Materials Sciences and Applicatio ns, 2011, 2, 12251232 doi:10.4236/msa.2011.29166 Published Online September 2011 (http://www.SciRP.org/journal/msa) Copyright © 2011 SciRes. MSA 1225 Thermodynamic Properties of Semiconductors with Defects Vu Van Hung, Le Dai Thanh Department of Physics, Hanoi National Univ ersity of Education, Hanoi, Vietnam. Email: bangvu57@yahoo.com Received April 13th, 2011; revised May 22nd, 2011; accepted June 3rd, 2011. ABSTRACT Thermodynamic properties of diamond cubic and zincblende semiconductors with point defects are considered by the statistical moment method (SMM). The thermal expan sion coefficient, the specific h eats at constan t volume and those at constant pressure, CV and CP, and the isothermal compressibility are derived analytically for semiconductors with de fects. The SMM calculated thermodynamic quantities of the Si, and GaAs semiconductors with defects are in good agreement with the exp erimental results. Keywords: Anharmonic Defective Semiconductor, Statistical Moment Method 1. Introduction Recently, there has been a great interested in the study of bulk semiconductor, semiconductor heterostructures and nanodevices [14] since they provide us a wide variety of academic problems as well as the technological applica tions. The physical characteristics of semiconductors are determined both by the properties of the host crystal and by the presence impurities and crystalline defects. Crys tal lattice defects or other impurities also modify the properties of the semiconductor and thus may make a semiconductor unsuitable for its intended applications. The point defects in semiconductors including the va cancies play an important role in many properties of material. Understanding these defects will lead to im proved semiconductor devices for the technological ap plications. First principles (or ab initio) electronic structure com putations have been performed on semiconductor com pounds and the results compared with experiment [57]. More recently a large number of highquanlity calcula tions have been performed on groupIV, IIIV, and IIVI materials. Modern calculations allow the accurate re laxation of structures to their minimum energy configu rations and the incorporation of temperature effects. One can also study the melting of solids and phase transitions in Si using firstprinciples moleculardynamics method [8,9]. Such calculations are computationally expensive and, currently, simulations can only be run for periods of tens of picoseconds, which is not long enough for some of the processes of interest. In our previous papers [10,11] the statistical moment method was used to investigate the thermodynamic quantities of the elemental perfect semiconductors, tak ing into account the anharmonicity effects of thermal lattice vibrations. The thermal expansion coefficients, elastic moduli, specific heats at constant volume and those at constant pressure, CV and CP, are derived ana lytically for diamond cubic semiconductors. The purpo se of the present article is to investigate the temperature dependence of the thermodynamic proper ties of the semiconductors with defects using the analytic statistical moment method (SMM) [1216]. The ther modynamic quantities are derived from the Helmholtz fnee energy of semiconductors with defects. 2. Theory 2.1. Atomic Displacements of Semiconductor To derive the temperature dependence of the thermody namic properties of semiconductors, we use the statisti cal moment method. This method allows us to take into account the anharmonicity effects of thermal lattice vi brations on the thermodynamic quantities in the analytic formulations. The essence of the SMM scheme can be summarized as follows: for simplicity, we derive the thermodynamic quantities of crystalline materials with cubic symmetry, taking into account the higher (fourth) order anharmonic contributions in the thermal lattice vibrations going be
