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GaN heterostructures
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
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Pyroelectric effect and lattice thermal conductivity of InN/GaN heterostructures Gopal Hansdah, Bijay Kumar Sahoo * Department of Physics, N.I.T, Raipur, Raipur 492010, India
A R T I C L E I N F O
A B S T R A C T
Keywords: InN/GaN heterostructures Built-in-polarization Lattice thermal conductivity Pyroelectricity Transition temperature
The built-in-polarization (BIP) of InN/GaN heterostructures enhances Debye temperature, phonon mean free path and thermal conductivity of the heterostructure at room temperature. The variation of thermal conductivities (kp: including polarization mechanism and k: without polarization mechanism) with temperature predicts the existence of a transition temperature (Tp) between primary and secondary pyroelectric effect. Below Tp, kp is lower than k; while above Tp, kp is significantly contributed from BIP mechanism due to thermal expansion. A thermodynamic theory has been proposed to explain the result. The room temperature thermal conductivity of InN/ GaN heterostructure with and without polarization is respectively 32 and 48 W m-1 K1. The temperature Tp and room temperature pyroelectric coefficient of InN has been predicted as 120 K and 8.425μC m2 K1, respectively which are in line with prior literature studies. This study suggests that thermal conductivity measurement in InN/GaN heterostructures can help to understand the role of phonons in pyroelectricity.
1. Introduction InN/GaN heterostructure is a promising optoelectronics material for fabrication of light-emitting diodes (LED) and laser diodes (LD) [1]. The InN/GaN heterostructure system has several advantages. The advantages include high rate of optical phonon emission (2.5 1013 s1), operating in THz frequency range, high value of electron drift velocity (5 107 cm s1) and large conduction band offset [2]. The difference in the growth temperature between InN (~500 C) and GaN (~750 C) is large. So, fabrication of high quality InN/GaN heterostructures is a challenging task. In addition, the lattice mismatch between InN and GaN is about 11% causing huge defects at the interface. Krupanidhi et al. [3] have grown InN/GaN heterostructures using PAMBE system where InN films of 300 nm thick were grown epitaxially on 4 μm-GaN/Al2O3 (0 0 0 1) templates at different growth temperatures. An important feature of InN/GaN heterostructures is the existence of spontaneous (sp) and piezoelectric (pz) polarizations at the interface [4]. The sp polarization arises due to non-center symmetric nature of hexagonal unit cell of InN and GaN; and pz polarization is induced due to lattice mismatch strain [5]. These polarizations generate a strong built-in-polarization (BIP) electric field whose intensity reaches several MV/cm [6,7]. During the operation of InN/GaN heterostructure LED or LD, temperature at interface varies/rises. Due to temperature variation,
this BIP changes its behavior which modifies significantly optical, thermal and electrical properties of heterostructures [8]. So, it is required to explore what happens to polarization mechanism under heating and its effect on performance of InN/GaN heterstrucure devices. The change of the spontaneous polarization with temperature causes the pyroelectric effect which is measured by its pyroelectric coefficient [9–14]. This property arises naturally in low dimensional structures like InN/GaN heterostructure due to symmetry breaking. This special effect of InN/GaN heterostructure is promising for applications in surveillance, night vision, thermal radiation detectors and missile tracking and guidance and in high temperature pyroelectric sensors [5]. The pyroelectricity is attributed to the quasiharmonic thermal shifts of internal strains (internal displacements of cations and anions carrying their Born effective charges) [15]. The pyroelectric coefficient is divided into two parts: primary and secondary pyroelctric coefficients [9–11]. The primary pyroelectricity (zero-external-strain) dominates at low temperatures, while the secondary pyroelectricity (from external thermal strains) becomes comparable with the primary pyroelectricity at high temperatures [15]. Contributions of acoustic and optical phonon modes to the primary pyroelectric coefficient are described by Debye function and Einstein function, respectively [12,13]. The pyroelectric coefficients of GaN and AlN at room temperature are respectively 4.8 and 6 to 8.7 μC m2 K1 [14–17]. However, the pyroelectric coefficient of InN is
* Corresponding author. E-mail address: [email protected] (B.K. Sahoo). https://doi.org/10.1016/j.jpcs.2018.02.018 Received 9 October 2017; Received in revised form 24 January 2018; Accepted 9 February 2018 Available online 14 February 2018 0022-3697/© 2018 Elsevier Ltd. All rights reserved.
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
Where, constants A and B are amplitude coefficients of vibration of acoustic and optical modes, respectively. θD and θE are Debye and Einstein temperatures, respectively; andx ¼ ℏω=kβ T. Here ℏ ¼ h=2π ; h is Planck's constant and kβ is Boltzmann's constant. Debye temperature is defined by θD ¼ ℏ ωD = kβ and Einstein temperature is defined byθE ¼ ℏ ω=kβ where ωD is Debye frequency and ω is Einstein frequency. According to Einstein theory of lattice specific heat, atoms vibrate independently with same frequencyω; and can vibrate up to any frequency without cut-off limit. According to Debye theory of lattice specific heat, atoms are coupled and vibrate collectively with a continuously varying frequency having a cut-off frequency ωD known as Debye frequency. A number of solids has ωD < ω which shows that θD <θE . Debye temperature of GaN, InN and AlN are 600, 660 and 980 K, respectively. So, Einstein temperatures of above semiconductors are higher than their corresponding Debye temperature values. The terms represented by eqs. (1) and (2) are same in form, but their temperature dependence are different. The temperature dependence of the primary pyroelectric coefficient of the wurtzite nitrides is expressed as
still missing in the literature [18]. A number of groups have studied pyroelectric effect; however the transition temperature between primary and secondary pyroelectricity has not been investigated [9–15]. In this work, this transition temperature has been explored in InN/GaN heterostructures. Thermal conductivity changes with temperature; so the polarization mechanism under heating and its effect in InN/GaN heterostructures could be well explored by investigating the lattice thermal conductivity of InN/GaN heterostructures taking the BIP electric field of the heterostructures into account. 2. Theoretical model The Psp of InN and GaN are very close, roughly considered as 0.042 C m-2 and -0.041 C m-2, respectively [19–21]. So, Psp of InN/GaN heterostructure is Psp ¼ 0.001 C m-2; and pz polarization of InN/GaN heterostructure is Ppz ¼ ½e31 21 þ e32 22 þ e33 23 Cm2. Hereeij represents piezoelectric coefficients of InN and 21 ¼ 22 ¼ ðaInN aGaN Þ=aGaN and 23 ¼ ðcInN cGaN Þ=cGaN represent strain in InN layer due to growth of InN on GaN which generates biaxial starin in InN. Growing InN on GaN produces elastic strains in both materials, but when, as is usually the case for the single heterojunction, the GaN layer is much thicker than InN layer, the GaN layer is unstrained and all the strain occurs in the InN layer. The direction of Psp is along [0001], i.e, towards substrate where as Ppz is along [0001] i.e, along growth direction of InN/GaN heterostructure [7] (see Fig. 1). The material parameters of InN and GaN are taken from Refs. [4,5,19, 20]. Total built-in-polarization is P¼Psp þ Ppz which is given by 0.001þ ðe31 21 þ e32 22 þ e33 23 ÞCm2. The conventional pyroelectric coefficient is defined as χ ¼ dPSP =dT[5].This coefficient consists of two partsχ ¼ χ p þ χ s . The first part is primary pyroelectric coefficient whereas second part is secondary pyroelectric coefficient. The primary pyroelectric coefficient χ p is induced by atomic rearrangements inside the unit cell [15]. According to the dynamical theory of crystal lattices, the primary pyroelectric coefficient of the wurtzite nitrides can be expressed as the sum of two terms which denote the contributions from acoustic modes and optical modes. The temperature dependence of contribution of acoustic modes to the primary pyroelectric coefficient of the wurtzite nitrides is approximately described by Ref. [12].
χ acoustic ¼
A 3
T θD
3
θ =T
∫ 0D
x4 ex dx 1Þ2
ðex
χ P ¼ χ acoustic þ χ optical "
χ optical ¼
# 2 θE eθE =T B T ðeθE =T 1Þ2
A 3
T θD
3
θ =T
∫ 0D
x4 ex dx þ B x ðe 1Þ2
# 2 θE eθE =T T ðeθE =T 1Þ2
(3)
It means primary pyroelectric coefficient can be expressed as sum of the Debye function and Einstein function. The contribution of the acoustic modes to the primary pyroelectric coefficient is dominant from 0 K to around 100 K and the contribution of optical modes is very small. When the temperature is close to 600 K, the contributions of the optical modes and acoustic modes are comparable. It indicates that the contributions of the acoustic and optical modes to the primary pyroelectric coefficient change with temperature. It is substantially different from that of conventional ferroelectrics-based pyroelectrics which are described only by the optical modes. According to pyroelectric theory, at low temperature the primary pyroelectric coefficient is proportional to specific heat CV of Debye–Einstein model which is given by Ref. [6]. " CV ¼
# 3 2 T x4 ex θE eθE =T θ =T 3N k 3 ∫ 0D dx þ 3N k θD T ðeθE =T 1Þ2 ðex 1Þ2 (4)
(1)
It considers both Debye specific heat and Einstein specific heat. At higher temperature, pyroelectric coefficient is proportional to Cp which includes thermal expansion of the InN. In order to obtain respective contributions of acoustic modes and optical modes to the primary pyroelectric coefficient of the wurtzite nitrides at different temperature, constants A and B need to be determined. The experimental data of Cp of GaN has been reported by Danilencko et al. [22]. Yan et al. [12] have used eq. (4) to fit with the experimental data to evaluate the constants A and B. The temperature dependence of the primary pyroelectric coefficient of binary nitrides obtained by Yan et al. for GaN is given by Ref. [12].
