MAGNETO-THERMOELECTRIC COEFFICIENTS OF HEAVILY DOPED N-TYPE Mg 2 Si MATERIAL

In contrast to parabolic band model typically used in understanding electronic properties in general, thermoelectric and magneto-thermoelectric in particular, this study confirms non-parabolic band model results in better understanding of Seebeck coefficient and Nernst coefficient in the presence of magnetic field for Mg 2 Si. The magneto Seebeck coefficient was found significantly enhanced from its non-zero field value as a function of magnetic field for different electron concentrations in the range 0.6-12×10 25 /m 3 on the average at room temperature for non-parabolic model compared to parabolic band model. The result for Nernst coefficient shows increasing trend as function of magnetic field except for certain electron concentrations for parabolic band model while it is decreasing with magnetic field on average for non-parabolic band model.


INTRODUCTION
According to [1] Mg 2 Si have been investigated both by experimental and theoretical approaches.By contrast, transport properties have never been determined so far using calculation methods so that the present study is among such few studies.The crystal structure data revealed that Mg 2 Si has space group of Fm3m with Mg occupied sites in 8c (1/4,1/4,1/4) coordinates and Si occupied sites in 4a (0,0,0) coordinates [2].
In n-type Mg 2 Si crystal doped with more than 6×10 16 /cm 3 or more shallow donors concentration, the donor band merges with the conduction band [3].The effect of heavily doping on electronic properties of semiconductors becomes important.There are a number of papers in literature [4,5] which show bit of data on electrical properties of heavily doped semiconductors.Recently, considerable interest has been aroused in the properties of heavily doped semiconductors because of their use in diodes and thin solar cells [6,7].Heavily doped semiconductor based thermoelectric devices as a single device have been widely utilized as non-contact temperature sensors, flow sensors, gas sensors, accelerometers, and power generators while array of them as infrared (IR) imaging devices and micro-spectrometers [8].It can be used for CNT application [9] and Lithium battery components [2].
Thermoelectric effects are in general dependent on magnetic field with the principal parameters, used in thermoelectric, the Seebeck and Peltier effects having the corresponding thermomagnetic coefficients the Nernst and Ettingshausen effects, respectively [12].
In [13] reported on the role of impurities on the lattice dynamics of nanocrystalline Si produced by a bottom-up process with respect to thermoelectric applications.[14] accounts the fact that when the doping density increases there are at least four phenomena occurring simultaneously, on one hand the conduction band and valence band rigidly shift towards one another due to increasing interactions among free carriers, and between free carriers and dopant atoms, causing a reduction in the band gap, in addition to this the two band edges are perturbed and band tails are formed, arising from the random impurity distribution and the resulting potential energy fluctuation.The next phenomena occurred is the shallow dopant band, located slightly above the valence band and the last one is Fermi level shift.
The review work by [15] discusses the nature of main factors such as impurities, defects, electrically active thermal donors and other complexes that are formed due to direct impurity scattering, acoustic phonon scattering, and optical phonon scattering etc. do influence the energy band structure formation in semiconductors, basic electrophysical, optical, thermoelectrical, even mechanical properties.[16] suggested the following expression for density of states of nonparabolic band having band tails where, and For high density limit the potential energy fluctuation reduces to a Gaussian with the standard deviation of We have a screened coulomb potential of impurity atoms with  (dielectric constant) of the given semiconductor.The Thomas-Fermi screening length for all ionized dopants ( ≅  ) according to [17] is The density of states function given by equation ( 1) is very complicated and thus is not useful for making any calculation.Slotboom [18] has however; suggested the following approximation for y(z) as ; for z > 0.601, ,  > 0.85ẟ We obtain the following expression of the electron concentration, where, The corresponding expressions for parabolic density of state and the electron concentration are and where, the standard Fermi-Dirac integral is In terms of the reduced energy and Fermi energy x = and = , respectively.Thus, F 1/2 corresponds to i = ½.
It is more convenient to introduce normalized electron concentration n n given by In this study the semi-classical and quantum treatments are applied in the calculations of scattering mechanisms under the assumptions of the electron concentrations from 0.6×10 18 -12×10 20 /cm 3 and in the temperature range 77 -300K.

