Effect of nano-crystallization and atomic substitution on electronic and heat properties of Bi2Te3

The specific heat of nano-crystalline (NC) Bi2Te3 (20 nm) in the temperature range of 150 K to 300 K is theoretically studied and compared with the specific heat of the bulk Bi2Te3 materials. Carrier concentration and Hall coefficient are also evaluated using free electron model and compared with experimental results. Lattice (phonon) specific heat has been observed with an overlap repulsive potential with the help of Debye model. For Nano Crystalline materials having high interface volume ratio, the Debye temperature and phonon frequencies are reasonably less at the interfaces than at the core of nano-crystal. Such softening of phonon frequencies at interfaces results in enhancement of Specific Heat in nano-structures. Total specific heat would include interface, core and electronic specific heat. The temperature derivative of the internal energy yields the electronic contribution to specific heat. The present analysis based on the softening of phonon frequencies mechanism is sufficient to describe the enhancement in specific heat by nano-crystallization.


Introduction
Nanomaterials are being widely studied by the researchers across the globe because of their extreme physical and mechanical properties.The electrical and thermal properties of such nano-crystalline (NC) materials strongly depends on the temperature, pressure, and size of the nanoparticles [1][2][3].Nanocrystallization has been developed as a unique technique to achieve the desired physical properties those are expected for the specific applications [4,5].Experimental reports revealed that the Specific Heat (Cp) of many nanomaterials observed higher in comparison to those observed for bulk forms of the same composition [6][7].Study of specific heat revels very important information about electronic structure, phonon structure and various phonon frequency modes.Charge carrier contribution to specific heat is related with carrier concretion and Fermi energy (Ef) whereas the enhancement in specific heat by nano crystallization sheds important light on the size of nanoparticles and interface to volume ratio of the nanostructures.It is also interesting to study the Hall effect in these materials as it is related to electron or hole mobility in the system.
Bi2Te3 is an interesting composition which exhibits very high thermal, electrical, and thermoelectric properties.These materials are very useful for thermoelectric generation and cooling applications [8].Various alloys of Bi2Te3 are intensively studied to analyse their electronic and thermal properties [9][10].Atomic substitution by appropriate doping and nano-crystallization are the efficient techniques to achieve the desired physical properties in these compounds.In the current investigation, we propose a phenomenological model to impropriate the enhancement in specific heat Cp by nanocrystallization.The proposed method is based on softening of phonon frequencies at the grain boundaries in nanoparticles.We Also estimated and compared the carrier concentration, density of ststes and Hall effect for the same compound using free electron mode and found good agreement with experimental data.

Electronic properties
We proceed to calculate the carrier concentrations at any temperature T from the density of states in energy and the statistics of the carriers.
The free electron model suitably defines density of states effective mass of hole (݉ * ) and effective mass of electron (݉ * ).First, we consider the density of states to obtain the concentration of electron in the conduction band and then we consider the density of states to obtain the concentration of hole in the valance band.[11] The densities of the hole and electron written in the form of electron (ne) and hole (nh) concentration as follows: Where , here the effective mass of the electron is m ୣ * = .58× m ୣ Similarly hole concentration is expressed as follows: Where , the effective mass of the hole is m ୦ * = 1.08.×m ୣ .

Hall coefficient
Hall coefficient is expressed in terms of carrier concentration R ୌ = ଵ ୬ୣ (3) Where n is the carrier concentration and e are the charge of electron

Electronic specific heat
The charge carrier contribution to heat capacity is given as the temperature derivative of internal energy [12] Cel = మ ଷ k ଶ N(E )T=γT (4) Here, N (EF) denotes the density of state near Fermi level and γ is Somerfield constant.

Lattice specific heat
As the temperature range of study falls in the vicinity of the Debye temperature, we used Debye model to estimate the phonon contribution to specific heat capacity.Frequency of acoustic modes can be calculated from effective ion charge −Ze.Coulomb interactions between nearby atoms is expressed as inverse-power overlap repulsion [13] Φ f is repulsion force parameter, K is elastic force constant which is expressed as follows: here, s represents the index number related with the overlap repulsive potential.The elastic force constant and atomic mass M are used to calculate the acoustic phonon frequency, where ZD is the Debye frequency and N is number of atoms in a unit cell In case of nanoparticles, the lattice specific heat (phonon contribution) can be estimated in two parts: One term denotes the specific heat contributions by the atoms presents at the interfaces (C ୦ ୍ ) and another term is the specific heat contribution by atoms present at the core of nanoparticles ൫C ୮୦ େ ൯. [12] Here, A ୍ denotes the interface to volume ratio in nanomaterials, and associated with the size of nanoparticles, whereas A େ is related with A ୍ (A େ + A ୍ = 1).C ୦ ୍ and C ୦ େ are functions of Debye temperature at interfaces (ϴ ୈ ୍ )and in the core of nanocrystal ϴ ୈ େ ,respectively.Total specific heat includes the lattice and electronic contribution.
The next section presents and discusses the results obtained from present model.