Thermodynamic Properties of Semiconductors with Defects 1226 yond the quasiHa monic (QH) approximation. The exten tions for the SMM formalism to noncubic systems is straightforward. The basic equations for obtaining ther modynamic quantities of the given crystals are derived in a following manner: the equilibrium thermal lattice ex pansions are calculated by the force balance criterion and then the thermodynamic quantities are determinded for the equilibrium lattice spacings. The anharmonic contri butions of the thermodynamic quantities are given ex plicitly in terms of the power moments of the thermal atomic displacements. Let us first define the lattice displacements We demote il the vector defining the displacement of the ith atom in the lth unit cell, from its equilibrium po sition. The po tential energy of the whole crystal is expressed in terms of the positions of all the atoms from the sites of the equilibrium lattice. We use the theory of small atomic vibrations, and expand the potential energy Uas a power series in the cartesian components, u il Uu i u of the displacement vector around this point. il For the evaluation of the anharmonic contributions to the free energy u , we consider a quantum system, which is influenced by supplemental forces i in the space of the generalized coordinates . i For simplicity, we only discuss monatomic systems and hereafter omit the indices on the sublattices Then, the Hamiltonian of the crystalline system is given by q l 0ˆ ii i Hq (1) where 0 denote the crystalline Hamiltonian without the supplementary forces i and upper huts repre sent operrators. The supplementary forces i are acted in the direction of the generalized coordinates The thermodynamic quantities of the anharmonic crystal (harmonic Hamiltonian) will be treated in the Einstein approximation. . i q After the action of the supplementary forces i the system passes into a new equilibrium state. If the 0th atom in the lattice is affected by a supplementary force p p , then the total force acting on it must be zero, and one gets the force balance relation as 23 00 ,,, 40 ,,, 11 24 10 12 ii ii ii ii iii eq eq iiii iiiii uu uu uuu uuu p uuuu i u (2) The thermal averages of the atomic displacements ii uu and iii uuu (called as second and third order moments) at given site i can be expressed in terms of the first moment i R u with the aid of the re curence formula [1214]. Then Equation (2) is trans formed into the new differential equation: 2 2 d y 32 22 2 31 10 dy yXy dp dy kyp dp m dp yX m (3) where j yu, coth xx , and ; B kT 2 x . In the above Equation (3), ,k and are defined by 22 0 6 i eq uu 0 2 44 0 2 2 30 1 2 1 12 i iieq i iiii eq i ijx jy u u uu 4 eq jz km u (4) In deriving Equation (3) we have imposed the symme try criterion for the thermal averages in the diamond cu bic lattices as jj j pppp uu uy j u Let us introduce the n ew variable y in the above Equa tion (3) 3 yy (5) Then we have the new differential equation instead of Equation (3) 2 32 2 dd 31 dd 0 yy yy X k pp Ky p y (6) where 2 2 2 ;; 3 212 1 273 3 Kkp pK k KX kk (7) For higher temperatures, the relation xcothx1 holds and Equation (6) is reduced to 2 23 2 dd 30 dd yy yyKyp pp (8) Copyright © 2011 SciRes. MSA
Thermodynamic Properties of Semiconductors with Defects 1227 The nonlinear differential equ ation of Equation (8) can be solved in the following manner: We expand the solu tion in terms of the “force” up to the second order as y* p * 01 2 *2 yApAp (9) where 1 and 2 are the constants [12]. The above Equation (8) is solved as 2 03 2 3 y K A (10) with 2233 44 123 468 55 66 56 10 12 4 aaa KKK aa KK a (11) and 12 34 32; 332;51;4583;aa aa 56 1589 3;1633.aa Here, 0 represents the atomic displacement for the case when the force is zero. The general solution of Equation (3) is solved as y* p * 00 22 2 04 3 16 12 1 332 ppK yy y k yK K7k (12) Once the thermal expansion 0 of the lattice is found, one can get the Helmholtz free energy of the system in the following form y 00 U1 (13) where 0 denotes the free energy in the harmonic ap proximation and 1 the anharmonic contribution to the free energy. The Helmholtz free energy of our system can be derived from the Hamiltonian H of the following form: 0 HV where 0 denote the Hamiltonian of the harmonic ap proximation, the parameter and V the anharmonic vibrational contributions. Following exactly the general formular in the SMM formulation [12], one can get the free energy of the system as 00 0 dUV (14) where V represents the Hamiltonian corresponding t o the anharmonicity contribution. Then the free energy of the system is given by 23 21 00 2 24 22 2112 3 3 3 2 2 2 2 2 32 13 32 412211 32 2 33 27 311 1(1)3 27 36 NX UX kk XX N X k NM kKN M kk X KKK k kK NM K 2 2 332 23 24 2 21 3 2 22 111 3( 33 6 2 31 27 9 231 3(1) 18 31 6 k NX KKk K kk k NM M KK k NaX Kk K k NM K 1) (15) where 1 2 1 3 2 3 a MK , and the second term of above Equation (15) denotes the harmonic contribution to the free energy 2 03ln1e x Nx (16) with the aid of the “real space” free energy formula ,ETS one can find the thermodynamic quantities of given systems. The thermodynamic quantities such as specific heats and elastic modul at temperature T are directly derived from the free energy of the system. 2.2. Thermodynamic Properties of Semiconductors with Defects The Gibbs free energy of crystals consisting of N atoms and vacancies has the form: nN 0 ,, f V GTPG TPngTS C (17) where 0,GTP is the Gibbs free energy of the perfect crystals consisting N atoms, , f V TP is the Gibbs en ergy change on forming a single vacancy, the en tropy of mixing: C S ! ln !! CB Nn Sk Nn (18) Copyright © 2011 SciRes. MSA