BOLTZMANN TRANSPORT EQUATION WITH RELAXATION TIME APPROXIMATION
We define a probability density function f(r,k,t) that describes the probability of finding an electron at position r with a wave vector k at time t.The distribution function can be written as where f 0 is the thermal equilibrium distribution and f 1 is a small perturbation.EEJP. 2 (2023) The Boltzmann transport equation is and assumption that ∇  , ∇  as well as the electric field E are first order quantities and on equating the first order terms in equation ( 14) to obtain where, In the presence of Lorenz force consisting of magnetic field that created second order perturbation where f 1 is assumed to behave as a negative scalar function of the energy  as where, when B = 0 the general vector G reduces to F for electric field only.
We can say the collision process can be described such that the loss of momentum P/τ is described by the Lorenz equation For spherically symmetric   with  =  =  , and making appropriate coordinate transformations We can express Eq.( 20) as For the case of transverse magnetic field  perpendicular to the field E = E x and thermal gradient in the x-y plane, the components of ϕ are Solving for ϕ x and ϕ y gives Where ω = * is cyclotron frequency.

DERIVATION OF CONDUCTIVITY FOR PARABOLIC AND NON-PARABOLIC BAND STRUCTURES
In reality there is a dynamic equilibrium maintained between energy input in the form of both heat and the electromagnetic forces and the scattering processes which tend to randomize momentum and relax energies.In the derivation of conductivities, we have assumed that (i) the electrons are behaving identically with a uniform velocity.
(ii) the ionized impurity scattering is the most important for the electronic transport processes in heavily doped semiconductor and it is energy dependent (τ = τ0ε 3/2 ) since most phonon modes would be suppressed at low temperatures in the range 77-300K and τ 0 is approximated to be 0.01 ns [19].f 1 is responsible for the electric current density and the calculation of the electric current density lies within the first Brillouin zone with   = 1/4π 3 is the local density of states in k-space to get after some steps.
and in the same way In the general case, the electrical and thermoelectric conductance in a transverse magnetic field are expressed in terms of the tensor components  and  as in the equations ( 26) and ( 27).
The most important consequence of the assumptions asserted previously is that all elements of the conductivity tensor are expressed as weighted averages over the density of states ρ(ε), and zero field distribution, f(ε) in the coming derivations in accordance with We use Onsager symmetry [20] for diagonal  ∥;  ∥ and off-diagonal  ;  elements of electrical and thermoelectric conductivities to get where for n-type semiconductor df 0 /dε = -f 0 /k B T. Solving equations ( 26) and ( 27) for E x and E y gives and It can be written in vector form [19] as where, we have electric resistivity (), hall effect factor (R H ) that appears due to the application of the magnetic field that leads to a decrease in longitudinal conductivity and induces an off-diagonal conductivity, which is called the hall effect.This is commonly called the magneto-resistance.We have also Seebeck coefficient(S) and Nernst coefficient (N).When a conductive solid is placed under a longitudinal temperature gradient and a transverse magnetic field, two types of thermoelectric responses occur, i.e., the magneto-Seebeck effect in the longitudinal direction and the Nernst effect in the other transverse direction.
We can get magneto-Seebeck coefficient by setting j x = j y = 0 and eliminating E y in equation (33) under isothermal condition ∇  = 0 as where the mobility  = * and the integral functions in equation ( 36) are expressed with the help of the reduced energy x = ε/k B T and τ = τ 0 x 3/2 accordingly other quantities which appeared in the derivations were reduced to dimensionless for sake of unit consistency. and Furthermore, Nernst coefficient is obtained by setting j x = j y = 0 and eliminating E x in equation (33) under isothermal condition ∇  = 0 We can obtain the corresponding expression for magneto-Seebeck coefficient for the case of modified density of states based on standard model with parabolic density of states (which doesn't incorporate the effect of band tails) by substituting equation ( 1) for modified density of states and by extending the integration limits from -∞ to ∞. where, and the integral functions in equation ( 40) are obtained by setting  = 5/2; 7/2; 4; 5 into this equation respectively.
The corresponding expression for Nernst coefficient is