Similarly, the temperature dependence of the contribution of the optical modes to the primary pyroelectric coefficient of the wurtzite nitrides is approximately expressed as [12]. "
¼
(2)
"
χP ¼
# 3 2 T x4 ex θE eθE =T θ =T 10:03 ∫ 0D dx þ 3:34 θD T ðeθE =T 1Þ2 ðex 1Þ2 (5)
In this work, same expression has been used for InN. The numerical values may change material to material. Since the values of A and B are unavailable for InN, we have used same numerical values expecting that this should not differ much for other same group III nitrides. The secondary pyroelectric coefficient χ s is the contribution induced by coupled effects of thermal deformation and piezoelectricity [15]. Therefore, the secondary pyroelectric coefficient is calculated from piezoelectric coefficients and thermal expansion coefficients of InN and is calculated by e31 α1 þ e32 α2 þ e33 α3 . αi are the thermal expansion
Fig. 1. InN/GaN heterostructure. 112
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
h
coefficients of InN. However, it is reported that piezoelectric constants of GaN and InN are uncertain by as much as 30% [21]. Therefore, the calculated χ should be considered as rough estimates. From Eq. (1), it is clear that χ p follows the form of specific heat. Therefore, χ p vanishes as T3 at low temperatures and saturates at high temperatures. Above room temperature, the secondary pyroelectric effect is comparable with the primary effect [12]. For GaN, disagreement in the experimentally measured pyroelectric coefficients is reported [15–17], possibly due to the piezoelectric contribution from the strain introduced by the substrates. In semiconductors, the thermal energy is carried by acoustic phonons. Optical phonon's group velocity is small in comparison to acoustic phonons so their contribution to thermal energy transport is negligible. Thus Einstein's term has negligible contribution in both primary pyroelectric coefficient and specific heat at low temperature. The kinematic expression for the thermal conductivity is [22]. k ¼ Cp v l 3 ¼ CP v2 τc 3;
1 1 1 1 1 τc ¼ 1 = τ1 n þ τ u þ τ p þ τ d þ τ b þ τ ep
h
Polarization mechanism modifies the thermal parameters. The revised parameters such as elastic constant, phonon velocity and Debye temperature of InN have been computed taking into account the built-inpolarization field. In this work, the material parameters and spontaneous and piezoelectric coefficients of InN and GaN are taken from Refs. [4,5, 19,20]. Calculation finds that elastic constant including polarization C44, is 61 GPa, which is around 28% higher than C44 p (48 GPa).1 GPa ¼ 109 Pa. The average phonon group velocity with polarization vp is 10% higher than velocity without polarization, v. The percentage change is defined by [(vp - v)/v] x 100. The Debye temperature is enhanced by 10%. The contribution of polarization to thermal conductivity has been computed. The Gruneisen parameter has been chosen as 0.58 [31]. Fig. 2 shows the simulated thermal conductivities as a function of temperature for with and without BIP field (k: thermal conductivity without BIP field and kp: including BIP field). A strong reduction of the thermal conductivity is clearly observed in the Fig. 2 [26]. This is due to point defect (mass difference) scattering. At high temperature, Umklapp phonon scattering dominates. Hence thermal conductivity shows exponential decrease. In this study dislocation density has been kept lower than 1011 cm2. Higher than this value, thermal conductivity shows significant decrease [32–34].The boundary scattering has been computed using thickness of InN as L ¼ 300 nm [3]. However, if L is taken lower than 200 nm, then phonon thermal conductivity decreases more than 20% [35]. Fig. 3 shows thermal conductivity of InN layer as a function of thickness. It can be seen that when InN thickness is equal to or above 1000 nm, thermal conductivity begins to approach a constant value of bulk. However, when thickness is lower than 1000 nm, there is a regular decrease of thermal conductivity. As thickness become lower than 600 nm, thermal conductivity decreases fast. Below 300 nm, thermal conductivity shows a significant drop. In thin films, when the characteristic length (film thickness) is comparable to the phonon mean free path, the boundary or interface scattering becomes important [36]. It is to be noted that Roul et al. [3] have grown InN/GaN heterostructure by plasma-assisted molecular beam epitaxial (PAMBE) technique. They have grown InN films of 300 nm thick epitaxially on 4 μm-GaN/Al2O3 (0 0 0 1) templates at different growth temperatures under nitrogen rich condition and reported that thickness controls transport properties significantly.
(6)
e15 2 þ e31 2 þ e33 2 þ psp 2
.
ε0 ε
(9)
3. Results and discussion
where v, l andτc are phonon group velocity, phonon mean free path and relaxation time, respectively. From above discussions, it can be concluded that thermal conductivity is a function of pyroelectric coefficient and specific heat. This suggests that thermal conductivity measurement can reveal pyroelectric property of a material as both show similar type of variation when temperature changes. Thus, study of thermal conductivity including contribution of BIP electric field could give information regarding pyroelectric coefficient of InN/GaN heterostructure. ! ! ! The polarization P generates BIP electric field E ¼ P =ε0 εr [5,6]. This field interacts with strain field of the InN/GaN heterostructure and modifies elastic constant which changes phonon velocity, Debye temperature and phonon mean free path of the InN [23]. This enhances thermal conductivity of the material [24]. The elastic constant with polarization field is [25]. C44; p ¼ C44 ; p þ
i
i (7)
where C44 is elastic constant of InN without polarization effect. The
1 ; average velocity v is defined by v ¼ 1 3ð1=vT;1 þ 1=vT;2 þ 1=vL Þ =
where vL and vT are the longitudinal and transverse sound velocities, respectively. The transverse branches are computed by vT ¼ ½C44 =ρ1=2 and the longitudinal velocity is determined byvL ¼ ½C33 =ρ 1=2 . The Debye frequency is determined by the relation ωD ¼ v½3N=4π V0 1=3 :Here, N is the number of atoms present in the unit cell of volume V0. The Debye temperature θD is calculated byθD ¼ h ωD = kβ . The thermal conductivity in a solid is written as sum of phonon and electron contribution k ¼ kph þ ke . Experimentally, it has been known that ke ¼ 103 kph . Thus, electronic contribution is neglected and for a semiconductork ffi kph . Thermal conductivity is calculated by Callaway formula [24]. k ffi kph ¼
kβ 4 T 3 θD =T τc x4 ex ∫ dx 2π 2 ℏ3 v 0 ðex 1Þ2
(8)
The phonon scattering processes are divided into normal scattering (τn ) and resistive scattering (τr ).The normal scattering process describes phonon-phonon scatterings where momentum is conserved. In resistive processes phonon momentum is not conserved. The resistive processes are Umklapp scattering (τu 1 ), point-defect scattering (τp 1 ), dislocation scattering (τd 1 ), boundary scattering (τb 1 ) and phonon-electron scattering (τep 1 ). The inverse of scattering rate is phonon relaxation time which is elaborated in Ref. [28]. The combined phonon relaxation time is given by Refs. [26–31].
Fig. 2. Thermal conductivity of InN/GaN heterostructure as a function of temperature. 113
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
and is given by CðPÞ Cð0Þ ¼
"
T
dP dT
ε0 εr
2 þ P
# d2 P dT 2
(10)
As polarization phenomenon enhances the phonon velocity and mean free path; so, phonon group velocity and mean free path with polarization mechanism can be written as v þ dv and l þ dl, where dv and dl are respective polarization contributions. According to eq. (3), the change in thermal conductivity of the heterostructure due to polarization mechanism can be written as kp k ¼
T
"
dP dT
3 ε0 εr
2
# d2 P þ P 2 dv dl dT
(11)
This expression can be used to explain Fig. 2. (1). From Fig. 2, thermal conductivities (with and without polarization) are equal, indicating that polarization is not contributing to k at Tp i. e, kp ¼ k ; at T ¼Tp. It can be observed from Fig. 2 that, Tp is close to 120 K. Hence eq. (11) can be written as
Fig. 3. Thermal conductivity as a function of thickness of InN.
