On the q-deformed Thermodynamics and q-deformed Fermi Level in Intrinsic Semiconductor

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1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 5, HIKARI Ltd, htts://doi.org/ /ast On the q-deformed Thermodynamics and q-deformed Fermi Level in Intrinsic Semiconductor Won Sang Chung 1, Eun Ji Jang and Jae Yoon Kim Deartment of Physics and Research Institute of Natural Science College of Natural Science Gyeongsang National University, Jinju , Korea Coyright c 2016 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim. This article is distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Abstract In this aer the q-deformed thermodynamic laws are investigated and q-deformed Fermi level in intrinsic semiconductor is discussed. Keywords: q-deformed thermodynamics; q-deformed Fermi level; intrinsic semiconductor 1 Introduction Statistical mechanics is the theoretical aaratus with which one studies the roerties of macroscoic systems. systems made u of many atoms or molecules. and relates those roerties to the system s microscoic constitution. There is a growing interest in generalizing the Boltzman-Gibbs statistical mechanics. Because the entroy lays a fundamental role in the statistical hysics, the entroy should be deformed so as to construct a new ( deformed theory. The first attemt has been accomlished by Tsallis [1, 2]. Based on the fact that Boltzman-Gibbs theory is not adequate for various comlex, natural, artificial and social system, he introduced the non-extensive entroy is given by 1 Corresonding author S q = k(σ W i 1 q i 1/q, (q > 0 (1

2 214 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim where we relaced q 1 q in the original exression given in the Refs [1,2]. The non-extensive Boltzman-Gibbs entroy has attracted much interest among hysicists, chemists and mathematicians who study thermodynamics of comlex system [3-4]. When the deformation arameter q goes to 0, the nonextensive entroy reduces to the ordinary one. Tsallis entroy can be written as S q = k W i=1 1 q i ln q i = k where q-logarithmic function is defined as W i=1 i ln q 1 i, (2 ln q x = { x q 1, q (x > 0, q 0 ln x (q = 0 and its inverse is given by { (1 + qx 1/q, (x, q R, q 0 e q (x = e x (q = 0 (3 (4 This entroy gives the MaxEnt robability distribution (MPD when we imose the following constraints where W i = 1, U q = i=1 i = 1 Z q e q ( βe i, (5 Z q = W i=1 E i 1 q i = const, (6 W e q ( βe i (7 i Recently, Chung [7] roosed another tye of deformed Boltzmann factor: i = 1 Z q [e q (E i ] β (8 This is different from the eq.(5. In this aer from the q-deformed thermodynamic laws, we derive the eq.(8 and use it to discuss the q-deformed Fermi level in intrinsic semiconductor. This aer is organized as follows: In section 2, we discuss the q-deformed thermodynamics. In section 3, we construct the q-deformed grand artition function and q-deformed Fermi-Dirac distribution. In section IV we discuss the modification of the Fermi level for an intrinsic semiconductor.

3 On the q-deformed thermodynamics The q-deformed thermodynamic laws The q-oerations that emerges from non-extensive statistical mechanics seems to rovide a natural background for its mathematical formulation. The definitions of q-sum and q-difference [ 5, 6] is given by x y = x + y + qxy x y = x y 1 + qy It can be easily checked that the oeration satisfies commutativity and associativity. For the oerator, the identity additive is 0. The q-exonential and q-logarithm have the following roerties: ln q (xy = ln q x ln q y e q (xe q (y = e q (x y ln q (x/y = ln q x ln q y e q (x/e q (y = e q (x y (10 In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat caacity as to maintain constant temerature, T, for the combined system. If our system is in state s 1, then there would be a corresonding number of microstates available to the reservoir. Denote this number by Ω R (s 1. Because the combined system is isolated, all microstates are equally robable. Let P (s i be the robability that our system is in state s i. Then, we have (9 P (s 1 P (s 2 = Ω R(s 1 Ω R (s 2. (11 Here, more generally, the entroy of the reservoir can be assumed to be exressed in terms of the q-logarithm: From the eq.(11 and the eq.(12, we have S R = ln q (Ω R (12 P (s 1 P (s 2 = e q(s R (s 1 e q (S R (s 2 = e q (S R (s 1 S R (s 2 (13 In the q-deformed thermodynamics, we assume that the q-deformed entroy is not additivity but q-additivity, which imlies that for two subsystem A, B, we have the following total entroy S = S A S B (14 If we define (x q = 1 ln(1 + qx (15 q