RESULTS
The calculation of Seebeck coefficient (S) for zero field used the expressions as − and − √ for parabolic and non-parabolic bands respectively [21].Similarly in the presence of magnetic field and temperature gradient equation (36) for Seebeck(S) and equation (39) for Nernst coefficient(N) in parabolic band and the corresponding equations equations ( 40) and (43) in non-parabolic are carried out under the assumption that the electron concentrations vary from 0.6 -12×10 19 /cm 3 and in the temperature range 77 -300K.The effective mass of Mg 2 Si is m n * = 0.46m 0 (m 0 is mass of free electron) and its dielectric constant is ϵ = 20 as well as the parameters a and  were determined in the way as in [4].Numerical values were obtained by using Matlab software in such a way that the reduced value of electron concentration n n = n/(10 25 m -3 ) is used to get the corresponding reduced Fermi-energy η.A number of steps are required to get the write value of η for a given electron concentration through iterative method.Equation (11) for parabolic band was used and Equation ( 7) was for non-parabolic band in this iterative method in such a way that the integral on the LHS is evaluated for a given η and next one should get the write value which equates the given value of n n on the RHS.That is how the calculated values were obtained.These values are the basis for the calculations of the thermomagnetic coefficients in the study in the presence of magnetic field varying in the range from 0.2 -1T.Note that in this study electron mobility is taken to be featureless and the relaxation time is typically taken to be 10 -11 s without losing the generality as in [19,22,23].
Figure 1 is plotted with the calculated values of Seebeck coefficient for zero field case S(0) and S(B) in the presence of magnetic field that are used to determine Δ = S(B) -S(0) for parabolic and non-parabolic bands respectively.The change is significant for non-parabolic band consideration that is about 360μV/K on average compared to its value for parabolic band that is about 135.6 μV/k on average for magnetic field ranging in the interval 0.2 -1T.In both cases there is enhancement of Seebeck coefficient due to magnetic field.As it is the case for this study, we need the modern approaches for better understanding of the thermomagnetic property of Mg2Si-based materials based on the consideration of non-parabolic band of heavily doped semiconductor rather than for that of parabolic band consideration.Due to this reason the value 360μV/K is reliable value that shows there is significant enhancement of Seebeck due to magnetic field for Mg 2 Si-based materials.The enhancement of Seebeck coefficient 360 μV/K is two orders of magnitude greater than the experimental uncertainty level ⁓ 10 μV/K [24].Therefore, one would expect significant effect of magnetic field in heavily doped n-type Mg 2 Si.The result in Figure 2 provides the value of Nernst coefficient(N) as a function of magnetic field ranging in the interval 0.2 -1T for electron concentration in heavily doping range 0.6 -12×10 25 /m 3 for parabolic band consideration.There is slow variation of Nernst coefficient with its negative values for certain electron concentrations such as 2×10 25 /m 3 and 4×10 25 /m 3 compared to relatively rapid variations for the rest values of electron concentrations.For the parabolic band consideration one can see the Nernst coefficient increases with magnetic field for the latter case.The result in figure 3 is the value of Nernst coefficient as a function of magnetic field ranging in the interval 0.2 -1T due to resolution problem in the graph it shows the average values for electron concentration in the heavily doping range 0.6 -12×10 25 /m 3 .In contrast to the result in Figure 2 for parabolic band the Nernst coefficient decreases with magnetic field for non-parabolic band consideration with its magnitude in the order of 10 -9 V/K compared to the order of magnitude 10 -6 V/K for parabolic band consideration as shown in Figure 2.There is significant difference for order of magnitude of Nernst coefficient as a function of magnetic field for parabolic band consideration as compared to non-parabolic consideration.We are once again recommending using the more modern approaches in the consideration of non-parabolic band consideration as compared to parabolic consideration.The same behavior as in figure 1 and figure 2 are expected to be observed for similar experimental arrangements.Therefore, there is a huge potential for further experimental explorations and investigations on such behaviors.These values are not meant to be good fits to measurements.Note that the results as shown are what we obtained with the help of Matlab software.Seebeck and Nernst coefficients should vary much with the magnetic field, so the results for these coefficients are closely agree with the first principle calculations as in [25] that show the importance of non-parabolic effects, in the thermoelectric properties of semiconductors.The same magneto Seebeck and Nernst behavior were experimentally observed according to experimental measurements [26] for polycrystalline Wely semimetal NbP.

CONCLUSION
Our theoretical calculation confirms that the material under study shows the general trend of exhibiting higher values of the magneto-Seebeck coefficients for non-parabolic band consideration compared to parabolic band consideration.Our theoretical results also show that the magnitude of the Nernst coefficient is larger at lower temperatures, lower carrier concentrations, and magnetic fields with the exception of that for non-parabolic consideration where Nernst coefficient shows decreasing trend with magnetic field.These observations could prove useful in evaluating the Nernst coefficient in other materials having similar band structures as that of Mg2Si, while it is true that the relaxation times may differ with carrier concentration and magnetic field dependent mobility for our result to well agree with experimental result these factors should be considered in future studies for better result.

Figure 1 .
Figure 1.Change in Seebeck coefficient from non-zero field value S(0) to S(B) due to magnetic field for parabolic and non-parabolic bands

Figure 2 .
Figure 2. Nernst coefficient against magnetic field for different electron concentrations in parabolic band