Thermodynamic Properties of Semiconductors with Defects 1228 From the minimization condition of t rgy of the crystal with point defectshe Gibbs free ene , we obtain the eq uilibrium concentration of the vacancies as [15,16] , exp f PT n (19) where is the change in the Gibbs fre the foon of a vacancy and can be given by e energy due to mati * 00 , f V TP uPV (20) It should be noted that pressure affects the diffusivity through both the free energies, * 0 , and change, resulting from the formatof the point defect, In the volume ion V. This change is due to the PV work done by the pressure medium against the volume change associated with defect formation and migrati o n. the above Equation (20), 00 11 () ,, iiijoij o uWrrr r rep , 23 ii j resent the inter nal energy associated with atom 0 and io the effective interaction energies between the oth and ith atoms, * o denotes the change in the Helmholtzenergy free of the central atom which creates the vacancy, by moving itself to a certain sink site in the crystal, and is given by ** 1 oo C (21) where 0 denotes the free energy of the central atom after mo a certain sink sites i simply regarded as a numerical factor. In the previous 7] oving tn the crystal, C is paper [1, we t a ke the ave rag e value for C as * 12o o u C (22) From Equations (20), (21), and (22 pression of the Gibbs energy change vacancy: ) we obtain the ex on forming a single 0 , f V u 2 TP PV (23) Form Equations (17) and (23) it is easy to ob expression for the Helmholtz free energ y o defects: tain the f crystals with 0 def C u nN TS (24) 2 Applying the GibbsHelmholtz relation and using Equation (24) we find the expression for t crystal with defects and so the specific heat at constant vo he energy of a lume CV has the form 00 200 2 2 022 2 def V VV Nn CC 11 ff f fVVV V ff VV ff V fV V B gu gg gTu TT uugg TT T gg g uk (25) In the case of zero pressure, 0 ,0 2 f V u gT def V Cof crystal with , the specific heat at constant volume fe de cts has the simple form: 00 00 00 2 20000 2 2 00 00 22 2 def VB VV Nn k CC 1(1 ) 22 2 2 1 22 uu uu uu uuuu uu uu (26) where CV is the specific heat at constant volume of per fect crystal [10]. The equation of states of the system with defects at fi nite temperature T is now obtained from Equation (24) and the pressure P of the system is given by the deriva tive of the free energy with respect to volume as 3 def def TT a PVVa or .3 def T a Pv Na (27) where v is the atomic volume. From the Equations (24) and (27) equation of states of the crystal with defects at zero pres ion one can find the sure in the harmonic approximat Copyright © 2011 SciRes. MSA
Thermodynamic Properties of Semiconductors with Defects 1229 00 1 22 V TT Nn uu aa (28) From the Equation (28) one can find the average near estneighbor distance (NND), of ato tal at zero pressure and temperuation (28) can be (0, )aT ature T. Eq ls with ms in crys solved using a computational program to find out the values of the NND of the crystadefects, (0, )aT . Let us now consider the compressibility of the solid phase (diamond and zincblende structures). The iso thermal compressibility can be given as 3 2 (,) 3(,0) 1 T aPT aP V 2 2 (,) 23( ,) oTdef T VP aP T PVPT a (29) where 22 200 22 2 2 00 1 22 1 1 24 def V TT T V T Nn uu aa a Nn uu a (30) In the case of zero pressure, the expression of t thermal compressibility for crystals with defects is given as he iso 3 def T 0 22 2 2 200 00 2 2 2 3 3 1 111 24 2 T V T T T a a a Va Nn uuuu a a a (31) or 2 20000 2 2 2 1 11 2 2 def T T V T T T Nnuu uu a a a 1 4 (32) with is the isothermal compressibility of perfective crystals at zero pressure T 3 0 22 2 3 3 T T a a a Va (33 The specific heat at constant pressure, ) C of cr ef ystal with dects is determined from the well known thermo dynamic relations 2 9 def