The role of the electron–phonon coupling is relatively small for bulk material. Neglecting the electron–phonon coupling may lead to over estimation of the thermal transport for ultra-thin films. The electron concentration ne is taken as 1019cm3. However, if electron concentration ne is above 1020 cm3, then thermal conductivity is decreased by 30% [37–39]. Electrons in the conduction band scatter low-energy phonons effectively where surface modes dominate, resulting in a smaller thermal conductivity. Our study points out that phonon scattering rates are suppressed by built in polarization mechanism; even strong phonon scattering from point defect is suppressed by polarization field. The influence of polarization is clearly observed in the low and high temperature ranges. The peak value of thermal conductivity decreases due to polarization effect. This is explained by the fact that polarization effect significantly changes the phonon density of states. This redistribution of phonons results as phonons gain energy from polarization effect and the effect improves the energy difference between the adjacent modes of phonons [40]. From Fig. 2, it can also be seen that room temperature thermal conductivity is enhanced by built-in-polarization field [25,41]. This is due to high Debye temperature which extends the upper limit of thermal conductivity integration and longer phonon mean free path at room temperature. A noticeable feature observed in an experimental study [31,41] was that the thermal conductivity of the InN film shows higher thermal conductivity than the theoretical prediction at room temperature. We propose that one reason of this high thermal conductivity of this InN film at room temperature could be due to the contribution of built-in-polarization field to thermal conductivity. It can be seen from Fig. 2 that thermal conductivity with BIP field (kp) is lower than thermal conductivity without BIP field (k) up to a certain characteristic temperature, Tp. The thermal conductivities can be written as kp k ¼ ve for T Tp; kp k ¼ þ ve for T > Tp and kp k ¼ 0 for T ¼ Tp. At Tp both thermal conductivities show cross over. This happens due to change in polarization with temperature which changes thermal conductivity k. This signifies pyroelectric nature of the InN/GaN heterostructure. Thus, thermal conductivity measurement can reveal pyroelectricity of InN/GaN heterostructure. A thermo dynamical theory has been proposed to explain this result. The change in free energy of the heterostructure material under built-in! ! polarization electric field is dF ¼ P :d E . Here F stands for Gibb's free energy, P and E, respectively stands for polarization and built-in-polari! ! zation electric field. This can be written asdF ¼ ð P : d P Þ=ε0 εr . The increase in the free energy of the system isFðPÞ Fð0Þ ¼ P2 =ð2ε0 εr Þ . F(P) and F(0), respectively represents free energy with and without polarization. The change in entropy is ð∂F=∂TÞand it becomes ðP=ε0 εr Þð∂P=∂TÞ . The change in specific heat of the system is T ð∂S=∂TÞ
0 ¼
T
"
3 ε0 εr
dP dT
2 þ P
# d2 P dv dl atT ¼ Tp; dT 2
(12)
This equation requires either dv ¼ 0 or dl ¼ 0 or ½ðdP=dTÞ2 þ Pðd2 P= dT 2 Þ ¼ 0. Irrespective of the temperature, polarization always contributes to phonon velocity and mean free path; so, dv and dl cannot be taken as zero. The only choice remain is ½ðdP=dTÞ2 þ Pðd2 P=dT 2 Þ ¼ 0. One trivial solution of this is P ¼ 0 which immediately implies that its contribution to phonon velocity and mean free path is also zero i.e, dv ¼ dl ¼ 0. Thus, Psp þ Ppz ¼ 0 which implies 0.001þ ðe31 21 þ e32 22 þ e33 23 Þ ¼ 0 takes place close to Tp. This can be explained by the fact that sp and pz are equal and opposite to each other in the InN/GaN heterostructure and they balance each other close to Tp. Let us examine if P ¼ 0 close to Tp. The piezoelectric polarization has been calculated using temperature dependent form of lattice constants and thermal expansion coefficients of InN and GaN.21 ¼ ½aInN ð1 þ αaInN TÞ aGaN ð1 þ αaGaN TÞ = ½aGaN ð1 þ αaGaN TÞ. The values of thermal expansion coefficients of InN and GaN are taken from Refs. [42,43]. 23 ¼ ½cInN ð1 þ αcInN dTÞ cGaN ð1 þ αcGaN TÞ = ½cGaN ð1 þ αcGaN TÞ. The variation of sp, pz and total P as a function of temperature has been plotted in Fig. 4. It can be observed from Fig. 4 that total polarization P has a zero crossover (X-axis zero line) at a temperature below 130 K. Thus, the solutions k¼kp and P ¼ 0 are falling within a
Fig. 4. Total polarization as a function of temperature. 114
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
computed up to 130 K; and above 130 K, Ppz will be taken in the calculation for secondary pyroelectric effect (χ S ). In InN/GaN heterostructure,
1 dT 130 ∫0 . This yields primary Psp ¼ 0.001 C m-2. So, χ p ¼ ðPsp Þ pyroelectric coefficient roughly equal to 7.692 C m2K1 at 130 K. Well above room temperature, secondary pyroelectric effect is comparable with the primary pyroelectric effect; however at room temperature it is small. We computedχ S at temperature 300 K from e31 α1 þ e32 α2 þ e33 α3 . It is around - 0.733μC m2 K1. So, pyroelectric coefficient at 300 K is roughly equal to 8.425μC m2 K1. The pyroelectric coefficients of GaN and AlN at room temperature are measured and found, respectively as 4.8 and 6 to 8.7 C m2 K1 [14–17]. It can be argue that this is minimum value of pyroelectric coefficient at room temperature as minimum spontaneous polarization difference between InN and GaN has been considered. The room temperature thermal conductivity of InN/GaN heterostructure with and without polarization is predicted, respectively as 32 and 48 W m-1 K1.This shows polarization phenomenon enhances thermal conductivity at room temperature [47–50]. From Fig. 2, it can be seen that above temperature Tp, thermal conductivity with and without BIP field decrease, however at different rates with temperature. This decrease at different rate suggests that thermal conductivity measurement can reveal pyroelectric behavior of a semiconductor [16].
temperature range of 120–130 K. Both k¼kp and P ¼ 0 are not falling at a particular temperature is attributed to uncertainty in values of material parameters, polarization constants and thermal expansion coefficients of InN and GaN. Romanov et al. [44] have demonstrated that for strained InxGa1x N and AlyGa1y N layers lattice matched to GaN, the piezoelectric polarization becomes zero for nonpolar orientations and also at another point ~ 45 tilted from the c plane. The zero crossover has only a very small dependence on the In or Al content of the ternary alloy layer. With the addition of spontaneous polarization, the angle at which the total polarization equals zero increases slightly for InxGa1xN, but the exact value depends on the In content. For AlyGa1y N mismatched layers the effect of spontaneous polarization becomes important by increasing the crossover point to ~70 from c-axis oriented films. In this work, we have shown that total polarization (sp plus pz) can be made zero at a particular temperature in InN/GaN heterostructure and if InN layer is alloyed with GaN this temperature of zero crossover can be tailored according to wish. (2). From Fig. 2, kp k ¼ ve for T Tp. That is BIP field reduces thermal conductivity of heterostructure below Tp. This is explained by the fact that below this temperature thermal contraction of lattices in InN begins. This exhibits a decrease in thermal expansion coefficient (αi ) with decreasing T at low temperatures. This behavior is associated with the change in normal mode frequency due to change in volume which is described by Gruneisen parameter. This results in reduction of Grunessen parameter with decreasing temperature. In the case of Si, αi becomes negative at 120 K [45]. The Gruneisen parameters for certain low frequency transverse acoustic modes are negative and produce the negative range of αi . Due to above reasons Ppz becomes negligible small; and Psp dominates over Ppz at low temperature below Tp. InN has negative thermal expansion coefficients over a range of low temperatures. Recent study [46] has reported that thermal expansion coefficient of InN is very small and negative thermal expansion coefficient begins below 120 K for InN. It can be seen from Fig. 2 that temperature Tp for InN is close to 120 K. (3). From Fig. 2, kp k ¼ þ ve for T > Tp. That is BIP field enhances thermal conductivity of heterostructures above Tp. This is explained by the fact that above temperature Tp thermal expansion of lattices in InN becomes appreciable. Above Tp, pz starts dominating over sp due to thermal expansion which generates more lattice mismatch resulting in significant contribution of polarization to thermal conductivity [25,40,41]. Thus, Tp can be considered as transition temperature between primary and secondary pyroelectric effect in InN as above Tp thermal expansion takes place which is reason of secondary pyroelectric effect. Now it is possible to predict pyroelectric coefficient of the InN at room temperature. It is calculated in the following way. From eq. (12) we take ½χ 2 þ Pðdχ =dTÞ ¼ 0; here, χ ¼ dP=dTas other factors appearing in eq. (12) cannot be taken as zero. This can be written as
dχ
χ2
χ ¼
T
∫0
¼
dT P which
dT þC ðPsp þ Ppz Þ
4. Conclusions The pyroelectric property of InN/GaN heterostructure has been investigated theoretically from effect of built in polarization on lattice thermal conductivity. The built-in-polarization (BIP) enhances thermal conductivity of InN/GaN heterostructure at room temperature. The variation of thermal conductivity (k) with temperature for including and excluding BIP mechanism in the heterostructure predicts a transition temperature between primary and secondary pyroelectric effects. Below this temperature, kp with BIP field is lower than k without BIP field. This is due to decrease of Grunessen parameter below transition temperature while above this temperature kp is significantly contributed by BIP mechanism due to thermal expansion. The room temperature thermal conductivity of InN/GaN heterostructure with and without polarization is predicted, respectively as 32 and 48 W m-1 K1. The transition temperature Tp between primary and secondary pyroelectric effect; and room temperature pyroelectric coefficient of InN has been predicted as 120 K and 8.425μC m2 K1, respectively which are in line with prior literature studies and argued that this is minimum value of pyroelectric coefficient of InN at room temperature. This study suggests that thermal conductivity measurement can reveal role of phonons in pyroelectricity in InN. Acknowledgement Author BKS acknowledges with thanks to DST-SERB, Govt. of India for financial support through Grant No. EMR/2016/001019.GH acknowledges University Grants Commission, India for financial support (Award letter No: F1-17-1/2016-17/NFST-2015-17-ST-772/SA-III/ website).
