Polarization electric field effect on cross-plane thermal conductivity of GaN/InxGa1-xN/GaN superlattice (x ≤ 0.3)

GaN/InxGa1-xN/GaN superlattice (SL) has electric field at interfaces termed as interfacial polarization electric (IPE) field. In this work, effect of this electric field on cross-plane thermal conductivity (kcp) of the SL is investigated for Indium content x ≤ 0.3. IPE field revises phonon velocity which enhances interfacial scattering and thermal boundary resistance (TBR). This is due to unequal changes in specific heat and phonon velocity which leads to decrease in phonon transmission and more mismatches of acoustic properties of material. This reduces kcp. Room temperature kcp in presence (absence) of IPE field for GaN (10nm)/InxGa1-xN (5nm) SL are 4.652 (5.720) and 4.282 (5.221) Wm−1K−1 respectively, for x=0.1 and 0.3. This work demonstrates that electric field of nitride SL can be utilized to reduce k for optimum thermoelectric power production.


Introduction
Efficiency of a thermoelectric (TE) material is generally quantified by dimensionless figure of merit (ZT) which is directly proportional to electrical conductivity ( ) and Seebeck coefficient (S); but inversely proportional to k of the material [1]. The main paradigm to achieve material's high ZT is to enhance their power factor (S 2 σ) and/or reduce their k. A small reduction in k can improve TE efficiency significantly. Various approaches have been undertaken to reduce k of SL. Doping, alloying, period width variation, lattice mismatch, nano-particle substitution, interfacial roughness and atomic disorder etc [2][3][4][5]. Many of these studies have proved that k of SL is lower than the corresponding k of bulk material. Thus, more approaches must be explored to reduce k of SL. In this work, polarization property of nitride SL has been exploited to reduce k of SL.
Recent study reported that GaN /InxGa1-xN/ GaN SL can be a promising contender for TE devices due to its lowest k within the III-V nitride alloys [6]. Its ZT can approach best TE material Bi and Te. This motivates to explore methods to enhance ZT of GaN /InxGa1-xN SL reducing k and increasing , S of the SL further. This SL has a unique advantage called polarization mechanism which produces an interfacial polarization electric (IPE) field at each interface of the SL due to difference in spontaneous (SP) and lattice mismatch piezoelectric (PZ) polarizations of layers [7]. Sztein

IPE field and its effect
Figure1 shows structure of a GaN/InxGa1-xN/GaN SL. Total polarization in a layer is P=P sp + P pz . In this SL, GaN is barrier while InxGa1-xN performs as well. The interfacial electric field due to total polarization in a well Ew (or in barrier Eb) layer is [7]: Here lw (lb), Pw(Pb) and w  ( b  ) are width, polarization and dielectric constant of well (barrier) material, respectively. The IPE field is ~ 1 MV/cm. Nitrides are piezoelectric semiconductors; so, as per Heckmann principle, the coupling between the strain field and electric field will induce additional electric polarization. This will contribute to elastic constant of the alloy. The elastic constant of InxGa1-xN in the presence of IPE field has been found as [8]   , respectively; where  is material density. The average phonon velocity is obtained as Debye frequency is determined by the expression Here, N is the number of atoms