4 216 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim we can rewrite the eq.(14 as (S q = (S A q + (S B q (16 Consider two systems in thermal contact, A and B. Here we assume that the q-deformed total energy is not additive but q-additive: U = U A U B (17 or (U q = (U A q + (U B q (18 Two systems will exchange the q-deformed energy (with U not changing until the q-deformed entroy of the combined system is as big as ossible, at which oint we have equilibrium. This means that the derivative of (S q with resect to either (U A q or (U B q must vanish at equilibrium. We have d(s q = (S A q (U A q d(u A q + (S B q (U B q d(u B q = 0 (19 Using the fact that (U q = (U A q + (U B q remain invariant, we have d(u q = d(u A q + d(u B q = 0, (20 we have d(u B q = d(u A q (21 Inserting the eq.(21 into the eq.(19, we get (S A q (U A q = (S B q (U B q = β = 1 kt (22 In a canonical ensemble ( more recisely, generalized canonical ensemble, the deformed first law of thermodynamics is deformed into (S R (s 1 q (S R (s 2 q = β[(u R (s 1 q (U R (s 2 q ] (23 where U R (s i and E(s i denote the energies of the reservoir and the system at s i, resectively. Using the q-deformed conservation of energy U R (s i E(s i = 0, we have (S R (s 1 q (S R (s 2 q = β[(u R (s 1 q (U R (s 2 q ] = β[(e(s 2 q (E(s 1 q ] (24

5 On the q-deformed thermodynamics 217 Substituting the eq.(24 into the eq.(13, we have P (s 1 P (s 2 = e q (S R (s 1 S R (s 2 = ex ((S R (s 1 q (S R (s 2 q = ex [β((e(s 2 q (E(s 1 q ] = (ex [((E(s 2 q (E(s 1 q ] β = [e q(e(s 2 ] β [e q (E(s 1 ] β = [e q(e(s 1 ] β [e q (E(s 2 ] β (25 which imlies, for any state s of the system P (s = 1 Z [e q(e(s] β, (26 where Z is given by Z = s [e q (E(s] β, (27 and the index s runs through all microstates of the system. 3 Grand artition function The grand canonical ensemble is a generalization of the canonical ensemble where the restriction to a definite number of articles is removed. This is a more realistic reresentation of hysical systems than the canonical ensemble since one can rarely fix the total number of articles in a macroscoic system. In a grand canonical ensemble ( more recisely, generalized grand canonical ensemble, we assume that the q-deformed total article number is not additive but q-additive: N = N A N B (28 and q-deformed total energy is also q-additive: U = U A U B (29 We consider the case that the q-deformed total article number and q-deformed total energy remain fixed. Two systems will exchange energy and article (with U and N not changing until the q-deformed entroy of the combined system is as big as ossible, at which oint we have equilibrium. This means that the

6 218 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim derivative of (S q with resect to (U A q, (U B q, (N A q and (N B q must vanish at equilibrium. We have d(s q = (S A q (U A q d(u A q + (S B q (U B q d(u B q + (S A q (N A q d(n A q + (S B q (N B q d(n B q = 0 Using the fact that (U q = (U A q + (U B q remain invariant, we have we have Inserting the eq.(32 into the eq.(30, we get (30 d(u q = d(u A q + d(u B q = 0, (31 d(u B q = d(u A q (32 (S A q (U A q = (S B q (U B q = β = 1 kt Using the fact that (N q = (N A q + (N B q remain invariant, we have we have Inserting the eq.(35 into the eq.(33, we get (33 d(n q = d(n A q + d(n B q = 0, (34 d(n B q = d(n A q (35 (S A q (N A q = (S B q (N B q = µβ = µ kt (36 where µ is a chemical otential. In a grand canonical ensemble ( more recisely, generalized grand canonical ensemble, the deformed first law of thermodynamics is (S R (s 1 q (S R (s 2 q = β((e(s 1 q (E(s 2 q + βµ((n(s 1 q (N(s 2 q (37 From the eq.(37, we have P (s 1 P (s 2 which imlies, for any state s of the system where Z is given by = [e q(e(s 1 ] β [e q (N(s 1 ] µβ [e q (E(s 2 ] β [e q (N(s 2 ] µβ (38 P (s = 1 Z [e q(e(s] β [e q (N(s] µβ (39 Z = s [e q (E(s] β [e q (N(s] µβ (40 and the index s runs through all microstates of the system.