yields
1 :
(13)
References
At this point, it is not possible to find out value of C, however a closer look to eq. (5) suggests that χ p follows the form of specific heat. Thereforeχ p vanishes as T3 at low temperatures and saturates at high temperatures. For rough estimate of primary pyroelectric coefficientχ p , we can set C ¼ 0 by assuming that at T ¼ 0, there is no polarization and hence there is no pyroelectricity in the material. In other words, as T→0, Cv →0 which is third law of thermodynamics. Using this assumption, we have estimated pyroelectric coefficient at room temperature (300 K). From Fig. 2, Psp dominates up to temperature 130 K; and above 130 K, Ppz becomes appreciable. So, primary pyroelectric coefficient will be
[1] H. Morkoc, Nitride Semiconductor Devices, Wiley-VCH Verlag GmbH & Co., Weinheim, 2013. [2] J. Wu, J. Appl. Phys. 106 (2009), 011101. [3] B. Roul, M. Kumar, M.K. Rajpalke, T.N. Bhat, S.B. Krupanidhi, J. Phys. D Appl. Phys. 48 (2015), 423001. [4] O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, B. Schaff, L.F. Eastman, J. Appl. Phys. 87 (2000) 334. [5] O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, B. Schaff, L.F. Eastman, J. Phys. Condens. Matter 14 (2002) 3399. 115
G. Hansdah, B.K. Sahoo [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Journal of Physics and Chemistry of Solids 117 (2018) 111–116 [29] S. Krukowskia, A. Witeka, J. Adamczyk, J. Jun, M. Bockowski, I. Grzegory, B. Lucznika, G. Nowaka, M. Wroblewskia, A. Presz, S. Gierlotka, S. Stelmacha, B. Palosza, S. Porowski, P. Zinnt, J. Phys. Chem. Solid. 59 (1998) 289. [30] A.X. Levander, T. Tong, K.M. Yu, J. Suh, D. Fu, R. Zhang, H. Lu, W.J. Schaff, O. Dubon, W. Walukiewicz, D.G. Cahill, J. Wu, Appl. Phys. Lett. 98 (2011), 012108. [31] T. Tong, D. Fu, A.X. Levander, W.J. Schaff, B.N. Pantha, N. Lu, B. Liu, I. Ferguson, R. Zhang, J.Y. Lin, H.X. Jiang, J. Wu, David G. Cahill, Appl. Phys. Lett. 102 (2013), 121906. [32] A. Jezowski, B. Danilchenko, M. Bockowski, I. Grzegory, S. Krukowski, T. Suski, T. Paszkiewicz, Solid State Commun. 128 (2003) 69. [33] G. Slack, L. Schowalter, D. Morelli, J. Freitas, J. Cryst. Growth 246 (2002) 287. [34] D. Kotchetkov, J. Zou, A.A. Balandin, D.I. Florescu, F.H. Pollak, Appl. Phys. Lett. 79 (2001) 4316. [35] G. Pozina, Appl. Phys. Lett. 107 (2015), 251106. [36] A. Minnich, Appl. Phys. Lett. 107 (2015), 183106. [37] G. Tang, J. Appl. Phys. 110 (2011), 046102. [38] B. Liao, Phys. Rev. Lett. 114 (2015), 115901. [39] W. Wan, J. Phys. Condens. Matter 24 (2012), 295402. [40] L. Zhu, H. Luo, Theor. Appl. Mech. Lett. 6 (2016) 277–281. [41] A. Jezowski, O. Churiukova, J. Mucha, T. Suski, I.A. Obukhov, B.A. Danilchenko, Mater. Res. Express 2 (2015), 085902. [42] K. Wang, R.R. Reeber, Appl. Phys. Lett. 79 (2001) 1602. [43] S. Figge, H. Kroncke, D. Hommel, B.M. Epelbaum, Appl. Phys. Lett. 94 (2009), 101915. [44] A.E. Romanov, T.J. Baker, S. Nakamura, J.S. Speck, J. Appl. Phys. 100 (2006), 023522. [45] M. Balkanski, R.F. Wallis, Semiconductor Physics and Application, Oxford University Press Inc., New York, 2000. [46] L. Xu, R. Wang, X. Yang, H. Yan, J. Appl. Phys. 110 (2011), 043528. [47] W. Liu, A.A. Balandin, C. Lee, H.Y. Lee, Phys. Stat. Sol.(a) 202 (2005) R135–R137. [48] W. Liu, A.A. Balandin, J. Appl. Phys. 97 (2005), 123705. [49] A. Jezowski, et al., Mater. Res. Express 2 (2015), 085902. [50] A. Pansari, V. Gedam, B.K. Sahoo, J. Phys. Chem. Solid. 87 (2015) 177.
C. Ren, Material Sci. Technol. 32 (2016) 418. F. Bernardini, V. Fiorentini, Appl. Surf. Sci. 166 (2000) 23. V.N. Davydov, Russ. Phys. J. 57 (2015) 1648. J. Zook, S.T. Liu, J. Appl. Phys. 49 (1978) 4604. T.B. Bahder, R. Tober, J. Bruno, Superlattice. Microst. 14 (1993) 149. N. Cho, Y. Poplavko, S.J. Noh, J. Kor. Phys. Soc. 42 (2003) 803. W. Yan, R. Zhang, Z. Xie, X. Xiu, Y. Zheng, Z. Liu, S. Xu, Z. He, Appl. Phys. Lett. 94 (2009), 242111. W. Yan, R. Zhang, X. Xiu, Z. Xie, P. Han, R.L. Jiang, S.L. Gu, Y. Shi, Y.D. Zheng, Appl. Phys. Lett. 90 (2007), 212102. V. Fuflyigin, E. Salley, A. Osinsky, P. Norris, Appl. Phys. Lett. 77 (2000) 3075. J. Liu, M.V. Fernandez-Serra, P.B. Allen, Phys. Rev. B 93 (R) (2016), 081205. A.D. Bykhovski, V.V. Kaminski, M.S. Shur, Q.C. Chen, M.A. Khan, Appl. Phys. Lett. 69 (1999) 3254. Sven Jachalke, Patrick Hofmann, Gunnar Leibiger, Frank S. Habel, Erik Mehner, Tilmann Leisegang, Dirk C. Meyer, Thomas Mikolajick, Appl. Phys. Lett. 109 (2016), 142906. L.C. Lew Yan Voon1, M. Willatzen, J. Appl. Phys. 109 (2011), 031101. I. Vurgaftman, J.R. Meywer, J. Appl. Phys. 94 (2003) 3675. M. Feneberg, K. Thonke, J. Phys. Condens. Matter 19 (2007) 403201. C.E. Dreyer, A. Janotti, C.G. Van de Walle, D. Vanderbilt, Phys. Rev. X 6 (2016), 021038. B. Danilchenko, T. Paszkiewicz, A. Jezowski, T. Plackowski, Appl. Phys. Lett. 89 (2006), 061901. B.K. Sahoo, J. Mater. Sci. 47 (2012) 2624. A. Jezowski, O. Churiukova, J. Mucha, T. Suski, I.A. Obukhov, B.A. Danilchenko, Mater. Res. Express 2 (2015), 085902. B.K. Sahoo, S.K. Sahoo, S. Sahoo, J. Phys. Chem. Solid. 74 (2013) 1669. J. Zou, D. Kotchetkov, A.A. Balandin, D.I. Florescu, F.H. Pollak, J. Appl. Phys. 92 (2002) 2534. A. Sztein, J.E. Bowers, S.P. DenBaars, S. Nakamura, J. Appl. Phys. 113 (2013), 183707. V. Gedam, A. Pansari, B.K. Sahoo, Indian J. Phys. 90 (9) (2016) 991.
