Thermal conductivity of amorphous and crystalline GeTe thin film at high temperature: Experimental and theoretical study
Kanka Ghosh, Andrzej Kusiak, Pierre Noé, Marie-Claire Cyrille, Jean-Luc Battaglia
TThermal conductivity of amorphous and crystalline GeTe thin film at hightemperature: Experimental and Theoretical study
Kanka Ghosh, ∗ Andrzej Kusiak, Pierre No´e, Marie-Claire Cyrille, and Jean-Luc Battaglia University of Bordeaux, CNRS, Arts et Metiers Institute of Technology,Bordeaux INP, INRAE, I2M Bordeaux, F-33400 Talence, France CEA-LETI, MINATEC, Grenoble, France
Thermal transport properties bear a pivotal role in influencing the performance of phase changememory (PCM) devices, in which, the PCM operation involves fast and reversible phase changebetween amorphous and crystalline phases. In this article, we present a systematic experimentaland theoretical study on the thermal conductivity of GeTe at high temperatures involving fast-change from amorphous to crystalline phase upon heating. Modulated photothermal radiometry(MPTR) is used to experimentally determine thermal conductivity of GeTe at high temperaturesin both amorphous and crystalline phases. Thermal boundary resistances are accurately taken intoaccount for experimental consideration. To develop a concrete understanding of the underlyingphysical mechanism, rigorous and in-depth theoretical exercises are carried out. For this, first-principles density functional methods and linearized Boltzmann transport equations (LBTE) areemployed using both direct and relaxation time based approach (RTA) and compared with that ofthe phenomenological Slack model. The amorphous phase experimental data has been describedusing the minimal thermal conductivity model with sufficient precision. The theoretical estimationinvolving direct solution and RTA method are found to retrieve well the trend of the experimentalthermal conductivity for crystalline GeTe at high temperatures despite being slightly overestimatedand underestimated respectively compared to the experimental data. A rough estimate of vacancycontribution has been found to modify the direct solution in such a way that it agrees excellentlywith the experiment. Umklapp scattering has been determined as the significant phonon phononscattering process. Umklapp scattering parameter has been identified for GeTe for the whole tem-perature range which can uniquely determine and compare umklapp scattering processes for differentmaterials.
I. INTRODUCTION
Chalcogenide alloys have been evolved as excellent can-didates for the purpose of electronic nonvolatile memorystorage (phase change memories-PCM)[1–7]. This appli-cation involves a fast and reversible alteration betweenamorphous and crystalline phase on heating. PCM cellsconsist of a nanoscale volume of a phase change (PC)material, normally a Tellurium (Te) based alloy, whichundergoes a reversible change between amorphous andcrystalline states, possessing a contrasting electrical resis-tivity and thus enabling the PCMs to be used for binarydata storage [1, 4]. In a PCM device, crystallization byheating the amorphous PC alloy above its crystallizationtemperature with electric current pulses is called SET op-eration while amorphization of the crystalline region bymelting and quenching using higher and shorter electriccurrent pulses is called RESET operation [7]. Germa-nium telluride (GeTe) is one of the promising candidateswithin the phase change materials due to its notably highcontrast in electrical resistance as well as a stable amor-phous phase with a higher crystallization temperatureupon doping for the data retention process [7, 8]. Thecrystallization temperature for GeTe is ≈ ◦ C (453 K)[9, 10]. GeTe has also been implemented with a super-lattice configuration as GeTe-Sb Te that are extensively ∗ Corresponding author: [email protected] used for their application in optical as well as PCM stor-age devices [11]. Further, the interface between the GeTeand Sb Te in the superlattice configuration is found tocontrol the phase transition, accompanied by a reducedentropy loss, which helps in making fast and efficientPCMs [12]. Doped GeTe with either N or C has beenstudied as a way to postpone the phase change for hightemperature applications [7, 10]. The effect of dopingis also found to reduce the thermal conductivity ( κ ) ofGeTe significantly [10]. Thermal conductivity ( κ ) servesas a crucial parameter for PCM operations as heat dissi-pation, localization and transport can significantly affectthe SET/RESET processes and can therefore consider-ably influence the performance of PCMs in terms of cy-clability, switching time and data retention.While considerable amount of investigations have beenreported on the electronic transport properties of GeTe[13, 14], very few reports on the thermal conductiv-ity of GeTe starting from room temperature to high-temperature range are found to exist in the literature.P. Nath et al. [15] characterized the thermal propertiesof thick GeTe films in both crystalline and amorphousphases. While amorphous phase showed negligible elec-tronic contribution in the thermal conductivity ( κ ), thecontribution in crystalline phases were found to be nearly25% of the measured value of κ [15] at the room tem-perature. E.M. Levin et al. [14] observed high thermalconductivity of GeTe at 300 K and 720 K which they at-tributed mostly to free charge carriers. Non-equilibriummolecular dynamics simulation (NEMD) studies by Da- a r X i v : . [ c ond - m a t . m t r l - s c i ] J un vide Campi et al. [5] showed that a 3% of Ge vacancieseffectively reduce the bulk lattice thermal conductivity ofcrystalline GeTe from 3.2 Wm − K − to 1.38 Wm − K − at 300 K, justifying a large spread of the experimentallymeasured thermal conductivities. First principles calcu-lations [6] also revealed this large variability of experi-mentally measured bulk thermal conductivity due to thepresence of Ge vacancies. Very recently, frequency do-main thermoreflectance study [16] was carried out fordetermining thermal conductivity in GeTe thin films asa function of film thickness for both amorphous and crys-talline phases.Though the aforementioned studies dealt with thethermal transport of GeTe in a broader sense, there areplenty of open questions that still remain. A thoroughand systematic understanding of the thermal conduc-tivity of GeTe as a function of temperatures, rangingfrom room temperature to a temperature that is higherthan the crystallization temperature, in the context ofdifferent scattering processes involved, is one amongstthem. This systematic understanding involves multipleapproaches: (a) An accurate experimental estimationof the thermal conductivity of GeTe thin films by tak-ing into account the significant contributions of thermalboundary resistances at the boundaries of the GeTe layer.