Result & Discussion
In order to further understand the effect of softening of phonon frequencies and its effect on specific heat capacity along with charge carrier contribution to heat capacity in Bi2Te3 we extend our analysis to estimate carrier concentration and Hall effect.
For estimation of carrier concentration, we use effective mass as a material dependent parameter within the temperature range from 0K to 300K.The effect can be observed with equation (1).The value of Nc is increasing with increase in the effective mass of electron.Figure 1 shows the temperature dependence of carrier concentration within temperature ranges from 0 to 300 K. Above results shows that as doping concertation increase the carrier concentration decreases.
Carrier concertation of undoped Bi2Te3 shows negative temperature dependence and it decreases with increase in temperature as shown by black dots in figure 1.For doping concertation 0.01 and 0.05, carrier concertation is almost temperature independent.Whereas, for x=0.08 carrier concertation increases with increase in temperature (see down triangles in figure 1).Further the hall coefficient R ୌ = ଵ ୬ୣ , is calculated using carrier concentration (n) and charge of electron (e).Hall coefficient has an inverse relation with carrier concentration.The above estimated results of carrier concertation and Hall coefficient shows good agreement if compared with experimental results [14].Now we proceed to estimate the specific heat of bulk and nano-crystalline form of Bi2Te3 compound.The approach of estimating the parameters is the softening of phonon frequency at interface in nano materials.For this model we first estimated Debye temperature (θD) and Debye frequency (߱ =݇ ܶߐ /ℏ) to estimate the phonon specific heat at interface ‫ܥ(‬ ூி ).Also we estimated the phonon specific heat at the core of nano-crystal ‫ܥ(‬ ே ).Finally, we evaluated the phonon specific heat with the equation ‫ܥ(‬ = ‫ܣ‬ ூி ‫ܥ‬ ூி + ‫ܣ‬ ே ‫ܥ‬ ே ) .Here ‫ܣ‬ ூி is the interface to volume ratio in the nano material and ‫ܣ‬ ே is related to ‫ܣ‬ ூி as follows A ୍ + A େ = 1 (11) The volume fractional relation is given by AIF =3d/D, here d is the average width of grain boundaries and D is the diameter of the nano-crystal.
Electronic contribution to the specific heat estimated using the density of state at Fermi level N( ) and the Somerfield constant ࢽ.Here density of states are related with effective mass of carriers and carrier concentration.With the addition of electronic contribution the total specific heat found to be Here C ୦ is the phonon contribution to specific heat which generally varies with βT ଷ in nanostructures.The electron contribution towards heat capacity Cel of bulk (Nano Crystals) is shown in the figure 3(4) which shows linear temperature dependence.For the calculation of Cel we used the Sommerfield constant which is proportionally constant here (γ = 1.4 mJ mol -1 k -2 ) The numeric value of γ dependent on density of states of charge carriers that is a function of the effective mass and carrier concentration of the carrier.Specific heat (Cp) of bulk Bi3Te3 has been estimated theoretically and compared with the available experimental data.Present approach evaluates the specific heat capacity by considering the effect of softening of phonon frequency at the interface in nano structured materials.The complete understanding of specific heat can be done by estimating independent contribution from phonons present at the core of nano particle and phonons activated at the interfaces in addition with the charge carrier contribution to the Cp.For the phonon contribution to specific heat of bulk Bi2Te3 first we calculated θD=165K, which is matching with the experimental value of θD=160K and θD=155k [15,16].Generally, θD varies with temperature and also depends on the technique of measurement, hence it has a standard deviation of around θD±15K.Bi2Te3 has rhombohedral structure, The number of atoms in a unit cell N=6.The phonon contribution to specific heat Cph is calculated and plotted in Figure 1.For the NC material there are two components first one is within the nanocrystal and second is at the grain boundaries.At interfaces θD observed to be low as compared to θD at the core of nanoparticles.Since in nano materials interatomic distance is bit larger than interatomic distance in the core of crystal so the phonon frequencies are smaller in the NC materials due to reduction of θD at the interface.This effect is understood as softening of phonon frequencies and force constants (k).Theoretically calculated Cp is compared with experimentally reported data [16].
There should be two components which contribute to the total specific heat of the nanocrystalline material that is specific heat at interface ‫ܥ‬ ூி ( ߐ ூி ) and the specific heat at the core of NC C ୦ େ (ߐ ே ).We have taken Debye temperature as ߐ ூி = ‫ܭ58‬ for interface in Bi2Te3 nano crystals.
The value of ‫ܣ‬ ூி = ଷௗ య for the spherical nanoparticles where d is width of grain boundaries and D is the mean diameter of the nanoparticles.The estimated parameter related to interface volume ratio in nano structures is A ୍ = 4.6 and A େ = 5.4 has been taken in our calculation.The temperature dependence of ‫ܥ‬ ூி ( ߐ ூி ) and C ୦ େ (ߐ ே ) has been observed with the help of fractional ratio parameter In current investigation we observe that phonon softening is the mechanism for increase in the specific heat of nano materials.The contribution to specific heat by the electrons Cel.Is very small as compared to specific heat obtain from phonon Cp.Due to phonon softening effect there has not been measurable change in the electronic contribution in specific heat Cel.Electron contribution along with phonon contribution using phonon softening effect can successfully explain the Specific heat phenomena in the nano materials as well as in the bulk form.