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Pyroelectric effect and lattice thermal conductivity of InN/GaN heterostructures Gopal Hansdah, Bijay Kumar Sahoo * Department of Physics, N.I.T, Raipur, Raipur 492010, India
A R T I C L E I N F O
A B S T R A C T
Keywords: InN/GaN heterostructures Built-in-polarization Lattice thermal conductivity Pyroelectricity Transition temperature
The built-in-polarization (BIP) of InN/GaN heterostructures enhances Debye temperature, phonon mean free path and thermal conductivity of the heterostructure at room temperature. The variation of thermal conductivities (kp: including polarization mechanism and k: without polarization mechanism) with temperature predicts the existence of a transition temperature (Tp) between primary and secondary pyroelectric effect. Below Tp, kp is lower than k; while above Tp, kp is significantly contributed from BIP mechanism due to thermal expansion. A thermodynamic theory has been proposed to explain the result. The room temperature thermal conductivity of InN/ GaN heterostructure with and without polarization is respectively 32 and 48 W m-1 K1. The temperature Tp and room temperature pyroelectric coefficient of InN has been predicted as 120 K and 8.425μC m2 K1, respectively which are in line with prior literature studies. This study suggests that thermal conductivity measurement in InN/GaN heterostructures can help to understand the role of phonons in pyroelectricity.
1. Introduction InN/GaN heterostructure is a promising optoelectronics material for fabrication of light-emitting diodes (LED) and laser diodes (LD) [1]. The InN/GaN heterostructure system has several advantages. The advantages include high rate of optical phonon emission (2.5 1013 s1), operating in THz frequency range, high value of electron drift velocity (5 107 cm s1) and large conduction band offset [2]. The difference in the growth temperature between InN (~500 C) and GaN (~750 C) is large. So, fabrication of high quality InN/GaN heterostructures is a challenging task. In addition, the lattice mismatch between InN and GaN is about 11% causing huge defects at the interface. Krupanidhi et al. [3] have grown InN/GaN heterostructures using PAMBE system where InN films of 300 nm thick were grown epitaxially on 4 μm-GaN/Al2O3 (0 0 0 1) templates at different growth temperatures. An important feature of InN/GaN heterostructures is the existence of spontaneous (sp) and piezoelectric (pz) polarizations at the interface [4]. The sp polarization arises due to non-center symmetric nature of hexagonal unit cell of InN and GaN; and pz polarization is induced due to lattice mismatch strain [5]. These polarizations generate a strong built-in-polarization (BIP) electric field whose intensity reaches several MV/cm [6,7]. During the operation of InN/GaN heterostructure LED or LD, temperature at interface varies/rises. Due to temperature variation,
this BIP changes its behavior which modifies significantly optical, thermal and electrical properties of heterostructures [8]. So, it is required to explore what happens to polarization mechanism under heating and its effect on performance of InN/GaN heterstrucure devices. The change of the spontaneous polarization with temperature causes the pyroelectric effect which is measured by its pyroelectric coefficient [9–14]. This property arises naturally in low dimensional structures like InN/GaN heterostructure due to symmetry breaking. This special effect of InN/GaN heterostructure is promising for applications in surveillance, night vision, thermal radiation detectors and missile tracking and guidance and in high temperature pyroelectric sensors [5]. The pyroelectricity is attributed to the quasiharmonic thermal shifts of internal strains (internal displacements of cations and anions carrying their Born effective charges) [15]. The pyroelectric coefficient is divided into two parts: primary and secondary pyroelctric coefficients [9–11]. The primary pyroelectricity (zero-external-strain) dominates at low temperatures, while the secondary pyroelectricity (from external thermal strains) becomes comparable with the primary pyroelectricity at high temperatures [15]. Contributions of acoustic and optical phonon modes to the primary pyroelectric coefficient are described by Debye function and Einstein function, respectively [12,13]. The pyroelectric coefficients of GaN and AlN at room temperature are respectively 4.8 and 6 to 8.7 μC m2 K1 [14–17]. However, the pyroelectric coefficient of InN is
* Corresponding author. E-mail address: [email protected] (B.K. Sahoo). https://doi.org/10.1016/j.jpcs.2018.02.018 Received 9 October 2017; Received in revised form 24 January 2018; Accepted 9 February 2018 Available online 14 February 2018 0022-3697/© 2018 Elsevier Ltd. All rights reserved.
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
Where, constants A and B are amplitude coefficients of vibration of acoustic and optical modes, respectively. θD and θE are Debye and Einstein temperatures, respectively; andx ¼ ℏω=kβ T. Here ℏ ¼ h=2π ; h is Planck's constant and kβ is Boltzmann's constant. Debye temperature is defined by θD ¼ ℏ ωD = kβ and Einstein temperature is defined byθE ¼ ℏ ω=kβ where ωD is Debye frequency and ω is Einstein frequency. According to Einstein theory of lattice specific heat, atoms vibrate independently with same frequencyω; and can vibrate up to any frequency without cut-off limit. According to Debye theory of lattice specific heat, atoms are coupled and vibrate collectively with a continuously varying frequency having a cut-off frequency ωD known as Debye frequency. A number of solids has ωD < ω which shows that θD <θE . Debye temperature of GaN, InN and AlN are 600, 660 and 980 K, respectively. So, Einstein temperatures of above semiconductors are higher than their corresponding Debye temperature values. The terms represented by eqs. (1) and (2) are same in form, but their temperature dependence are different. The temperature dependence of the primary pyroelectric coefficient of the wurtzite nitrides is expressed as
still missing in the literature [18]. A number of groups have studied pyroelectric effect; however the transition temperature between primary and secondary pyroelectricity has not been investigated [9–15]. In this work, this transition temperature has been explored in InN/GaN heterostructures. Thermal conductivity changes with temperature; so the polarization mechanism under heating and its effect in InN/GaN heterostructures could be well explored by investigating the lattice thermal conductivity of InN/GaN heterostructures taking the BIP electric field of the heterostructures into account. 2. Theoretical model The Psp of InN and GaN are very close, roughly considered as 0.042 C m-2 and -0.041 C m-2, respectively [19–21]. So, Psp of InN/GaN heterostructure is Psp ¼ 0.001 C m-2; and pz polarization of InN/GaN heterostructure is Ppz ¼ ½e31 21 þ e32 22 þ e33 23 Cm2. Hereeij represents piezoelectric coefficients of InN and 21 ¼ 22 ¼ ðaInN aGaN Þ=aGaN and 23 ¼ ðcInN cGaN Þ=cGaN represent strain in InN layer due to growth of InN on GaN which generates biaxial starin in InN. Growing InN on GaN produces elastic strains in both materials, but when, as is usually the case for the single heterojunction, the GaN layer is much thicker than InN layer, the GaN layer is unstrained and all the strain occurs in the InN layer. The direction of Psp is along [0001], i.e, towards substrate where as Ppz is along [0001] i.e, along growth direction of InN/GaN heterostructure [7] (see Fig. 1). The material parameters of InN and GaN are taken from Refs. [4,5,19, 20]. Total built-in-polarization is P¼Psp þ Ppz which is given by 0.001þ ðe31 21 þ e32 22 þ e33 23 ÞCm2. The conventional pyroelectric coefficient is defined as χ ¼ dPSP =dT[5].This coefficient consists of two partsχ ¼ χ p þ χ s . The first part is primary pyroelectric coefficient whereas second part is secondary pyroelectric coefficient. The primary pyroelectric coefficient χ p is induced by atomic rearrangements inside the unit cell [15]. According to the dynamical theory of crystal lattices, the primary pyroelectric coefficient of the wurtzite nitrides can be expressed as the sum of two terms which denote the contributions from acoustic modes and optical modes. The temperature dependence of contribution of acoustic modes to the primary pyroelectric coefficient of the wurtzite nitrides is approximately described by Ref. [12].
χ acoustic ¼
A 3
T θD
3
θ =T
∫ 0D
x4 ex dx 1Þ2
ðex
χ P ¼ χ acoustic þ χ optical "
χ optical ¼
# 2 θE eθE =T B T ðeθE =T 1Þ2
A 3
T θD
3
θ =T
∫ 0D
x4 ex dx þ B x ðe 1Þ2
# 2 θE eθE =T T ðeθE =T 1Þ2
(3)
It means primary pyroelectric coefficient can be expressed as sum of the Debye function and Einstein function. The contribution of the acoustic modes to the primary pyroelectric coefficient is dominant from 0 K to around 100 K and the contribution of optical modes is very small. When the temperature is close to 600 K, the contributions of the optical modes and acoustic modes are comparable. It indicates that the contributions of the acoustic and optical modes to the primary pyroelectric coefficient change with temperature. It is substantially different from that of conventional ferroelectrics-based pyroelectrics which are described only by the optical modes. According to pyroelectric theory, at low temperature the primary pyroelectric coefficient is proportional to specific heat CV of Debye–Einstein model which is given by Ref. [6]. " CV ¼
# 3 2 T x4 ex θE eθE =T θ =T 3N k 3 ∫ 0D dx þ 3N k θD T ðeθE =T 1Þ2 ðex 1Þ2 (4)
(1)
It considers both Debye specific heat and Einstein specific heat. At higher temperature, pyroelectric coefficient is proportional to Cp which includes thermal expansion of the InN. In order to obtain respective contributions of acoustic modes and optical modes to the primary pyroelectric coefficient of the wurtzite nitrides at different temperature, constants A and B need to be determined. The experimental data of Cp of GaN has been reported by Danilencko et al. [22]. Yan et al. [12] have used eq. (4) to fit with the experimental data to evaluate the constants A and B. The temperature dependence of the primary pyroelectric coefficient of binary nitrides obtained by Yan et al. for GaN is given by Ref. [12].