(b) A consistent and thorough theoretical investigation ofthe temperature variation of thermal conductivity start-ing from first-principles density functional theory as wellas solving the linearized Boltzmann transport equations(LBTE) and (c) Using simple phenomenological modelssuch as Slack model, which exhibit closed-form solutionsthat can easily identify the underlying physical mecha-nism involved in the thermal transport. As revealed inthe study by E. Bosoni et al. [17], thermal conductiv-ity ( κ ) calculated from the Slack model [18], was indeedfound in good agreement with the experimentally mea-sured value of κ at room temperature.In this article, we investigate the thermal conductiv-ity of GeTe films from room temperature up to 230 ◦ C(503 K) starting from the amorphous state. The contri-bution of the film thermal conductivity from the ther-mal resistance at the interfaces between the GeTe filmand lower and upper layers are clearly discriminated inthe whole temperature range. The modulated photother-mal radiometry (MPTR) is employed as the experimen-tal technique for the study. The Levenberg-Marquardt(LM) technique is used in order to identify the essen-tial parameters from phase measurements and a modelthat simulates the phase within the experimental con-figuration. In a stand-alone exercise of theoretical un-derstanding, thermal conductivity of GeTe is calculatedstarting from first-principles density functional theory(DFT) coupled with solving linearized Boltzmann trans-port equations (LBTE) by both direct method and relax-ation time approach. In order to explain the measuredchange in κ ( T ), distinct contributions coming from vari-ous scattering mechanisms are understood. Further, phe-nomenological models by Slack et al. [19] and Cahill et al. [20] are also employed to elucidate the physical mech-anisms in the heat transport process. These systematictheoretical and experimental investigations are found toprovide significant clarity and insight in understandingthe variation of thermal conductivity of GeTe for a widerange of temperature. This work is organized as follows:Section II deals with experimental measurements usingMPTR method. Computational details involving firstprinciples and thermal conductivity calculations are de-scribed in Section III. Section IV presents the theoreticalresults followed by summary and conclusions in SectionV. II. EXPERIMENTAL RESULTS
Amorphous GeTe films, with thicknesses of 200, 300and 400 nm, are deposited by magnetron sputtering inan Ar atmosphere on 200 mm silicon wafers covered bya 500 nm thick SiO top layer. The thicknesses of thedeposited films as well as their homogeneity are con-trolled by X-Ray Reflectivity (XRR). The detailed de-scription of the MPTR setup had been presented else-where [7]. The main principle consists of front face pe-riodic heating of the studied sample by a laser source.Since the GeTe layer is not opaque at the laser wave-length (1064 nm) and in order to prevent oxidation orevaporation of the GeTe at high temperature, a 100 nmthick platinum (Pt) layer is deposited by sputtering inorder to act as an optical to thermal transducer. Theperiodic heat flux φ ( ω ) is thus absorbed by the Pt layerdue to its high extinction coefficient at the laser wave-length, and the optical source is then transformed intoheat. On the other hand, thanks to its high thermalconductivity and low thickness, the Pt layer is assumedto be isothermal for the frequency range swept duringthe experiment. The thermal response of the sample atthe location of the heating area by the laser is measuredusing an infrared detector. As the temperature changeis low at the heated area, the linearity assumption ofheat transfer is fulfilled and the emitted infrared radia-tion from the sample surface is linearly proportional tothe temperature at the heated area. A lock-in ampli-fier is used to extract the amplitude and the phase fromthe signal of the IR detector as a function of the fre-quency. The thermal properties are thus obtained byfitting of the experimental phase by means of a thermalmodel which allows to describe the heat transfer withinthe sample. According to the film thickness and modula-tion frequency, the transient behaviour fulfils the Fourierregime of heat conduction. Since the heated area is muchlarger (laser spot of ∼ ω , the phase is definedas ψ ( ω ) = arg [ Z ( ω )] = arctan (Im (Z ( ω )) / Re ( Z ( ω ))),where the transfer function Z ( ω ) denotes the ratio be-tween the periodic temperature θ ( ω ) at the heated areaand φ ( ω ) as: Z ( ω ) = θ ( ω ) φ ( ω ) = BD (1)Parameters B and D are calculated from thequadrupoles formalism [21] as: (cid:20) A BC D (cid:21) = (cid:20) A GeT e B GeT e C GeT e D GeT e (cid:21) (cid:20) A SiO B SiO C SiO D SiO (cid:21) (cid:20) A Si B Si C Si D Si (cid:21) (2)Where A j = 1 + exp ( − γ i e i ) ; B j = (1 + exp ( − γ i e i )) γ i κ ∗ i (3) C j = (1 + exp ( − γ i e i )) γ i κ ∗ i ; D j = A j (4)With γ i = (cid:112) j ω/a ∗ i , where a ∗ i (= κ ∗ i /C p i ), κ ∗ i , C p i and e i are the effective thermal diffusivity, effective thermalconductivity, specific heat per unit volume and thicknessof layer i respectively. For the SiO and Si layers, theeffective thermal conductivity is equal to the real thermalconductivity, i.e., κ ∗ SiO = κ SiO and κ ∗ Si = κ Si . Onthe other hand, the effective thermal resistance for GeTe( R ∗ GeT e ) accounts with the intrinsic thermal conductivity κ GeT e of the GeTe layer and the thermal resistance at thetwo interfaces with Pt and SiO as: R ∗ GeT e = e GeT e κ ∗ GeT e = e GeT e κ GeT e + R P t − GeT e + R GeT e − SiO (cid:124) (cid:123)(cid:122) (cid:125) R i (5)where R i is the total interfacial thermal resistance. Itmust be precisely mentioned here that the GeTe layercannot be considered as thermally resistive for the high-est frequency values and especially for the high thick-ness of the layer. This is the reason why we considerheat diffusion within the quadrupole model. This can beeasily demonstrated by calculating the Fourier relatedquantity (cid:112) a GeT e /ω , using the known value of a GeT e at room temperature and comparing it to the thickness e GeT e . Considering the measured value Y φ ( ω i ) of thephase at different frequencies ω i ( i = 1 ..N ), the value of κ ∗ GeT e is estimated by minimizing the objective func-tion J = (cid:107) Y φ − Ψ (cid:107) , where Y φ = Y φ ( ω i ) i = .. N and Ψ = ψ ( ω i ) i =1 ..N are vectors with length N , related re-spectively to the measured and simulated phase at all theinvestigated frequencies. This minimization is achievedby implementing the Lavenberg-Marquardt (LM) algo-rithm [22]. Then, the effective thermal resistance R ∗ GeT e of GeTe thin film is plotted as a function of film thick-nesses ( e GeT e ) for different temperatures as shown in Fig1. A linear regression R ∗ GeT e = α e GeT e + β = e GeT e /κ GeT e + R i is found that allows to extract FIG. 1. Variation of thermal resistance ( R ∗ GeTe ) of GeTethin film as a function of film thickness ( e GeTe ) for differenttemperatures ranging from amorphous to crystalline phasechange. κ GeT e = 1 /α from the slope (shown in Fig 2.(b))and the sum of the two interfacial resistances R i = β by extrapolation to e GeT e = 0 (shown in Fig 2.