7 On the q-deformed thermodynamics q-deformed Fermi-Dirac statistics and Fermi level Now let us consider two-level system whose ground energy is zero and excited energy is E. In this case the q-deformed artition function is The internal energy is given by Z = 1 + [e q (E] β [e q (N] µβ (41 U = Ef q (E, (42 where the q-deformed Fermi-Dirac distribution function is f q (E = [e q (E] β [e q (N] µβ (43 We will examine how many allowed states are near an energy of interest, and the robability that those states will actually be filled with electrons. Density of states and article statistics concets are indisensable in study of bulk materials. The electrons at the bottom of a conduction band (and holes at the to of the valence band behave aroximately like free articles (with an effective mass m traed in a box. The conduction and valence band densities of states near the band edges in real materials, the mass m of the article is relaced by the aroriate carrier effective mass. The density of states for conduction band and valence band are given by N c (E = m n 2m n (E E c (E E π 2 3 c (44 m 2m (E v E N v (E = (E E π 2 3 v (45 Now let us assume that electrons and holes in each bands obey the q-deformed Fermi-Dirac distribution. In this case the q-deformed Fermi-Dirac distribution function is 1 f(e = ( (1 + qe β/q (1 + qe F β/q Then, the concentration of electrons in the conduction band at equilibrium is n 0 = f(en c (EdE = m n 2m n E Ec de π (1 + qe β/q (1 + qe F β/q E c E c (47

8 220 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim Because E c E F kt, we have n 0 m n 2m n de E E π 2 3 c (1 + qe β/q (1 + qe F β/q (48 E c If we relace u = E E c E E F (49 in the eq.(48, we get n 0 = m n 2m n (1 + qe π 2 3 F β/q (1 + qe c β/q (E c E F 3/2 1 0 du [ qe ] β/q F u u 1/2 (1 u β/q 5/2 1 + qe c = 2 mn 2m n (1+qE 3 π 2 3 F β/q (1+qE c β/q (E c E F 3/2 F 1 ( 3 2, 5 2 β q, β q, 5 2 ; 1, 1 + qe F 1 + qe c (50 where Aell s hyergeometric functions of two variables [11] is defined by F 1 (a, b, b, c; x, y = and the Pochhammer symbol is m=0 n=0 (a m+n (b m (b n (c m+n x m y n (51 (a n = a(a + 1(a + 2 (a + n 1, (n 1, (a 0 = 1 (52 The concentration of holes in the valence band at equilibrium is, 0 = Ev = m 2m Ev de E π 2 3 v E Because E F E v kt, we have (1 f(en v (EdE ( (1 + qe β/q (1 + qe F β/q (53 If we relace 0 m 2m π 2 3 in the eq.(54, we get Ev de E v E(1 + qe β/q (1 + qe F β/q (54 u = E v E E F E 0 = m 2m π 2 3 (55