Similarly, the temperature dependence of the contribution of the optical modes to the primary pyroelectric coefficient of the wurtzite nitrides is approximately expressed as [12]. "
¼
(2)
"
χP ¼
# 3 2 T x4 ex θE eθE =T θ =T 10:03 ∫ 0D dx þ 3:34 θD T ðeθE =T 1Þ2 ðex 1Þ2 (5)
In this work, same expression has been used for InN. The numerical values may change material to material. Since the values of A and B are unavailable for InN, we have used same numerical values expecting that this should not differ much for other same group III nitrides. The secondary pyroelectric coefficient χ s is the contribution induced by coupled effects of thermal deformation and piezoelectricity [15]. Therefore, the secondary pyroelectric coefficient is calculated from piezoelectric coefficients and thermal expansion coefficients of InN and is calculated by e31 α1 þ e32 α2 þ e33 α3 . αi are the thermal expansion
Fig. 1. InN/GaN heterostructure. 112
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
h
coefficients of InN. However, it is reported that piezoelectric constants of GaN and InN are uncertain by as much as 30% [21]. Therefore, the calculated χ should be considered as rough estimates. From Eq. (1), it is clear that χ p follows the form of specific heat. Therefore, χ p vanishes as T3 at low temperatures and saturates at high temperatures. Above room temperature, the secondary pyroelectric effect is comparable with the primary effect [12]. For GaN, disagreement in the experimentally measured pyroelectric coefficients is reported [15–17], possibly due to the piezoelectric contribution from the strain introduced by the substrates. In semiconductors, the thermal energy is carried by acoustic phonons. Optical phonon's group velocity is small in comparison to acoustic phonons so their contribution to thermal energy transport is negligible. Thus Einstein's term has negligible contribution in both primary pyroelectric coefficient and specific heat at low temperature. The kinematic expression for the thermal conductivity is [22]. k ¼ Cp v l 3 ¼ CP v2 τc 3;
1 1 1 1 1 τc ¼ 1 = τ1 n þ τ u þ τ p þ τ d þ τ b þ τ ep
h
Polarization mechanism modifies the thermal parameters. The revised parameters such as elastic constant, phonon velocity and Debye temperature of InN have been computed taking into account the built-inpolarization field. In this work, the material parameters and spontaneous and piezoelectric coefficients of InN and GaN are taken from Refs. [4,5, 19,20]. Calculation finds that elastic constant including polarization C44, is 61 GPa, which is around 28% higher than C44 p (48 GPa).1 GPa ¼ 109 Pa. The average phonon group velocity with polarization vp is 10% higher than velocity without polarization, v. The percentage change is defined by [(vp - v)/v] x 100. The Debye temperature is enhanced by 10%. The contribution of polarization to thermal conductivity has been computed. The Gruneisen parameter has been chosen as 0.58 [31]. Fig. 2 shows the simulated thermal conductivities as a function of temperature for with and without BIP field (k: thermal conductivity without BIP field and kp: including BIP field). A strong reduction of the thermal conductivity is clearly observed in the Fig. 2 [26]. This is due to point defect (mass difference) scattering. At high temperature, Umklapp phonon scattering dominates. Hence thermal conductivity shows exponential decrease. In this study dislocation density has been kept lower than 1011 cm2. Higher than this value, thermal conductivity shows significant decrease [32–34].The boundary scattering has been computed using thickness of InN as L ¼ 300 nm [3]. However, if L is taken lower than 200 nm, then phonon thermal conductivity decreases more than 20% [35]. Fig. 3 shows thermal conductivity of InN layer as a function of thickness. It can be seen that when InN thickness is equal to or above 1000 nm, thermal conductivity begins to approach a constant value of bulk. However, when thickness is lower than 1000 nm, there is a regular decrease of thermal conductivity. As thickness become lower than 600 nm, thermal conductivity decreases fast. Below 300 nm, thermal conductivity shows a significant drop. In thin films, when the characteristic length (film thickness) is comparable to the phonon mean free path, the boundary or interface scattering becomes important [36]. It is to be noted that Roul et al. [3] have grown InN/GaN heterostructure by plasma-assisted molecular beam epitaxial (PAMBE) technique. They have grown InN films of 300 nm thick epitaxially on 4 μm-GaN/Al2O3 (0 0 0 1) templates at different growth temperatures under nitrogen rich condition and reported that thickness controls transport properties significantly.
(6)
e15 2 þ e31 2 þ e33 2 þ psp 2
.
ε0 ε
(9)
3. Results and discussion
where v, l andτc are phonon group velocity, phonon mean free path and relaxation time, respectively. From above discussions, it can be concluded that thermal conductivity is a function of pyroelectric coefficient and specific heat. This suggests that thermal conductivity measurement can reveal pyroelectric property of a material as both show similar type of variation when temperature changes. Thus, study of thermal conductivity including contribution of BIP electric field could give information regarding pyroelectric coefficient of InN/GaN heterostructure. ! ! ! The polarization P generates BIP electric field E ¼ P =ε0 εr [5,6]. This field interacts with strain field of the InN/GaN heterostructure and modifies elastic constant which changes phonon velocity, Debye temperature and phonon mean free path of the InN [23]. This enhances thermal conductivity of the material [24]. The elastic constant with polarization field is [25]. C44; p ¼ C44 ; p þ
i
i (7)
where C44 is elastic constant of InN without polarization effect. The
1 ; average velocity v is defined by v ¼ 1 3ð1=vT;1 þ 1=vT;2 þ 1=vL Þ =
where vL and vT are the longitudinal and transverse sound velocities, respectively. The transverse branches are computed by vT ¼ ½C44 =ρ1=2 and the longitudinal velocity is determined byvL ¼ ½C33 =ρ 1=2 . The Debye frequency is determined by the relation ωD ¼ v½3N=4π V0 1=3 :Here, N is the number of atoms present in the unit cell of volume V0. The Debye temperature θD is calculated byθD ¼ h ωD = kβ . The thermal conductivity in a solid is written as sum of phonon and electron contribution k ¼ kph þ ke . Experimentally, it has been known that ke ¼ 103 kph . Thus, electronic contribution is neglected and for a semiconductork ffi kph . Thermal conductivity is calculated by Callaway formula [24]. k ffi kph ¼
kβ 4 T 3 θD =T τc x4 ex ∫ dx 2π 2 ℏ3 v 0 ðex 1Þ2
(8)
The phonon scattering processes are divided into normal scattering (τn ) and resistive scattering (τr ).The normal scattering process describes phonon-phonon scatterings where momentum is conserved. In resistive processes phonon momentum is not conserved. The resistive processes are Umklapp scattering (τu 1 ), point-defect scattering (τp 1 ), dislocation scattering (τd 1 ), boundary scattering (τb 1 ) and phonon-electron scattering (τep 1 ). The inverse of scattering rate is phonon relaxation time which is elaborated in Ref. [28]. The combined phonon relaxation time is given by Refs. [26–31].
Fig. 2. Thermal conductivity of InN/GaN heterostructure as a function of temperature. 113
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
and is given by CðPÞ Cð0Þ ¼
"
T
dP dT
ε0 εr
2 þ P
# d2 P dT 2
(10)
As polarization phenomenon enhances the phonon velocity and mean free path; so, phonon group velocity and mean free path with polarization mechanism can be written as v þ dv and l þ dl, where dv and dl are respective polarization contributions. According to eq. (3), the change in thermal conductivity of the heterostructure due to polarization mechanism can be written as kp k ¼
T
"
dP dT
3 ε0 εr
2
# d2 P þ P 2 dv dl dT
(11)
This expression can be used to explain Fig. 2. (1). From Fig. 2, thermal conductivities (with and without polarization) are equal, indicating that polarization is not contributing to k at Tp i. e, kp ¼ k ; at T ¼Tp. It can be observed from Fig. 2 that, Tp is close to 120 K. Hence eq. (11) can be written as
Fig. 3. Thermal conductivity as a function of thickness of InN.