(a)).The standard deviation of κ ∗ GeT e is calculated fromthe covariance matrix at the end of the iterative min-imization process as: σ ( κ ∗ GeT e | Y ) ∼ cov ( Θ ) E / √ N where the covariance matrix is: cov ( Θ ) = (cid:0) S T S (cid:1) − with vector S = [ S Q ( α i )] N with S Q ( α i ) = (cid:104) ∂ψ ( ω i ) i =1 ,N /∂κ ∗ GeT e (cid:105) κ ∗ GeTe =ˆ κ ∗ GeTe denoting the sensi-tivity function of the phase according to κ ∗ GeT e calcu-lated for κ ∗ GeT e = ˆ κ ∗ GeT e where ˆ κ ∗ GeT e is the optimalvalue for κ ∗ GeT e . Finally the residual vector is E = Y φ − Ψ ( κ ∗ GeT e = ˆ κ ∗ GeT e ). The standard deviation on R ∗ GeT e is obtained starting from σ ( κ ∗ GeT e ) by applica-tion of the law of propagation of uncertainties. Finally,the standard deviations on κ GeT e and R i are expressedaccording to residual variance of linear fitting on R ∗ GeT e points. We mention here that the grain size of GeTe atthe time of the crystallization is found to be 40 nm asreported in our earlier work [7] and it increases while in-creasing the annealed temperature (Fig 4 in [7]). Sincethe MPTR investigates a very large area, it is thus ex-pected to measure the average thermal conductivity thatis given by κ avL + κ el , where κ avL = κ x + κ z , where, κ x and κ z stand for the lattice thermal conductivities alonghexagonal a and c axes respectively.The phase change occurs well within the range of theexpected temperature ( ∼ ◦ C = 453 K). We note that R i for amorphous state is difficult to estimate becauseof the very low values of κ GeT e . For crystalline phase,we retrieve a consistent behavior with the value of R i being increased as the temperature is lowered (Fig 2.(a)).In the high temperature regime, diffuse mismatch model(DMM) has been found to describe R i quite satisfactorilyfor crystalline solids [23]. According to DMM model, FIG. 2. (a) Variation of total interfacial thermal resistance R i = R Pt − GeTe + R GeTe − SiO with temperature. Phase changeoccurs around 180 ◦ C = 453 K, which is shown via dotted line. (b) Experimentally measured thermal conductivity κ GeTe ( T ) ofthe GeTe thin film as a function of temperature. The rightward arrows denote the forward cycle exhibiting phase change fromamorphous to crystalline phase while the leftward arrow defines the backward cycle where GeTe exists in crystalline phase. asymptotic behavior of R i at high temperatures dependsinversely on the heat capacity as [23] R i = (cid:80) j c − ,j (cid:16)(cid:80) j c − ,j + (cid:80) j c − ,j (cid:17) (cid:88) j c ,j − C ( T ) (6)where, C ( T ) is the heat capacity of material 1 at T and c l,j is the velocity of phonon mode j in material l . Hereall the parameters except C ( T ) are temperature inde-pendent. Since C ( T ) increases with temperature, theabove relation shows that R i decreases with increasingtemperature. In Fig 2.(b), we observe a monotonicallydecreasing trend of κ GeT e as a function of T. Indeed, athigh temperatures (T (cid:29) Θ D ), where Θ D is Debye tem-perature, umklapp scattering is the dominating phonon-phonon scattering process associated with high momen-tum change in phonon-phonon collisions [24]. Glen A.Slack et al.[19] approximated umklapp relaxation time as τ − U = AT ω exp ( − Θ D / T ) which becomes τ − U = AT ω when T (cid:29) Θ D . This relaxation time estimation leads to κ ∝ τ − U ∝ T . III. COMPUTATIONAL DETAILS
Phonon density of states (PDOS) of crystalline GeTe(space group R3m) has been calculated employing thedensity functional perturbation theory (DFPT) [25] us-ing the QUANTUM-ESPRESSO [26] suite of programs.As the first step, self-consistent calculations, within theframework of density functional theory (DFT), are car-ried out to compute the total ground state energy of thecrystalline R3m-GeTe. For this purpose, Perdew-Burke-Ernzerhof (PBE) [27] generalized gradient approximation (GGA) is used as the exchange-correlation functional.The spin-orbit interaction has been ignored due to itsnegligible effects on the vibrational features of GeTe asmentioned in literature [6, 28]. Electron-ion interactionsare represented by pseudopotentials using the frameworkof projector-augmented-wave (PAW) method [29]. TheKohn-Sham (KS) orbitals are expanded in a plane-wave(PW) basis with a kinetic cutoff of 60 Ry and a chargedensity cutoff of 240 Ry as prescribed by the pseudopo-tentials of Ge and Te. The Brillouin zone integrationfor self consistent electron density calculations are per-formed using a 12 × ×
12 Monkhorst-Pack (MP) [30] k-point grid.For phonon calculations, a hexagonal 2 × × FIG. 3. 2 × × structure of crystalline GeTe as stacking bilayers isshown in Fig 3. To study the phonon density of states,linear response theory is applied via DFPT, to the Kohn-Sham equations to solve the electronic charge density( ρ n ) under small perturbations. As the force constantsare connected to the derivatives of ρ n with respect toatomic displacements, harmonic force constants are cal-culated by diagonalizing the dynamical matrix in recip-rocal space. Phonon density of states (PDOS) are thenevaluated by the inverse Fourier transform of the inter-atomic force constants (IFC) to real space from that ofthe dynamical matrices, using a uniform 5 × × q -vectors.The thermal conductivity of GeTe can be separatedinto two distinct contributions, one from electronic trans-port and the other from the phonon transport or the lat-tice contribution, such that κ = κ el + κ L , where κ L isthe lattice thermal conductivity and κ el is the electronicthermal conductivity. κ el has been obtained from first-principles calculations by solving semi-classical Boltz-mann transport equation (BTE) for electrons. Constantrelaxation time approximation (CRTA) and rigid bandapproximation (RBA) are employed as implemented inBoltzTraP code [32]. The energy projected conductivitytensor is calculated using: σ αβ ( (cid:15) ) = 1 N (cid:88) i, k σ αβ ( i, k ) δ ( (cid:15) − (cid:15) i, k ) d(cid:15) (7)Therefore, the transport tensors, or more specifically theelectrical conductivity tensor in this study, can be ob-tained from σ αβ ( T ; µ ) = 1Ω (cid:90) σ αβ ( (cid:15) ) (cid:20) − ∂f µ ( T ; (cid:15) ) ∂(cid:15) (cid:21) d(cid:15) (8)where, N is the number of k -points sampled, i is theband index, (cid:15) i, k are band energies, Ω is the volume ofunit cell, f µ is the Fermi distribution function and µ isthe chemical potential. The code computes the Fermi in-tegrals and returns the transport coefficients for differenttemperature and Fermi levels.For getting lattice thermal conductivity κ L , linearizedphonon Boltzmann transport equation (LBTE) is solvedusing both direct method introduced by L. Chaput et al.[33] as well as the single mode relaxation time approxima-tion or relaxation time approximation (RTA) employingPHONO3PY [34] software package. Initially, the super-cell approach with finite displacement of 0.03 ˚A is appliedto calculate the harmonic (second order) and the anhar-monic (third order) force constants, given byΦ αβ ( lκ, l (cid:48) κ (cid:48) ) = ∂ Φ ∂u α ( lκ ) ∂u β ( l (cid:48) κ (cid:48) ) (9)andΦ αβγ ( lκ, l (cid:48) κ (cid:48) , l (cid:48)(cid:48) κ (cid:48)(cid:48) ) = ∂ Φ ∂u α ( lκ ) ∂u β ( l (cid:48) κ (cid:48) ) ∂u γ ( l (cid:48)(cid:48) κ (cid:48)(cid:48) ) (10)respectively. First principles calculations usingQUANTUM-ESPRESSO [26] are implemented to calcu-late the forces acting on atoms in supercells. Using finite difference method, harmonic force constants are approx-imated as [34]Φ αβ ( lκ, l (cid:48) κ (cid:48) ) (cid:39) − F β [ l (cid:48) κ (cid:48) ; u ( lκ )] u α ( lκ ) (11)where F [ l (cid:48) κ (cid:48) ; u ( lκ )] is atomic force computed at r ( l (cid:48) κ (cid:48) )with an atomic displacement u ( lκ ) in a supercell. Simi-larly, anharmonic force constants are obtained using[34]Φ αβγ ( lκ, l (cid:48) κ (cid:48) , l (cid:48)(cid:48) κ (cid:48)(cid:48) ) (cid:39) − F γ [ l (cid:48)(cid:48) κ (cid:48)(cid:48) ; u ( lκ ) , u ( l (cid:48) κ (cid:48) )] u α ( lκ ) u β ( l (cid:48) κ (cid:48) ) (12)where F [ l (cid:48)(cid:48) κ (cid:48)(cid:48) ; u ( lκ ), u ( l (cid:48) κ (cid:48) )] is atomic force computedat r ( l (cid:48)(cid:48) κ (cid:48)(cid:48) ) with a pair of atomic displacements u ( lκ )and u ( l (cid:48) κ (cid:48) ) in a supercell. These two sets of linear equa-tions are solved using Moore-Penrose pseudoinverse as isimplemented in PHONO3PY [34].We use a 2 × × × × × × − a.u. for supercell calculations. For lattice thermalconductivity calculations employing both the direct solu-tion of LBTE and that of the RTA, q -mesh of 24 × × IV. THEORETICAL RESULTS ANDDISCUSSIONSA. Phonon density of states