9 On the q-deformed thermodynamics 221 (1+qE v β/q (1+qE F β/q (E F E v 3/2 1 m 2m π 2 3 (1+qE v β/q (1+qE F β/q (E F E v 3/2 F 1 0 [ du qe ] β/q F u u 1/2 (1 u β/q 5/2 1 + qe v ( 3 2, β q, β q, 5 2 ; 1, 1 + qe F 1 + qe v = 2 3 (56 Although we can exress n 0 and 0 in a closed from with a hel of the Aell s hyergeometric functions of two variables, we will consider the case of sufficiently small q case for the theory of intrinsic semiconductor where the concentration of electrons and concentration of holes are same. For small q, u to a first order in q, we have the following aroximation: [e q (E] β = (1 + qe β/q e βe ( qβe2 (57 The electron and hole concentrations in equilibrium described by derived formulas are valid whether the material is intrinsic or doed, which are aroximately given by n 0 = N c e E F Ec kt, 0 = N v e Ev E F kt (58 where and ( 2πm 3/2 [ ( N c = 2 n kt q h 2 8 kt + 3 ] 2 E c + E2 c EF 2 2kT ( 2πm kt 3/2 [ ( N v = q 15 h 2 8 kt E v + E2 F ] E2 v 2kT (59 (60 For intrinsic material, Fermi level E F lies at some intrinsic level E i near the middle of the band ga. The intrinsic electron and hole concentrations are: n i = Ñce E i Ec kt, i = Ñve Ev E i kt (61 where ( 2πm 3/2 [ ( Ñ c = 2 n kt q h 2 8 kt + 3 ] 2 E c + E2 c Ei 2 2kT (62 and ( 2πm kt 3/2 [ ( Ñ v = q 15 h 2 8 kt + 3 ] 2 E v + E2 i Ev 2 2kT (63 For intrinsic ( undoed semiconductor, the electron concentration n i and hole concentration i are same: n i = i = Ñ c Ñ v e Eg/kT (64

10 222 Won Sang Chung, Eun Ji Jang and Jae Yoon Kim The Fermi level for an intrinsic semiconductor is obtained by equating n i = i : E i = E F = E c + E v + 3 ( m kt 2 4 m n [ +q 1 4 (E2 c + Ev (E c + E v (kt E gkt 3 16 ( m m n ] kt (65 At room temerature kt is very small, so the above relation is aroximated as ( Ec + E v E i = E F = + 3 ( m ( kt 1 q q (E2 c + Ev 2 (66 m n Thus, the Fermi level for an intrinsic semiconductor in the q-deformed theory increases ( or decreases when q is negative ( or ositive. 5 Conclusion In this aer we formulated the q-deformed thermodynamics where the Boltzmann factor is given by i = 1 Z q [e q (E i ] β. We relaced the ordinary logarithm with the q-logarithm in defining the entroy. We deformed the thermodynamic law by using q-addition and q-difference, hence we obtained the abovementioned q-deformed Boltzmann factor. We constructed the q-deformed grand artition function and q-deformed Fermi-Dirac distribution. We alied them to the semiconductor hysics to obtain the modification of the Fermi level for an intrinsic semiconductor. We found that the Fermi level for an intrinsic semiconductor in the q-deformed theory increases ( or decreases when q is negative (or ositive. Acknowledgements. This work was suorted by the National Research Foundation of Korea Grant funded by the Korean Government (NRF- 2015R1D1A1A and by the Gyeongsang National University Fund for Professors on Sabbatical Leave, References [1] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52 (1988, htts://doi.org/ /bf [2] E. Curado, C. Tsallis, Generalized statistical mechanics: connection with thermodynamics, Journal of Physics A: Mathematical and General, 24

11 On the q-deformed thermodynamics 223 (1991, L69-L72. htts://doi.org/ / /24/2/004 [3] A. Cho, A fresh take on disorder, or disorderly science?, Science, 297 (2002, htts://doi.org/ /science [4] Sumiyoshi Abe, A. K. Rajagoal, A. Plastino et al., Revisiting disorder and tsallis statistics, Science, 300 (2003, htts://doi.org/ /science d [5] L. Nivanen, A. Le Mehaute, Q. Wang, Generalized algebra within a nonextensive statistics, Re. Math. Phys., 52 (2003, htts://doi.org/ /s ( x [6] E. Borges, A ossible deformed algebra and calculus insired in nonextensive thermostatistics, Physica A: Statistical Mechanics and its Alications, 340 (2004, htts://doi.org/ /j.hysa [7] Won Sang Chung, Some ossible q-exonential tye robability distribution in the non-extensive statistical hysics, Mod. Phys. Lett. B, 30 (2016, htts://doi.org/ /s Received: December 4, 2016; Published: February 25, 2017