Thermodynamic Properties of Semiconductors with Defects

Thermodynamic properties of diamond cubic and zinc-blende semiconductors with point defects are considered by the statistical moment method (SMM). The thermal expansion coefficient, the specific heats at constant volume and those at constant pressure, CV and CP, and the isothermal compressibility are derived analytically for semiconductors with defects. The SMM calculated thermodynamic quantities of the Si, and GaAs semiconductors with defects are in good agreement with the experimental results.


Introduction
Recently, there has been a great interested in the study of bulk semiconductor, semiconductor heterostructures and nanodevices [1][2][3][4] since they provide us a wide variety of academic problems as well as the technological applications.The physical characteristics of semiconductors are determined both by the properties of the host crystal and by the presence impurities and crystalline defects.Crystal 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 vacancies play an important role in many properties of material.Understanding these defects will lead to improved semiconductor devices for the technological applications.
First principles (or ab initio) electronic structure computations have been performed on semiconductor compounds and the results compared with experiment [5][6][7].More recently a large number of high-quanlity calculations have been performed on group-IV, III-V, and II-VI materials.Modern calculations allow the accurate relaxation of structures to their minimum energy configurations and the incorporation of temperature effects.One can also study the melting of solids and phase transitions in Si using first-principles molecular-dynamics 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, taking 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, C V and C P , are derived analytically for diamond cubic semiconductors.
The purpose of the present article is to investigate the temperature dependence of the thermodynamic properties of the semiconductors with defects using the analytic statistical moment method (SMM) [12][13][14][15][16].The thermodynamic quantities are derived from the Helmholtz fnee energy of semiconductors with defects.

Atomic Displacements of Semiconductor
To derive the temperature dependence of the thermodynamic properties of semiconductors, we use the statistical moment method.This method allows us to take into account the anharmonicity effects of thermal lattice vibrations 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-yond the quasi-Hamonic (QH) approximation.The extentions for the SMM formalism to non-cubic systems is straightforward.The basic equations for obtaining thermodynamic quantities of the given crystals are derived in a following manner: the equilibrium thermal lattice expansions are calculated by the force balance criterion and then the thermodynamic quantities are determinded for the equilibrium lattice spacings.The anharmonic contributions of the thermodynamic quantities are given explicitly 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 position.The potential 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 U as a power series in the cartesian components, 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 where  0 H denote the crystalline Hamiltonian without the supplementary forces i  and upper huts  represent 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 , , , The thermal averages of the atomic displacements   (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 recurence formula [12][13][14].Then Equation (2) is transformed into the new differential equation: where , and ; In the above Equation (3), , In deriving Equation (3) we have imposed the symmetry criterion for the thermal averages in the diamond cubic lattices as Let us introduce the new variable y in the above Equation (3) Then we have the new differential equation instead of Equation (3) where For higher temperatures, the relation xcothx 1 holds and Equation ( 6) is reduced to The nonlinear differential equation of Equation ( 8) can be solved in the following manner: We expand the solution in terms of the "force" up to the second order as where 1 A and 2 A are the constants [12].The above Equation ( 8) is solved as  Here, 0 represents the atomic displacement for the case when the force is zero.The general solution of Equation ( 3) is solved as Once the thermal expansion 0 of the lattice is found, one can get the Helmholtz free energy of the system in the following form where 0  denotes the free energy in the harmonic approximation 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: where  0 H denote the Hamiltonian of the harmonic approximation,  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 where  V  represents the Hamiltonian corresponding t o the anharmonicity contribution.Then the free energy of the system is given by where , and the second term of above Equation (15) denotes the harmonic contribution to the free energy with the aid of the "real space" free energy formula , E TS    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.

Thermodynamic Properties of Semiconductors with Defects
The Gibbs free energy of crystals consisting of N atoms and vacancies has the form: where   0 , G T P is the Gibbs free energy of the perfect crystals consisting N atoms,  , f V  g T P is the Gibbs energy change on forming a single vacancy, -the entropy of mixing: From the minimization condition of t rgy of the crystal with point defects he Gibbs free ene , we obtain the eq uilibrium concentration of the vacancies as [15,16]   , exp where f g  is the change in the Gibbs fre the fo on of a vacancy and can be given by e energy due to mati It should be noted that pressure affects the diffusivity through both the free energies, * From Equations ( 20), (21), and (22 pression of the Gibbs energy change vacancy: ) we obtain the exon forming a single Form Equations ( 17) and ( 23) it is easy to ob expression for the Helmholtz free energy o defects: tain the f crystals with Applying the Gibbs-Helmholtz 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 C V has the form 0 0 In the case of zero pressure, , the specific heat at constant volume fe dects has the simple form: where C V is the specific heat at constant volume of perfect crystal [10].
The equation of states of the system with defects at finite temperature T is now obtained from Equation (24) and the pressure P of the system is given by the derivative of the free energy with respect to volume as where v is the atomic volume.
From the Equations ( 24) and ( 27) equation of states of the crystal with defects at zero presion one can find the sure in the harmonic approximat From the Equation (28) one can find the average nearest-neighbor distance (NND), of ato tal at zero pressure and temper uation (28) can be (0, ) a T ature T. Eq ls with ms in cryssolved using a computational program to find out the values of the NND of the crysta defects, (0, ) a T .Let us now consider the compressibility of the solid phase (diamond and zinc-blende structures).The isothermal compressibility can be given as where In the case of zero pressure, the expression of t thermal compressibility for crystals with defects is given as he iso- or is the isothermal compressibility of perfective crystals at zero pressure The specific heat at constant pressure, ) P C of cr ef ystal with d ects is determined from the well known thermodynamic relations where the thermal expansion coefficient def  of defective crystal is given as

Results and D def iscussion
To calculate the thermodynamic quanti GaAs crystals with defects, we will use the many-body bo tions ties of Si and potential [18], which include both the two-dy and the three-body atomic interac , 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 manybody potential for monoatomic (A), binary (A-B are given in Tables 1 and 2, respectively. In Table 3, we compare the calculation results of the specific heats at constant pressure, C P 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-  and co e pressure, C P for GaAs defective crystal with the experimental results [20].
The linear thermal expansion coefficient of Si crystal is calculated using the many-body potenti e many-body potential gives reas the many-body potentials, as a function of the temperature.One can see in Figures 2 and 3 that the specific heats at constant pressure, C P increase with the temperature, in agreement with the experimental results [19,20].
lues of thermal expansion coefficient compared with the experimental results [19].In Figures 2 and 3, we present the temperature de specific heat at constant pressure C P of Si and GaAs crystals with defects, by dashed lines, in comparison with the corresponding experimental results

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 zincblende semiconductors with defects taking into account d 3 show the SMM specific heats at constant pressure, C P (dashed lines) of the diamond cubic Si and zincblende GaAs crystals with defects, calculated by using change is due to the P V  work done by the pressure medium against the volume change associated with defect formation and migration.theabove Equation (20), with atom 0 and io  the effective interaction energies between the o th and i th atoms, creates the vacancy, by moving itself to a certain sink site in the crystal, and is given by

Figure 1 .
Figure 1.Temperature dependence of the linear thermal expansion coefficient of Si crystal with defects.

Figure 2 .Figure 3 .
Figure 2. Temperature dependence of the specific heat at constant pressure C P of Si crystal with defects.