The structural parameters are optimized via DFT cal-culations and the optimized lattice parameter (a = 4.23˚A) and unit cell volume (56.26 ˚A ) of GeTe, are found tobe quite consistent with the values presented in literature[6, 35]. It is quite well established that at normal con-ditions GeTe crystallizes in trigonal phase (space groupR3m) with 2 atoms per unit cell. This structure gives riseto a 3+3 coordination of Ge with three short stronger in-trabilayer bonds and three long weaker interbilayer bonds[5, 6]. The bond lengths (shorter bonds = 2.85 ˚A, longerbonds = 3.25 ˚A) are also found to be consistent with thestudies done by Davide Campi et al. [6].To investigate the effect of phonons in the heat transferprocesses, we study the phonon density of states and the FIG. 4. (a) Calculated phonon density of states (PDOS) for GeTe crystal (R3m). The dotted line denotes the separationfrequency between acoustic and optical modes ( ∼
96 cm − = 2.88 THz) which is consistent with previous studies (see text).(b) Phonon dispersion relation of the rhombohedral (R3m) GeTe. Transverse and longitudinal phonon modes are denoted viasolid and dashed lines, respectively. The approximate separation frequency between acoustic and optical modes ( ∼
96 cm − )are shown via green dashed line. dispersion relation of crystalline rhombohedral (R3m)GeTe. Figure 4.(a) and (b) show the phonon densityof states and the phonon dispersion relation of undopedcrystalline (R3m) GeTe, calculated at ground state. Thedashed line in Fig 4.(a) serves as a separator between theacoustic and optical contribution of phonons. The acous-tic phonons extend from 0 to 96 cm − (= 2.88 THz) inthe frequency domain which is consistent with the obser-vations of Urszula D. Wdowik et al. [8]. The frequenciesof the two most prominent peaks in PDOS correspond-ing to acoustic ( ∼
80 cm − = 2.40 THz) and opticalphonons ( ∼
140 cm − = 4.20 THz) respectively, are alsofound to be close to the values observed by KwangsikJeong et al. [36]. Phonon dispersion relation (Fig 4.(b))along the high symmetry direction in the Brillouin zone(BZ) also shows similar trends to that of the earlier works[8, 17]. We observe the signature of LO-TO splittingas the discontinuities of the phonon dispersion at the Γpoint arising from the long range Coulomb interactions[8, 17]. The approximate separator between acoustic andoptical modes at 96 cm − is also shown by a horizontaldashed line in the phonon dispersion relation (Fig 4.(b)).Except a small and negligible contribution of transverseoptical modes at the Γ point, all frequencies <
96 cm − contribute to the acoustic modes. B. Electronic thermal conductivity
The calculation of the electronic thermal conductivity κ el rests generally on the use of the Wiedemann-Franzlaw κ el = Lσ e T where L is the Lorenz number, T is thetemperature and σ e is the electrical conductivity. Crys-talline GeTe is a p -type degenerate semiconductor [15, 37]with a high hole concentration and the Fermi level lyingwell inside the valence band that holds the charge carri-ers. Therefore it behaves similar to a metal. In that case,the value for L = π ( k B /e ) / . × − V . K − ( k B is the Boltzmann constant and e is the electroncharge) had generally been employed in most of the re-search works [15]. However, for highly doped materi-als at temperature higher than the Debye temperatureΘ D , the Hall mobility depends on the temperature as µ H ∝ T − (3 / r , where r is the scattering mechanismparameter, and decreases as η decreases with increas-ing temperature. Therefore, it is shown that the Lorenznumber is given by [38, 39]: L = (cid:18) k B e (cid:19) (cid:0) r + (cid:1) (cid:0) r + (cid:1) F r +5 / ( η ) F r +1 / ( η ) − (cid:0) r + (cid:1) F r +3 / ( η ) (cid:0) r + (cid:1) F r +1 / ( η ) (13)where η = ( E F − E V ) /k B T is the reduced Fermi energyfor p -type semiconductors and r = − / F n ( η ) = (cid:82) ∞ ( x n / (1 + e x − η )) d x . TABLE I. Electronic thermal conductivity ( κ el ) of the crys-talline R3m-GeTe is presented (column 4) as a function oftemperature (T) using Wiedemann-Franz law. The variationof electrical conductivity ( σ e ) and the Lorenz number (L) withtemperature are also shown in column 2 and column 3 respec-tively.T (K) σ e (Ω − m − ) L (WΩ K − ) κ el = Lσ e T (W/mK)322 9.78 × × − × × − × × − × × − × × − Following the method adopted by Gelbstein et al. [39],E. M. Levin and co authors [14] obtained that L for crys-talline GeTe varies between 2 . × − V . K − at 320 Kand 1 . × − V . K − at 720 K. Using this method, wecalculate the variation of L for GeTe as a function oftemperature within the investigated range in the presentstudy and we report the results in Table I.The electrical resistivity of GeTe films has been mea-sured experimentally using the Van der Pauw techniqueand is found to be ρ e = 1 /σ e = [8 . ± × − Ω . mat the room temperature (300 K), which corresponds forthe hole concentration to be 6.24 × cm − . Theconstant relaxation time approximation (CRTA) witha constant electronic relaxation time of 10 − s is usedfollowing the work of S. K. Bahl et al. [37]. Afteridentifying the hole concentration, we compute the elec-trical conductivity ( σ e ) by using DFT and solving theBoltzmann transport equation (BTE) for electrons, asimplemented in the BoltzTaP code [32] for the givenhole concentration at different temperatures. Consider-ing the values for L ( T ), the calculated electronic ther-mal conductivity varies linearly from 0.76 W/mK at 322K ( L = 2 . × − V . K − at 322 K) to 0.91 W/mKat 503 K ( L = 2 . × − V . K − at 503 K) (Table I).These results are consistent with the experimental onesby R. Fallica et al. [10]. C. Lattice thermal conductivity
In Fig 2.(b), total thermal conductivity of GeTe is seento manifest a fast change process with gradually increas-ing temperature, where a phase change from amorphous to crystalline phase is found to occur ∼ ◦ C or 453 K.We first focus on the thermal conductivity of the amor-phous phase of GeTe. Figure 5 shows lower values of κ for amorphous phase compared to that of the crystallinephase. The values of κ in amorphous phase are also ob-served to be almost constant throughout the temperaturerange studied in this work. Theoretically, the minimalthermal conductivity model derived by Cahill et al. [20]allows one to calculate κ for the amorphous materials as: κ min ( T ) = (cid:0) π (cid:1) / k B n / (cid:80) i =1 v i (cid:16) T Θ i (cid:17) (cid:82) Θ i /T x e x ( e x − d x (14)where n = ( k B Θ D / (cid:126) ) − / / (cid:0) π c s (cid:1) is the number ofphonons per unit volume (which can also be calculatedmore rigorously from the phonon DOS apart from usingthe values of Table II), c s is the speed of sound calculatedas c − s = (cid:0) v − L + 2 v − T (cid:1) / . s − [40] and Θ i isthe Debye temperature per branch. When T (cid:29) Θ D , thisrelation simplifies as κ min = 12 (cid:0) π n / (cid:1) / k B ( v L + 2 v T ) . (15)Using the required parameter values from Table II, theminimal thermal conductivity ( κ min ) is found