSpringer-Verlag, Berlin, 1984 w6x H Fritzche, M Pollak Žed., Hopping and Related Phenomena ŽSingapore: World Scientific, 1990 w7x M Pollak, B.I Shklovskii Žed., Hopping Transport in Solids