The role of the electron–phonon coupling is relatively small for bulk material. Neglecting the electron–phonon coupling may lead to over estimation of the thermal transport for ultra-thin films. The electron concentration ne is taken as 1019cm3. However, if electron concentration ne is above 1020 cm3, then thermal conductivity is decreased by 30% [37–39]. Electrons in the conduction band scatter low-energy phonons effectively where surface modes dominate, resulting in a smaller thermal conductivity. Our study points out that phonon scattering rates are suppressed by built in polarization mechanism; even strong phonon scattering from point defect is suppressed by polarization field. The influence of polarization is clearly observed in the low and high temperature ranges. The peak value of thermal conductivity decreases due to polarization effect. This is explained by the fact that polarization effect significantly changes the phonon density of states. This redistribution of phonons results as phonons gain energy from polarization effect and the effect improves the energy difference between the adjacent modes of phonons [40]. From Fig. 2, it can also be seen that room temperature thermal conductivity is enhanced by built-in-polarization field [25,41]. This is due to high Debye temperature which extends the upper limit of thermal conductivity integration and longer phonon mean free path at room temperature. A noticeable feature observed in an experimental study [31,41] was that the thermal conductivity of the InN film shows higher thermal conductivity than the theoretical prediction at room temperature. We propose that one reason of this high thermal conductivity of this InN film at room temperature could be due to the contribution of built-in-polarization field to thermal conductivity. It can be seen from Fig. 2 that thermal conductivity with BIP field (kp) is lower than thermal conductivity without BIP field (k) up to a certain characteristic temperature, Tp. The thermal conductivities can be written as kp k ¼ ve for T Tp; kp k ¼ þ ve for T > Tp and kp k ¼ 0 for T ¼ Tp. At Tp both thermal conductivities show cross over. This happens due to change in polarization with temperature which changes thermal conductivity k. This signifies pyroelectric nature of the InN/GaN heterostructure. Thus, thermal conductivity measurement can reveal pyroelectricity of InN/GaN heterostructure. A thermo dynamical theory has been proposed to explain this result. The change in free energy of the heterostructure material under built-in! ! polarization electric field is dF ¼ P :d E . Here F stands for Gibb's free energy, P and E, respectively stands for polarization and built-in-polari! ! zation electric field. This can be written asdF ¼ ð P : d P Þ=ε0 εr . The increase in the free energy of the system isFðPÞ Fð0Þ ¼ P2 =ð2ε0 εr Þ . F(P) and F(0), respectively represents free energy with and without polarization. The change in entropy is ð∂F=∂TÞand it becomes ðP=ε0 εr Þð∂P=∂TÞ . The change in specific heat of the system is T ð∂S=∂TÞ
0 ¼
T
"
3 ε0 εr
dP dT
2 þ P
# d2 P dv dl atT ¼ Tp; dT 2
(12)
This equation requires either dv ¼ 0 or dl ¼ 0 or ½ðdP=dTÞ2 þ Pðd2 P= dT 2 Þ ¼ 0. Irrespective of the temperature, polarization always contributes to phonon velocity and mean free path; so, dv and dl cannot be taken as zero. The only choice remain is ½ðdP=dTÞ2 þ Pðd2 P=dT 2 Þ ¼ 0. One trivial solution of this is P ¼ 0 which immediately implies that its contribution to phonon velocity and mean free path is also zero i.e, dv ¼ dl ¼ 0. Thus, Psp þ Ppz ¼ 0 which implies 0.001þ ðe31 21 þ e32 22 þ e33 23 Þ ¼ 0 takes place close to Tp. This can be explained by the fact that sp and pz are equal and opposite to each other in the InN/GaN heterostructure and they balance each other close to Tp. Let us examine if P ¼ 0 close to Tp. The piezoelectric polarization has been calculated using temperature dependent form of lattice constants and thermal expansion coefficients of InN and GaN.21 ¼ ½aInN ð1 þ αaInN TÞ aGaN ð1 þ αaGaN TÞ = ½aGaN ð1 þ αaGaN TÞ. The values of thermal expansion coefficients of InN and GaN are taken from Refs. [42,43]. 23 ¼ ½cInN ð1 þ αcInN dTÞ cGaN ð1 þ αcGaN TÞ = ½cGaN ð1 þ αcGaN TÞ. The variation of sp, pz and total P as a function of temperature has been plotted in Fig. 4. It can be observed from Fig. 4 that total polarization P has a zero crossover (X-axis zero line) at a temperature below 130 K. Thus, the solutions k¼kp and P ¼ 0 are falling within a
Fig. 4. Total polarization as a function of temperature. 114
G. Hansdah, B.K. Sahoo
Journal of Physics and Chemistry of Solids 117 (2018) 111–116
computed up to 130 K; and above 130 K, Ppz will be taken in the calculation for secondary pyroelectric effect (χ S ). In InN/GaN heterostructure,
1 dT 130 ∫0 . This yields primary Psp ¼ 0.001 C m-2. So, χ p ¼ ðPsp Þ pyroelectric coefficient roughly equal to 7.692 C m2K1 at 130 K. Well above room temperature, secondary pyroelectric effect is comparable with the primary pyroelectric effect; however at room temperature it is small. We computedχ S at temperature 300 K from e31 α1 þ e32 α2 þ e33 α3 . It is around - 0.733μC m2 K1. So, pyroelectric coefficient at 300 K is roughly equal to 8.425μC m2 K1. The pyroelectric coefficients of GaN and AlN at room temperature are measured and found, respectively as 4.8 and 6 to 8.7 C m2 K1 [14–17]. It can be argue that this is minimum value of pyroelectric coefficient at room temperature as minimum spontaneous polarization difference between InN and GaN has been considered. The room temperature thermal conductivity of InN/GaN heterostructure with and without polarization is predicted, respectively as 32 and 48 W m-1 K1.This shows polarization phenomenon enhances thermal conductivity at room temperature [47–50]. From Fig. 2, it can be seen that above temperature Tp, thermal conductivity with and without BIP field decrease, however at different rates with temperature. This decrease at different rate suggests that thermal conductivity measurement can reveal pyroelectric behavior of a semiconductor [16].
temperature range of 120–130 K. Both k¼kp and P ¼ 0 are not falling at a particular temperature is attributed to uncertainty in values of material parameters, polarization constants and thermal expansion coefficients of InN and GaN. Romanov et al. [44] have demonstrated that for strained InxGa1x N and AlyGa1y N layers lattice matched to GaN, the piezoelectric polarization becomes zero for nonpolar orientations and also at another point ~ 45 tilted from the c plane. The zero crossover has only a very small dependence on the In or Al content of the ternary alloy layer. With the addition of spontaneous polarization, the angle at which the total polarization equals zero increases slightly for InxGa1xN, but the exact value depends on the In content. For AlyGa1y N mismatched layers the effect of spontaneous polarization becomes important by increasing the crossover point to ~70 from c-axis oriented films. In this work, we have shown that total polarization (sp plus pz) can be made zero at a particular temperature in InN/GaN heterostructure and if InN layer is alloyed with GaN this temperature of zero crossover can be tailored according to wish. (2). From Fig. 2, kp k ¼ ve for T Tp. That is BIP field reduces thermal conductivity of heterostructure below Tp. This is explained by the fact that below this temperature thermal contraction of lattices in InN begins. This exhibits a decrease in thermal expansion coefficient (αi ) with decreasing T at low temperatures. This behavior is associated with the change in normal mode frequency due to change in volume which is described by Gruneisen parameter. This results in reduction of Grunessen parameter with decreasing temperature. In the case of Si, αi becomes negative at 120 K [45]. The Gruneisen parameters for certain low frequency transverse acoustic modes are negative and produce the negative range of αi . Due to above reasons Ppz becomes negligible small; and Psp dominates over Ppz at low temperature below Tp. InN has negative thermal expansion coefficients over a range of low temperatures. Recent study [46] has reported that thermal expansion coefficient of InN is very small and negative thermal expansion coefficient begins below 120 K for InN. It can be seen from Fig. 2 that temperature Tp for InN is close to 120 K. (3). From Fig. 2, kp k ¼ þ ve for T > Tp. That is BIP field enhances thermal conductivity of heterostructures above Tp. This is explained by the fact that above temperature Tp thermal expansion of lattices in InN becomes appreciable. Above Tp, pz starts dominating over sp due to thermal expansion which generates more lattice mismatch resulting in significant contribution of polarization to thermal conductivity [25,40,41]. Thus, Tp can be considered as transition temperature between primary and secondary pyroelectric effect in InN as above Tp thermal expansion takes place which is reason of secondary pyroelectric effect. Now it is possible to predict pyroelectric coefficient of the InN at room temperature. It is calculated in the following way. From eq. (12) we take ½χ 2 þ Pðdχ =dTÞ ¼ 0; here, χ ¼ dP=dTas other factors appearing in eq. (12) cannot be taken as zero. This can be written as
dχ
χ2
χ ¼
T
∫0
¼
dT P which
dT þC ðPsp þ Ppz Þ
4. Conclusions The pyroelectric property of InN/GaN heterostructure has been investigated theoretically from effect of built in polarization on lattice thermal conductivity. The built-in-polarization (BIP) enhances thermal conductivity of InN/GaN heterostructure at room temperature. The variation of thermal conductivity (k) with temperature for including and excluding BIP mechanism in the heterostructure predicts a transition temperature between primary and secondary pyroelectric effects. Below this temperature, kp with BIP field is lower than k without BIP field. This is due to decrease of Grunessen parameter below transition temperature while above this temperature kp is significantly contributed by BIP mechanism due to thermal expansion. The room temperature thermal conductivity of InN/GaN heterostructure with and without polarization is predicted, respectively as 32 and 48 W m-1 K1. The transition temperature Tp between primary and secondary pyroelectric effect; and room temperature pyroelectric coefficient of InN has been predicted as 120 K and 8.425μC m2 K1, respectively which are in line with prior literature studies and argued that this is minimum value of pyroelectric coefficient of InN at room temperature. This study suggests that thermal conductivity measurement can reveal role of phonons in pyroelectricity in InN. Acknowledgement Author BKS acknowledges with thanks to DST-SERB, Govt. of India for financial support through Grant No. EMR/2016/001019.GH acknowledges University Grants Commission, India for financial support (Award letter No: F1-17-1/2016-17/NFST-2015-17-ST-772/SA-III/ website).