to be con-sistent with the experimental data in amorphous phaseconsidering the error bars involved in the experimentalmeasurements (Fig 5). The reasonable agreement be-tween the experimental data and κ min based on Cahillmodel indicates that the dominant thermal transport inthe amorphous GeTe occurs in short length scales [41]between neighboring vibrating entities owing to the dis-order present in it.We then investigate the phonon contributions to thetotal thermal conductivity of crystalline GeTe. In orderto evaluate the lattice thermal conductivity ( κ L ) throughthe direct solution of LBTE, the method developed by L.Chaput [33] is adopted. According to this method, latticethermal conductivity is given as [33] κ αβ = (cid:126) k B T N V (cid:88) λλ (cid:48) ω λ υ α ( λ ) sinh ( (cid:126) ω λ k B T ) ω λ (cid:48) υ β ( λ (cid:48) ) sinh ( (cid:126) ω λ (cid:48) k B T ) (Ω ∼ ) λλ (cid:48) (16)where, Ω ∼ is the Moore-Penrose inverse of the collisionmatrix Ω, given by [33, 34]Ω λλ (cid:48) = δ λλ (cid:48) /τ λ + π/ (cid:126) (cid:88) λ (cid:48)(cid:48) | Φ λλ (cid:48) λ (cid:48)(cid:48) | [ δ ( ω λ − ω λ (cid:48) − ω λ (cid:48)(cid:48) ) + δ ( ω λ + ω λ (cid:48) − ω λ (cid:48)(cid:48) ) + δ ( ω λ − ω λ (cid:48) + ω λ (cid:48)(cid:48) )] sinh ( (cid:126) ω λ (cid:48)(cid:48) k B T ) (17)Here, Φ λλ (cid:48) λ (cid:48)(cid:48) denotes the interaction strength betweenthree phonon λ , λ (cid:48) and λ (cid:48)(cid:48) scattering [34]. However, adopting the relaxation time approximation (RTA) insolving LBTE, lattice thermal conductivity tensor κ L canbe written in a convenient and closed form as [34, 42] TABLE II. Theoretical and experimental values of different parameters used for thermal conductivity calculation using Slack[18] and Cahill [20] model.GeTe (R3m) Parameter description(s) Value(s) V [˚A ] Volume of the elementary cell 56.26 [35],[calculated from DFT] ρ [kg m − ] Density 5910 [35],[calculated from DFT] M Ge [g mol − ] Molar mass of Ge 72.63 M Te [g mol − ] Molar mass of Te 127.6Θ D [K] Debye temperature 180 [6] υ L [m s − ] Phonon group velocity (longitudinal) 2500 [40] υ T [m s − ] Phonon group velocity (Transverse) 1750 [40] G Gruneisen parameter 1.7 [17] E F [eV] Fermi energy 7.2552 [calculated from DFT] N [kg − ] number of phonon per unit mass 5 . × [calculated from PDOS] κ L = 1 N V (cid:88) λ C λ v λ ⊗ v λ τ λ (18)where N is the number of unit cells and V is the volumeof unit cell. The phonon modes ( q , j ) comprising wavevector q and branch j are denoted with λ . The modalheat capacity if given by C λ = k B (cid:18) (cid:126) ω λ k B T (cid:19) exp ( (cid:126) ω λ /k B T )[ exp ( (cid:126) ω λ /k B T ) − (19)Here, T denotes temperature, (cid:126) is reduced Planck con-stant and k B is the Boltzmann constant. v λ and τ λ repre-sent phonon group velocity and phonon lifetime respec-tively. We consider three scattering processes, namely normal, umklapp and isotope, denoted by N, U and Irespectively, in the theoretical study. For each of theseprocesses, the phonon lifetime has been realized usingMatthiessen rule as [41]1 τ λ = 1 τ Nλ + 1 τ Uλ + 1 τ Iλ (20)where τ Nλ , τ Uλ and τ Iλ are phonon lifetimes correspondingto the normal, umklapp and isotope scattering respec-tively.Generally, in harmonic approximation, phonon life-times are infinite whereas, anharmonicity in a crystalgives rise to a phonon self energy ∆ ω λ + i Γ λ . Thephonon lifetime has been computed from imaginary partof the phonon self energy as τ λ = λ ( ω λ ) from[34]Γ λ ( ω λ ) = 18 π (cid:126) (cid:88) λ (cid:48) λ (cid:48)(cid:48) ∆ ( q + q (cid:48) + q (cid:48)(cid:48) ) | Φ − λλ (cid:48) λ (cid:48)(cid:48) | { ( n λ (cid:48) + n λ (cid:48)(cid:48) +1) δ ( ω − ω λ (cid:48) − ω λ (cid:48)(cid:48) )+( n λ (cid:48) − n λ (cid:48)(cid:48) )[ δ ( ω + ω λ (cid:48) − ω λ (cid:48)(cid:48) ) − δ ( ω − ω λ (cid:48) + ω λ (cid:48)(cid:48) )] } (21)where n λ = exp ( (cid:126) ω λ /k B T ) − is the phonon occupationnumber at the equilibrium. ∆ ( q + q (cid:48) + q (cid:48)(cid:48) ) = 1 if q + q (cid:48) + q (cid:48)(cid:48) = G , or 0 otherwise. Here G represents re-ciprocal lattice vector. Integration over q -point tripletsfor the calculation is made separately for normal ( G = 0)and umklapp processes ( G (cid:54) = 0). For both direct methodand RTA, scattering of phonon modes by randomly dis-tributed isotopes [34] are also incorporated for compari-son. The isotope scattering rate, using second-order per-turbation theory, is given by Shin-ichiro Tamura [43] as τ Iλ ( ω ) = πω λ N (cid:80) λ (cid:48) δ ( ω − ω (cid:48) λ ) (cid:80) k g k | (cid:80) α W α ( k, λ ) W ∗ α ( k, λ ) | (22)where g k is mass variance parameter, defined as g k = (cid:88) i f i (cid:18) − m ik m k (cid:19) (23) f i is the mole fraction, m ik is relative atomic mass of i thisotope, m k is the average mass = (cid:80) i f i m ik and W is polarization vector. The database of the natural abun-dance data for elements [44] is used for the mass varianceparameters.For consistency check, we first simulate the lattice ther-mal conductivity ( κ L ) of crystalline rhombohedral GeTeat 300 K using RTA method and compare the value of κ L with the results of the work done by D. Campi et al.[6]. κ L , obtained from our study using RTA, is 2.29 W/mKusing the PBE functional in the DFT framework, whichis in good agreement with the results of D. Campi et al.[6]with κ L = 2.34 W/mK. After this consistency check, weuse the DFT and Boltzmann transport equation (BTE)to get the lattice thermal conductivity of GeTe at thetemperature regime studied in this work (322 K (cid:54) T (cid:54)
503 K).Figure 5 and Table III present the various contribu-tions of thermal conductivity, obtained theoretically, su-perimposed with the experimental data. Starting fromfirst principles calculations, both direct solution and RTAmethod are used to solve the LBTE to get the lat-tice thermal conductivity. Since κ L is anisotropic alonghexagonal c axis and a-b axes, the average lattice ther-mal conductivity is calculated as κ av = κ x + κ z [6, 45].Figure 5 and Table III show that the resulting theoretical κ tot for the direct solution of LBTE is slightly overesti-mated compared to the experimental results, specificallyat the higher temperatures ( T >
322 K). However, the di-rect method is found to capture the trend of κ ( T ) quitewell. Consistently, slightly lower values of κ are founddue to the incorporation of the effect of phonon modescattering due to isotopes. On the other hand, the resultsof thermal conductivity, obtained by solving Boltzmanntransport equation under the relaxation time approxima-tion (RTA), are found to be slightly underestimated ascompared to the experimental results. However, as T >
372 K, RTA results seem to be in better agreement withthe experimental data and the difference between exper-imental and RTA goes down from ≈
19 % at 322 K to ≈ κ ( T ) obtained through RTA,alike the direct solution, is found to be in good agreementwith the experimental trend. This trend of κ ( T ) impliesthat umklapp scattering seems to dominate the phonon-phonon scattering at higher temperatures, as predictedby experiments.Previously, the hole concentration of GeTe is foundto be 6.24 × cm − , indicating a significant role ofvacancies for the reported overestimation of the κ ( T ),obtained via direct solution of LBTE. Indeed, D. Campiet al. [6] found a considerable amount of lowering of κ L of crystalline GeTe at 300 K due to the vacancy presentin the sample. FIG. 5. Experimental and theoretical thermal conductivity ( κ ) of GeTe as a function of temperature are presented in a fastchange process. Phase change behavior is realized in the studied temperature range: 322 K (cid:54) T (cid:54)