k of GaN/InGaN/GaN SLs
The k of a semiconductor is written as sum of phonon and electron contribution . Thus, electronic contribution is neglected and for a semiconductor ph k k  . In solids, the thermal energy is carried by acoustic phonons. These phonons suffer scatterings due to other phonons, defects and interfaces. Here, normal scattering ( scattering rates are taken into calculation. The combined phonon scattering can be written as . Thermal conductivity k of a layer of the SL is determined by Callaway expression ( In order to determine k of a SL, effect of interfaces has to be included. Interfaces generates thermal boundary resistance (TBR) which is computed by acoustic mismatch model (AMM)and diffuse mismatch model (DMM). If the interface is specular, then phonon transmissivity from a layer i to layer j, Here Ci is specific heat of i th layer. With ( Using ij  determined from Eq.(5), we can estimate TBRs from material i to material j by Normally, k of SL is a tensor because of different materials and interfaces which generates inhomogeneity in thermal flow. Depending on the direction of imposed temperature gradients, k is divided into two parts: thermal flow parallel to a layer plane is termed as in-plane thermal conductivity (kip); whereas thermal flow across the layers is cross-plane thermal conductivity (kcp). The kcpof GaN/InxGa1-xN/GaN SLs with two layers per period is estimated by Here l1(l2) and k1(k2) are layer thickness and thermal conductivities of GaN(InGaN), respectively. R represents the TBRs between GaN and InGaN. Figure 2 shows kcp as a function of temperature for x=0.1 in the absence and presence of IPE field. The kcp has linear dependence on temperature from 20 <T< 120 K, whereas from 120 <T< 250K kcp has nonlinear dependence on temperature and kcp is relatively independent of temperature for T> 250 K [3]. From figure 2, it can be found that kcp in presence of IPE field is lower than kcp in the absence of IPE field. The RT kcp of GaN (10nm)/In0.1Ga0.9N(5nm)/GaN SL in the absence and presence of IPE field is 5.720and 4.652Wm -1 K -1 respectively. This implies that IPE field reduces RT kcp and the reason for decrease of kcp is the increased TBR and decreased k of layers due to IPE field. IPE field enhances phonon velocity and Debye temperature of the constituent material of the SL. Alloy effect raises scattering of high frequency phonons whereas boundary/interfaces enhance scattering of low frequency phonons. Low frequency phonons significantly contribute to k of SL. IPE field increases the strength of boundary/interfaces scattering in the SL resulting in enhanced scattering of low and mid frequency phonons further. Phonon-mode mismatch between layers, which scales with the lattice mismatch at the interface is significantly modifying the k. Tong et al. [10] experimentally determined k of In0.1Ga0.9N alloy as 13Wm -1 K -1 with 233 nm thicknesses. Our predicted value is well below this alloy limit. Xu et al. [5] observed kcp of GaN/In0.1 Ga0.9N/GaN nanoporus SL as 4.2 Wm -1 K -1 which they attributed to additional phonon scattering from porosity of the sample. Zhang et al. [4] determined kcp of GaN/In0.1 Ga0.9N/ SL as 4.8 Wm -1 K -1 which is closure to our predicted value. Figure 3 shows kcp of GaN/In0.3Ga0.7N/GaN SL as a function of temperature in the absence and presence of IPE field. Here also kcp curve can be divided into 3 regions: (i) linear dependence on temperature from 20 < T < 120 K which is attributed to coherent phonon transport through the layers and T 3 behaviour; (ii) nonlinear dependence on temperature from 120 < T < 250K which is due to the increased influence of incoherent effects and (iii) relatively independent of temperature for T > 250 K; but decreases very slowly with increasing temperature for T > 300 K [3][4][5].   Figure 3. kCP of GaN/In0.3Ga0.7N/GaN SL as a function of temperature.

k of GaN/In0.3Ga0.7N/GaN SL
The nature of kcp in these regions can be explained in a way similar to GaN/In0.1 Ga0.9N/GaNSL under the fact that the low-frequency phonons are not affected in the temperature region T < 120K, whereas the high-frequency phonons are greatly affected for elevated temperature T > 300K due to anharmonic (Umklap) scattering which makes kcp to saturate with T. Our result shows that room temperature kcp of GaN/In0.3Ga0.7N/GaN SL in the absence and presence of IPE field is 5.221 and 4.282Wm -1 K -1 respectively. This implies that IPE field reduces room temperature kcp and is due to enhanced TBR under influence of IPE field. From figure 3, it can be seen that a minute decrease of k is found for T> 300K due to increase of Umklap scattering. At room temperature authors such as Huxtableet al. [3], Tong et al. [10] and Saha et al. [11] have experimentally determined kcp as 4.6, 2.9 and 3.3 respectively for Si0.6Ge0.4/Si0.7Ge0.3 SL, In0.3Ga0.7N alloy and TiN/Al 0.7Sc0.3N SL. Low k in the alloy samples reveals presence of high amount of dislocations, impurities and inhomogeneities.

Conclusion
In this work, effect of interfacial polarization electric (IPE) field on cross-plane thermal conductivity kcp of GaN/InxGa1-xN/GaN SL has been explored for Indium content x≤ 0.3. Our result reveals that TBR is enhanced at interfaces under the action of IPE field leading to decreased phonon transmission and more mismatches of acoustic properties. This caused reduction of kcp of the SL. TBR is found to be (2.10 to 5.30) 10 -9 m 2 KW -1 which is closer to the reported value of similar SLs. Room temperature (RT) kcp in the presence (absence) of IPE field for GaN (10nm)/InxGa1-xN (5nm) SL are 4.652 (5.720) and 4.282 (5.221) Wm -1 K -1 which demonstrate more than 20% reduction and are closely agreeing with experimental kcp of similar SLs. This work shows that desired value of k can be achieved by tailoring electric field of nitride SL for maximum power production.

Acknowledgments
Authors acknowledge with thank to National Institute of Technology Raipur, Govt. of India for financial support through Cumulative Professional Development Allowance (CPDA). Expt. k CP of W 0.3 Ti 0.7 N/Sc 0.3 Al 0.7 N SL [11] Expt. k CP of In 0.3 Ga 0.7 N Alloy [10]