Conclusion
In the current investigation behaviour of specific heat Cp(T) in NC and bulk state of Bi2Te3 has been studied.The mechanism based on softening of phonon frequency have been explored to interpret the enhancement in Cp in NC materials.The specific heat has been calculated by Debye model with harmonic approximation using Debye temperature.Since the nanocrystalline materials having comparatively higher ratio of interface over the volume, the phonon frequencies correspondingly Debye temperature are less at interface as compared to at the core of the nanocrystals.The contribution to specific heat capacity by the atoms presents at interfaces (C ୮୦ ୍ ) is higher as compared to those present within the core of nanoparticle (C ୮୦ େ ), These two contributions are estimated separately to calculate the total phononic specific heat capacity.The characteristic material dependent parameters such as Debye temperature (θ ୈ ), elastic force constant (K), temperature derivative of the internal energy gives rise the measured specific heat (Cp) and its enhancement in nanomaterials.

Figure 1 .
Figure 1.Variation of carrier concentration nH with temperature T

Figure 2 .
Figure 2. Temperature-dependent Hall coefficient RH Figure2represents the temperature dependence of Hall coefficient RH in temperature range from 0 to 300 K.It has been observed that the value of Hall coefficient has been nearly constant, independent of temperature for undoped Bi2Te3 as well as for lower doping concertation at x=0.01 and 0.05.However, for doping concertation x=0.08, the hall coefficient RH shows noticeable change with temperature.It decreases linearly with increase in temperature till T = 150K and shows sudden fall above T = 150K as shown by down triangles in figure2.The above estimated results of carrier concertation and Hall coefficient shows good agreement if compared with experimental results[14].Now we proceed to estimate the specific heat of bulk and nano-crystalline form of Bi2Te3 compound.The approach of estimating the parameters is the softening of phonon frequency at interface in nano materials.For this model we first estimated Debye temperature (θD) and Debye frequency (߱ =݇ ܶߐ /ℏ) to estimate the phonon specific heat at interface ‫ܥ(‬ ூி ).Also we estimated the phonon specific heat at the core of nano-crystal ‫ܥ(‬ ே ).Finally, we evaluated the phonon specific heat with the equation ‫ܥ(‬ = ‫ܣ‬ ூி ‫ܥ‬ ூி + ‫ܣ‬ ே ‫ܥ‬ ே ) .Here ‫ܣ‬ ூி is the interface to volume ratio in the nano material and ‫ܣ‬ ே is related to ‫ܣ‬ ூி as follows A ୍ + A େ = 1 (11) The volume fractional relation is given by AIF =3d/D, here d is the average width of grain boundaries and D is the diameter of the nano-crystal.Electronic contribution to the specific heat estimated using the density of state at Fermi level N( ) and the Somerfield constant ࢽ.Here density of states are related with effective mass of carriers

3 Figure 3 .
Figure 3. Temperature-dependences of electronic specific heat of bulk Bi2Te3