503 K. Electronic ( κ el ) andphonon contributions ( κ L ) to the total thermal conductivity ( κ tot ) are shown. κ L is evaluated using RTA, direct solution ofLBTE and Slack model. N, U, I and V define normal, umklapp, isotope and vacancy scattering processes respectively.TABLE III. Experimental and theoretical total thermal conductivity ( κ ) of the crystalline R3m-GeTe is presented as a functionof temperature (T). The lattice contribution of thermal conductivity ( κ L ) is taken as an average of κ x and κ z as κ av = (2 κ x + κ z )/3 (see text). The unit of κ is Wm − K − . N, U, I and V represent normal, umklapp, isotope and vacancy scatteringprocesses respectively.T (K) Expt. Direct (N,U) Direct (N,U,I) Slack RTA(N,U) RTA(N,U,I) Direct (N, U, I, V)322 3.59 3.90 3.84 2.88 2.90 2.87 3.36372 3.09 3.53 3.48 2.69 2.66 2.64 3.06412 2.70 3.31 3.27 2.53 2.53 2.51 2.89452 2.62 3.12 3.09 2.40 2.41 2.39 2.75503 2.43 2.92 2.90 2.30 2.28 2.27 2.59 τ V = x (cid:18) ∆ MM (cid:19) π ω g ( ω ) G (cid:48) (24)where, x is the density of vacancies, G (cid:48) denotes the num-ber of atoms in the crystal and g ( ω ) is the phonon densityof states (PDOS). Using vacancies as isotope impurity, C.A. Ratsifaritana et al [46] denoted mass change ∆ M =3 M , where M is the mass of the removed atom. Eq. 24states that the phonon-vacancy relaxation time is tem-perature independent. D. Campi et al. [6] used thisphonon-vacancy scattering contribution for a GeTe sam-ple with hole concentration of 8 × cm − and foundan almost ≈ κ L at 300 K. As the holeconcentration is almost similar to that of our work (6.24 × cm − ) , in conjunction with the fact that τ − V is temperature independent, we estimate an overall ≈ κ L ( T ) throughout the temperaturerange studied as an effect of phonon-vacancy scattering.We find that the estimated κ , incorporating the vacancycontribution, in addition to the normal, umklapp and iso-tope scattering, agrees excellently with the experimentaldata for the whole temperature range in our study. Thisexercise strongly depicts the significant participation ofthe scattering between phonons and vacancy defects athigh temperatures.It is well known that the RTA, although describes thedepopulation of phonon states well, fails to rigorously ac-count the repopulation of phonon states [42]. While atlow temperatures, the applicability of RTA can be ques-tioned due to the dominance of momentum conservingnormal scattering and almost absence of umklapp scat-tering of phonons, in the high temperature regime of ourstudy, RTA is found to be a good trade off between accu-racy and the computational cost to describe the exper-imental results. This is primarily because of the highernumber of scattering events at higher temperature whichensures an isothermal repopulation of the phonon modes[47]. However, the difference from the total solution ofLBTE exists due to the over resistive nature of the scat-tering rates that effectively lowers the value compare tothe direct solution. This feature has been discussed inliterature [17, 47]. The reason of this underestimation isthat the RTA treats both umklapp and normal scatter-ing processes as resistive while the momentum conservingnormal scattering processes do not equally contribute tothe thermal resistance as that of the umklapp scattering[17].Simple phenomenological models can also serve a fastand efficient way to decipher the underlying physicalmechanism. The Slack model [18] expresses the latticethermal conductivity, when T > Θ D and the heat con-duction happens mostly by acoustic phonons, startingfrom the analytical expression of the relaxation time re- lated to umklapp processes as: κ L = C M Θ D δG n / c T (25)with: C = 2 . × − − . /G + 0 . /G (26)where n c is the number of atoms per unit cell, δ is vol-ume per atom ( δ is in angstrom in the relation), M isaverage atomic mass of the alloy and G is the Gruneisenparameter (see Table II). This relation between latticethermal conductivity and Gruneisen parameter in a solidis valid within a temperature range where only interac-tions among the phonons, particularly, anharmonic umk-lapp processes are dominant [18]. E. Bosoni et al. [17]found a good agreement between the lattice thermal con-ductivity of crystalline GeTe coming from the full solu-tion of BTE and that of the Slack model at room temper-ature. Lattice thermal conductivity due to Slack model( κ L (Slack)) is presented in Fig 5. Total thermal conduc-tivity is then realized by adding κ L (Slack) with κ el . Weretrieve an almost identical trend of both lattice ( κ L ) andtotal thermal conductivity ( κ tot ) in Slack model as thatof the RTA based solutions. This almost identical val-ues of κ L (Slack) and κ L (RTA) depicts that the opticalphonons contribute very little to κ L (RTA) as κ L (Slack)takes into account only acoustic phonon contributions.The values obtained for κ tot (Slack) are found to be lowerthan the experimental data for T <
412 K and graduallyseem to agree well for T (cid:62)
412 K (Fig 5 and Table III).Though RTA based solutions underestimate the exper-imental data, the trend of κ ( T ), which is almost identicalto phenomenological Slack model, is what needs atten-tion to elucidate the underlying heat transport mecha-nism at high temperatures. Further, RTA based solutionsare straightforward and can reveal the distinct role ofeach contributing parameters appearing in Eq.18. Conse-quently, in the following sections, we systematically studythe thermal transport properties of GeTe using both fre-quency and temperature variations using the results ob-tained from RTA solutions. D. Variation of lattice thermal conductivity ( κ L )with phonon frequency: Contribution of acousticand optical modes To investigate lattice thermal conductivity ( κ L ) in amore comprehensive manner, we calculate the cumula-tive lattice thermal conductivity as a function of phononfrequency for the high temperature regime defined as[34, 45] κ cL = (cid:90) ω κ L ( ω (cid:48) ) dω (cid:48) (27)1 FIG. 6. Cumulative lattice thermal conductivities ( κ L ) ofcrystalline GeTe are presented as a function of frequencies atfour different temperatures: (a) T = 322 K, (b) T = 372 K,(c) T = 452 K and (d) T = 503 K. Cumulative κ L , computedwithin the RTA framework, along hexagonal c axis ( κ z ), alongits perpendicular direction ( κ x ) and their average κ av = (2 κ x + κ z )/3 are shown. The derivatives of κ z and κ x with respectto frequencies are also shown for each temperature. where κ L ( ω (cid:48) ) is defined as [34, 45] κ L ( ω (cid:48) ) ≡ N V (cid:88) λ C λ v λ ⊗ v λ τ λ δ ( ω (cid:48) − ω λ ) (28)with N (cid:80) λ δ ( ω (cid:48) − ω λ ) is weighted density of states(DOS).Figure 6 shows the cumulative κ L of crystalline rhom-bohedral GeTe, along hexagonal c axis ( κ z ), along its per-pendicular direction ( κ x ) and their average κ av = (2 κ x + κ z )/3, as a function of phonon frequencies for four differ-ent temperatures using RTA framework. The derivativesof the cumulative values of κ z and κ x with respect tofrequencies are also shown for each temperature. It isfound that the lattice thermal conductivity ( κ L ) of GeTeis anisotropic with the value along z , parallel to the caxis in the hexagonal notation, is found to be smallerwith respect to that of the xy plane for the whole rangeof temperature studied. This picture is consistent withthe recent theoretical findings [6]. More details to theanisotropic aspect of κ L will be discussed in the nextsubsection.Figure 6 shows some distinct features in the cumulativelattice thermal conductivity ( κ cL ) of GeTe as a functionof both phonon frequency and temperature. As the tem-perature is increased, κ cL is found to saturate at gradually TABLE IV. Relative contributions of acoustic and opticalmodes to the total lattice thermal conductivity ( κ L ) for dif-ferent temperatures. The unit of κ L is W/mK.T (K) κ L (RTA) Contribution of Contribution ofacoustic modes (%) optical modes (%)322 2.14 77.1 22.9372 1.85 77.3 22.7452 1.53 77.1 22.9503 1.37 77.4 22.6 lower values, indicating gradual decrement of the lat-tice thermal conductivity with temperature. Further, thederivatives of κ cL with respect to phonon frequencies in-dicate the density of heat carrying phonons with respectto the phonon frequencies [48] and their contribution tothe κ cL . We note that this density of modes go to zero ata frequency where κ cL reaches a plateau. This frequency( ∼ ∼
96 cm − = 2.88 THz inFig 4). This correspondence imply that the phonon den-sity of states play a crucial role as a deciding factor tothe κ L . While in Fig 6, a significantly higher contribu-tion of these modes correspond to κ x is observed com-pared to that of the κ z in the acoustic frequency regime(frequency < κ x /d ω and d κ z /d ω are found in the optical frequencyregime (frequency > κ L . The values of κ cL for phonon frequencies < > κ L ) as a function of temperature. We ob-serve that a dominant 77 % of the contribution comesfrom acoustic modes compared to only around 23 % fromthe optical modes for the whole temperature range stud-ied. E. Anisotropy of lattice thermal conductivity ( κ L )of crystalline GeT e
As mentioned in the previous subsection, the latticethermal conductivity ( κ L ) of crystalline GeTe showsanisotropic behavior. The resulting κ L along z direction( κ L ( z )), parallel to the c axis in the hexagonal notation(along axis c in Fig 3) is found to be smaller than thatof the xy plane (a-b plane in Fig 3),( κ L ( x )), as shown inFig 7.To investigate the anisotropy, we study the ratio κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ), where κ cL ( x ) ( ω ) and κ cL ( z ) ( ω ) repre-sent the cumulative lattice thermal conductivities along2 FIG. 7. Lattice thermal conductivity ( κ L ) of GeTe along z direction ( κ L ( z ), along the c axis in the hexagonal notation)and along x − y direction (( κ L ( x ), along the a axis) are pre-sented as a function of temperature. N, U, I denote normal,umklapp and isotope scattering effects respectively. xy plane (a-b plane in Fig 3) and z direction (alongc axis in Fig 3) respectively. Figure 8.(a) shows thisratio as a function of phonon frequencies for the fourdifferent temperatures. We observe that despite thewide temperature range studied, the shape of the curvesremain independent of the temperature. The ratio κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ), initially starts with a low value, be-comes maximum to 1.5 at around 1.4 THz. With furtherincrease of frequencies, the ratio decreases and settles ata value of 1.37.The temperature independence of the anisotropy ratio κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ) can be further understood by study-ing the cumulative outer product of the phonon groupvelocities v λ ⊗ v λ , defined as W c ( ω ) ≡ N V (cid:88) λ v λ ⊗ v λ δ ( ω − ω λ ) (29)and W cx ( ω )/ W cz ( ω ), which is the ratio between the cumu-lative outer product of the phonon group velocities along x and z direction. FIG. 8. (a) Ratios of x and z components of cumulative lat-tice thermal conductivities, κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ) of crystallineGeTe as a function of phonon frequencies are presented forfour different temperatures. (b) Ratios of x and z compo-nents of cumulative direct vector products of group velocities, W cx ( ω )/ W cz ( ω ) of crystalline GeTe as a function of phononfrequencies for four different temperatures. Since phonon group velocities are almost tempera-ture independent, following that feature, we find thatthe ratio of W cx ( ω )/ W cz ( ω ) is strongly correlated with κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ). From zero frequency, W cx ( ω )/ W cz ( ω )starts increasing and reaches the ratio 1 around 0.7 THz,close to that of the κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ) (0.4 THz). Furtherincreasing frequency increases the anisotropy between thein-plane ( xy ) and out-of the plane ( z ) components ofthe cumulative outer product of phonon group velocitiesand finally saturates to a value of 1.23 at higher frequen-cies which is also comparable to the saturation value of κ cL ( x ) ( ω )/ κ cL ( z ) ( ω ), that is 1.37. Thus, the anisotropy as-sociated with the phonon group velocities, or more specif-ically, the cumulative outer product of phonon group ve-locities determines the anisotropy in the cumulative lat-tice thermal conductivity. F. Variation of lattice thermal conductivity ( κ L )with temperature: Phonon lifetime and phononmean free path Recalling Equation 18, in an alternate way to under-stand the frequency dependence of the different param-eters that contribute to the lattice thermal conductivity( κ L ), we study the variation of modal heat capacity ( C λ ),phonon group velocity ( v λ ) and the phonon lifetime ( τ )as a function of phonon frequency for the high temper-3ature regime studied in this work. The dominant con-tribution of phonon group velocities ( v λ ) are found toarise from the acoustic phonons and the optical modesare found to exhibit substantially lower group velocitiesthan the former. Furthermore, as expected, increasingtemperatures does not change the dependence of groupvelocities on phonon frequencies. Similar behavior is ex-pected from the modal heat capacity (C λ ) as a functionof phonon frequencies for different temperatures. Indeed,C λ stays nearly constant with a small ∼ k B which is consis-tent with the classical limit of C λ at high temperatures.This almost non varying patterns of phonon group ve-locities and mode heat capacity with temperature promptus to look closely on the frequency variation of phononlifetimes ( τ λ ) at these temperatures. We note here that τ λ defines modal phonon relaxation time or phonon life-time with λ denotes each mode. Figure 9 depicts thephonon lifetimes, coming from TA, LA and optical modesas a function of phonon frequency. We observe thatincreasing phonon frequency from 0 THz gives an ini-tial rise and then a quick decay of lifetimes within ∼ < FIG. 9. Phonon lifetimes of crystalline GeTe are shown as afunction of phonon frequencies for four different temperatures:(a) T = 322 K, (b) T = 372 K, (c) T = 452 K and (d) T= 503 K. Phonon lifetimes due to transverse acoustic (TA),Longitudinal acoustic (LA) and optical modes are presentedin blue, red and green points respectively. FIG. 10. Phonon mean free paths of crystalline GeTe areshown as a function of phonon frequencies for four differenttemperatures: (a) T = 322 K, (b) T = 372 K, (c) T = 452K and (d) T = 503 K. Phonon mean free paths along a-axisand c-axis are represented in red and gray dots respectively.