defdef PV def T TV CC (34) where the thermal expansion coefficient def of defec tive crystal is given as 22 0 33 def def BT a ka aV a (35) 3. Results and D def iscussion To calculate the thermodynamic quanti GaAs crystals with defects, we will use the manybody bo tions ties of Si and potential [18], which include both the twody and the threebody atomic interac ,,, 12 6 0 , , ij ijk ij ijk ij ijjk ki W r 0 2 ij ij rr r 3 13cos coscos. ij k ijk WZ rr r (36) The parameters were fitted to the bond lengths of the dimer and trimer and the lattice parameters and cohesive energy of the diamond structure. Parameters of the many body potential for monoatomic (A), binary (AB are given in Tables 1 and 2, respectively. In Table 3, we compare the calculation results of the specific heats at constant pressure, CP of Si crystal with defects obtained by using the SMM analytic formula with the experimental results of Ref. [19]. Here, it should be no 0 ) systems ted that the equilibrium concentration of the vacancies is very small at low temperature. At high temperature being near the melting one the contribution of the vacan Table 1. Paramete rs of many body potential r, Z, and for Si [18]. Quantity Si AA (eV) 2.81 r0AA (A0) 2.295 ZAAA(eVA09) 3484.0 Copyright © 2011 SciRes. MSA
Thermodynamic Properties of Semiconductors with Defects Copyright © 2011 SciRes. MSA 1230 Table 2. Paramete rs of many body potential r0, Z, and for GaAs [18]. Quantity GaAs AA (eV) 1.738 r0AA (A 0) 2.448 Z) 0.0 Z)9) .0 Z0)9) AAB (eVA09 (eV(A0190 460 ABB AA (eV(A 0 1826. A4 P. In Table 3, we also present the SMM calculations of the spe at constant volume, CV and V for Si crystal with defects present the SMM results quilib riumof the vacancies, the speats at con CV and Vmpare thcalculation results of th at constant al. The Figure 1 shown that thonable va pendence of the [19,20], by solid lines. Figures 2 an cies on the specific heats at constant volume, C V and those at constant pressure C of Si crystal is about 05%. cific heats def VV a CC C In e . Tble 4, wof the e concentration ecific h stant volume,def VV CC C , e specific heats and co e pressure, CP for GaAs defective crystal with the experi mental results [20]. The linear thermal expansion coefficient of Si crystal is calculated using the manybody potenti e manybody potential gives reas Figure 1. Temperature dependence of the linear thermal expansion coefficient of Si crystal with defects. the manybody potentials, as a function of the tempera ture. One can see in Figures 2 and 3 that the specific heats at constant pressure, CP increase with the tempera ture, in agreement with th e experimental results [19,20]. lues of thermal expansion coefficient compared with the experimental results [19]. In Figures 2 and 3, we present the temperature despecific heat at constant pressure CP of Si and GaAs crystals with defects, by dashed lines, in comparison with the corresponding experimental results 4. Conclusions The thermodynamic properties of semiconductors with defects have been studied using statistical moment method. We have presented the SMM formulation for the thermodynamic quantities of diamond cubic and zinc blende semiconductors with defects taking into account d 3 show the SMM specific heats at constant pressure, CP (dashed lines) of the diamond cubic Si and zinc blende GaAs crystals with defects, calculated by using Table 3. SMM calculated temperature dependence of T(K) 300 400 500 600 700 800 thermodynamic quantities for Si crystal with de fec t s. 900 1000 1100 1200 1300 1400 1500 6.3 6.8 4.5 7.3 1.5 7.7 1.7 2.0 1.5 8.1 3.4 1.2 3.3 nv 1038 1025 1020 1017 1014 1013 7.1 3.2 1.1 1.0 1.3 4.5 CV 1026 1018 1013 1010 108 107 def V C 2.369 3.577 4.318 4.793 5.1075.326 f 1011 1010 109 109 108 107 107 6.9 5.9 3.3 1.4 4.5 1.2 2.8 106 105 104 103 103 102 102 5.4845.603 5.696 5.768 5.826 5.868 5.895 2.39 3.63 4.41 4.92 5.28 5.53 5.73 5.88 6.01 6.12 6.21 6.30 6.36 de P C Cexp.[ 6.70 16]  5.33 5.63 5.83 5.98 6.10  6.30  6.47   Table 4M calculateperature ddenf thnc quantities foAs cal wifect T(K) 300 500 600 700 800 900 1000 1200 . SMd temepence oermodyamir Garystth des. 400 1.0 .7 2.84 2.0 5 7.6 2.3.9 1 2.5 1. nv 10 10 10 107 10 6.7 712. CV10 1085 102 10 5. 93 6.24 6.36.55 6.76 23 1018 14 1012 109 108 106 0 1.3 1.1.9 5.1 5. 3.6 1.5 1 17 1011 106 104 103 102 de P Cf 22 5.675.6.11 6 6.48 Cexp.[16] 5.65 5.79 5.93 6.07 6.21 6.35 6.49 6.63 6.91
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