yields
1 :
(13)
References
At this point, it is not possible to find out value of C, however a closer look to eq. (5) suggests that χ p follows the form of specific heat. Thereforeχ p vanishes as T3 at low temperatures and saturates at high temperatures. For rough estimate of primary pyroelectric coefficientχ p , we can set C ¼ 0 by assuming that at T ¼ 0, there is no polarization and hence there is no pyroelectricity in the material. In other words, as T→0, Cv →0 which is third law of thermodynamics. Using this assumption, we have estimated pyroelectric coefficient at room temperature (300 K). From Fig. 2, Psp dominates up to temperature 130 K; and above 130 K, Ppz becomes appreciable. So, primary pyroelectric coefficient will be
[1] H. Morkoc, Nitride Semiconductor Devices, Wiley-VCH Verlag GmbH & Co., Weinheim, 2013. [2] J. Wu, J. Appl. Phys. 106 (2009), 011101. [3] B. Roul, M. Kumar, M.K. Rajpalke, T.N. Bhat, S.B. Krupanidhi, J. Phys. D Appl. Phys. 48 (2015), 423001. [4] O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, B. Schaff, L.F. Eastman, J. Appl. Phys. 87 (2000) 334. [5] O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, B. Schaff, L.F. Eastman, J. Phys. Condens. Matter 14 (2002) 3399. 115
G. Hansdah, B.K. Sahoo [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Journal of Physics and Chemistry of Solids 117 (2018) 111–116 [29] S. Krukowskia, A. Witeka, J. Adamczyk, J. Jun, M. Bockowski, I. Grzegory, B. Lucznika, G. Nowaka, M. Wroblewskia, A. Presz, S. Gierlotka, S. Stelmacha, B. Palosza, S. Porowski, P. Zinnt, J. Phys. Chem. Solid. 59 (1998) 289. [30] A.X. Levander, T. Tong, K.M. Yu, J. Suh, D. Fu, R. Zhang, H. Lu, W.J. Schaff, O. Dubon, W. Walukiewicz, D.G. Cahill, J. Wu, Appl. Phys. Lett. 98 (2011), 012108. [31] T. Tong, D. Fu, A.X. Levander, W.J. Schaff, B.N. Pantha, N. Lu, B. Liu, I. Ferguson, R. Zhang, J.Y. Lin, H.X. Jiang, J. Wu, David G. Cahill, Appl. Phys. Lett. 102 (2013), 121906. [32] A. Jezowski, B. Danilchenko, M. Bockowski, I. Grzegory, S. Krukowski, T. Suski, T. Paszkiewicz, Solid State Commun. 128 (2003) 69. [33] G. Slack, L. Schowalter, D. Morelli, J. Freitas, J. Cryst. Growth 246 (2002) 287. [34] D. Kotchetkov, J. Zou, A.A. Balandin, D.I. Florescu, F.H. Pollak, Appl. Phys. Lett. 79 (2001) 4316. [35] G. Pozina, Appl. Phys. Lett. 107 (2015), 251106. [36] A. Minnich, Appl. Phys. Lett. 107 (2015), 183106. [37] G. Tang, J. Appl. Phys. 110 (2011), 046102. [38] B. Liao, Phys. Rev. Lett. 114 (2015), 115901. [39] W. Wan, J. Phys. Condens. Matter 24 (2012), 295402. [40] L. Zhu, H. Luo, Theor. Appl. Mech. Lett. 6 (2016) 277–281. [41] A. Jezowski, O. Churiukova, J. Mucha, T. Suski, I.A. Obukhov, B.A. Danilchenko, Mater. Res. Express 2 (2015), 085902. [42] K. Wang, R.R. Reeber, Appl. Phys. Lett. 79 (2001) 1602. [43] S. Figge, H. Kroncke, D. Hommel, B.M. Epelbaum, Appl. Phys. Lett. 94 (2009), 101915. [44] A.E. Romanov, T.J. Baker, S. Nakamura, J.S. Speck, J. Appl. Phys. 100 (2006), 023522. [45] M. Balkanski, R.F. Wallis, Semiconductor Physics and Application, Oxford University Press Inc., New York, 2000. [46] L. Xu, R. Wang, X. Yang, H. Yan, J. Appl. Phys. 110 (2011), 043528. [47] W. Liu, A.A. Balandin, C. Lee, H.Y. Lee, Phys. Stat. Sol.(a) 202 (2005) R135–R137. [48] W. Liu, A.A. Balandin, J. Appl. Phys. 97 (2005), 123705. [49] A. Jezowski, et al., Mater. Res. Express 2 (2015), 085902. [50] A. Pansari, V. Gedam, B.K. Sahoo, J. Phys. Chem. Solid. 87 (2015) 177.
C. Ren, Material Sci. Technol. 32 (2016) 418. F. Bernardini, V. Fiorentini, Appl. Surf. Sci. 166 (2000) 23. V.N. Davydov, Russ. Phys. J. 57 (2015) 1648. J. Zook, S.T. Liu, J. Appl. Phys. 49 (1978) 4604. T.B. Bahder, R. Tober, J. Bruno, Superlattice. Microst. 14 (1993) 149. N. Cho, Y. Poplavko, S.J. Noh, J. Kor. Phys. Soc. 42 (2003) 803. W. Yan, R. Zhang, Z. Xie, X. Xiu, Y. Zheng, Z. Liu, S. Xu, Z. He, Appl. Phys. Lett. 94 (2009), 242111. W. Yan, R. Zhang, X. Xiu, Z. Xie, P. Han, R.L. Jiang, S.L. Gu, Y. Shi, Y.D. Zheng, Appl. Phys. Lett. 90 (2007), 212102. V. Fuflyigin, E. Salley, A. Osinsky, P. Norris, Appl. Phys. Lett. 77 (2000) 3075. J. Liu, M.V. Fernandez-Serra, P.B. Allen, Phys. Rev. B 93 (R) (2016), 081205. A.D. Bykhovski, V.V. Kaminski, M.S. Shur, Q.C. Chen, M.A. Khan, Appl. Phys. Lett. 69 (1999) 3254. Sven Jachalke, Patrick Hofmann, Gunnar Leibiger, Frank S. Habel, Erik Mehner, Tilmann Leisegang, Dirk C. Meyer, Thomas Mikolajick, Appl. Phys. Lett. 109 (2016), 142906. L.C. Lew Yan Voon1, M. Willatzen, J. Appl. Phys. 109 (2011), 031101. I. Vurgaftman, J.R. Meywer, J. Appl. Phys. 94 (2003) 3675. M. Feneberg, K. Thonke, J. Phys. Condens. Matter 19 (2007) 403201. C.E. Dreyer, A. Janotti, C.G. Van de Walle, D. Vanderbilt, Phys. Rev. X 6 (2016), 021038. B. Danilchenko, T. Paszkiewicz, A. Jezowski, T. Plackowski, Appl. Phys. Lett. 89 (2006), 061901. B.K. Sahoo, J. Mater. Sci. 47 (2012) 2624. A. Jezowski, O. Churiukova, J. Mucha, T. Suski, I.A. Obukhov, B.A. Danilchenko, Mater. Res. Express 2 (2015), 085902. B.K. Sahoo, S.K. Sahoo, S. Sahoo, J. Phys. Chem. Solid. 74 (2013) 1669. J. Zou, D. Kotchetkov, A.A. Balandin, D.I. Florescu, F.H. Pollak, J. Appl. Phys. 92 (2002) 2534. A. Sztein, J.E. Bowers, S.P. DenBaars, S. Nakamura, J. Appl. Phys. 113 (2013), 183707. V. Gedam, A. Pansari, B.K. Sahoo, Indian J. Phys. 90 (9) (2016) 991.
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