Considering the temperature variation, we find thatacoustic modes induce smaller values of phonon lifetimeswith increasing temperature (Fig 9). Two prominent ob-servations evolve through this: (a) the trends in phononlifetimes ( τ λ ) as a function of frequency reassure the factthat acoustic phonons are the dominant carriers of heatwhich contributes to κ L , (b) as the phonon lifetime isdirectly proportional to κ L , the reduced contribution ofphonon lifetime with increasing temperature directs to-wards a gradual decrement of κ L with temperature. Thegradually reducing values of phonon lifetime with increas-ing temperature can be understood more prominentlyconsidering the mean free path picture. The modal meanfree path of phonons (Λ λ ) can be written asΛ λ = v λ τ λ (30)The transport of heat through phonons in the diffusiveregime (Λ (cid:28) L , L = linear dimension of the mediumof travelling phonons) undergoes several scattering pro-cesses namely scattering by electrons, other phonons, im-purities or grain boundaries [41]. At high temperatures,phonon-phonon scattering dominates along with impu-rity scattering to some extent. It is well understood thatanharmonic coupling on thermal resistivity leads to Λ ∝ /T at high temperatures [24]. To elaborate it fur-ther, we study the mean free paths of phonons (Λ λ ) asa function of phonon frequencies for different tempera-tures. As can be seen from Fig 10, the mean free pathsdecay quickly within ∼ ∝ /T .The anisotropic nature of κ L has also been understood bymeans of the mean free paths along a and c axes of R3m-GeTe. Separate contributions of the mean free pathsalong a-axis and c-axis are shown in Fig 10. Throughoutthe temperature range studied, phonon mean free pathscorrespond to the a-axis show higher values compared tothat of the c-axis, giving rise to an enhanced heat transferalong a-axis with higher values of κ L ( x ) compared to thec-axis with lower values of κ L ( z ). Thus, the anisotropicmean free path distribution of phonons is found to be thekey reason for exhibiting an anisotropic heat transfer incrystalline R3m-GeTe.To investigate the contribution of mean free paths ofdifferent lengthscales to κ L , cumulative lattice thermalconductivity (considering the κ ave = (2 κ x + κ z )/3) isstudied as a function of phonon mean free paths for thefour different temperatures (Fig 11). We observe fromFig 11 that the maximum values of the phonon meanfree paths (Λ max ) gradually decrease with temperature(T = 322 K, Λ max ∼
152 ˚A; T = 372 K, Λ max ∼ max ∼
108 ˚A; T = 503 K, Λ max ∼
97 ˚A), consistently with Λ ∝ /T . Moreover, a dominantcontribution ( ≈
67 %) to the lattice thermal conductivityis found to be coming from the phonon mean free paths (cid:54)
60 ˚A. As temperature increases, the contributions fromthe phonon mean free paths (cid:54)
60 ˚A are increased toalmost 93 % (Fig 11).Increasing temperature, thus, manifests in a way of in-creasing phonon-phonon scattering processes which acts
FIG. 11. Variation of cumulative lattice thermal conductivity( κ L ) of crystalline GeTe with phonon mean free path is shownfor four different temperatures: (a) T = 322 K, (b) T = 372K, (c) T = 452 K and (d) T = 503 K. The contributions fromthe mean free paths (cid:54)
60 ˚A to the κ L are mentioned for twoextremes of temperatures. on the lowering of mean free path of phonons. Withalmost similar v λ , the lowering of mean free path ofphonons can be associated with the decrements of τ λ . G. Contribution of transverse and longitudinalacoustic phonons
Up till now we understood the dominant contributionsof the acoustic modes to the thermal transport mecha-nisms of crystalline rhombohedral GeTe. In this section,we further systematically discriminate the relative contri-butions of transverse and longitudinal acoustic phononsto the lattice thermal conductivity of crystalline GeTefor the temperature range investigated in this study. Itis observed that the transverse modes (TA) contributeto almost 75 % while the longitudinal counterpart (LA)adds up the rest of 25 % to the lattice thermal conduc-tivity for the whole temperature range.Figure 12 presents the lattice thermal conductivities,obtained via Ab-initio DFT calculations coupled withRTA, due to both transverse (TA) and longitudinal (LA)acoustic phonons as a function of temperature. For con-sistency, we also plot the κ L values for acoustic modesfrom the experimentally measured values of κ . This hasbeen evaluated by subtracting the electronic thermal con-ductivity ( κ el ), measured from the simulation, as well asthe κ L due to optical phonons, computed from Ab-initioDFT and RTA, from the experimentally measured valuesof κ . Indicating a consistent picture from both experi-ment and theoretical calculations, as discussed earlier, anidentical trend is found (Fig 12) between the data calcu-lated from experiment and the simulated values of κ L dueto acoustic phonons (TA+LA) despite the differences FIG. 12. Lattice thermal conductivities, contributed solelyfrom acoustic modes are shown as a function of tempera-ture. ‘Experimental’ values of acoustic κ L are obtained bysubtracting κ el and optical mode contributed κ L , from theexperimental values of κ . RTA calculated κ L for transverseacoustic branch (TA), longitudinal acoustic branch (LA) andthe total acoustic contribution (LA+TA) are also shown. FIG. 13. Phonon lifetimes of GeTe corresponding to umklappscattering, along with the umklapp fitted parameter A areshown as a function of phonon frequencies for four differenttemperatures: (a) T = 322 K, (b) T = 372 K, (c) T = 452 Kand (d) T = 503 K. between the values, mostly for T <
412 K.Throughout this work, the trend of simulated κ ( T ) us-ing RTA has been found to be a straightforward andconvenient framework to describe the temperature de-pendence of the experimental results of κ for crystallineGeTe. Further, the exact similarity between RTA and thephenomenological Slack model, indicates that umklappscattering plays an important and significant phonon-phonon scattering mechanism in the high temperaturerange ( T (cid:29) Θ D ), Θ D being the Debye temperature. Toidentify the umklapp scattering parameter involved in theprocess, we fit the phonon relaxation time correspond tothe umklapp scattering, obtained from RTA, with theanalytical expression given by Glen A. Slack et al. [19] τ − U = AT ω exp (cid:18) − B Θ D T (cid:19) (31)with A ∝ (cid:126) G Mc Θ D and B ∼ M is averageatomic mass of the alloy and G is the Gruneisen pa-rameter. Being an empirical equation, it is necessary tofind the parameter A from the fitting procedure. Figure13 shows the phonon relaxation times as a function ofphonon frequency along with the fitted curve for all thefour temperatures. We observe that at higher frequen-cies the trend of phonon relaxation time indeed shows ω − dependence for all temperatures. The values of A ,thus retrieved from the fitting, are found to be indepen-dent of temperature with an average value of 1.1 × − ps K − .This quantification, via the parameter A , serves as animportant generic identification as this parameter canuniquely distinguish and compare the umklapp scatter-ing processes for different crystalline materials rangingfrom Θ D to an arbitrary high temperatures, within theoperational regime of umklapp scattering process. V. SUMMARY AND CONCLUSIONS
We have carried out a systematic experimental andtheoretical study on the thermal conductivity variation ofGeTe at high temperatures. The study involves fast andreversible phase change between amorphous and crys-talline phases of GeTe. Modulated photothermal radiom-etry (MPTR) as well as the Lavenberg-Marquardt (LM)technique are employed to determine thermal conductiv-ities of GeTe in both amorphous and crystalline phasesas a function of temperature. Thermal boundary resis-tances, coming from both Pt-GeTe and GeTe-SiO inter-faces, have been accurately taken into account for mea-suring κ experimentally. Van der Pauw technique as wellas Boltzmann transport equations are solved for electronsto estimate electronic thermal conductivity within theconstant relaxation time approximation (CRTA) frame-work.To compute lattice thermal conductivity ( κ L ), first-principles density functional theory (DFT) is used andthe solution to the linearized Boltzmann transport equa-tion (LBTE) has been realized via both direct methodand relaxation time approach (RTA). Normal, umklappand isotope effects are included in computing the phononrelaxation time in these approaches. While the directmethod is found to capture the trend of κ ( T ) quite well,the values are a bit overestimated compared to the exper-imental data. The hole concentration of 6.24 × cm − ,obtained using first principles calculations and BTEfor electrons, necessitates the incorporation of phonon-vacancy scattering to estimate κ L . Following a recentwork [6] on crystalline GeTe with almost same hole con-centration, vacancy contribution is incorporated to esti-mate more realistic values of κ . Indeed, the estimate of κ using direct method and adding the temperature inde-pendent phonon-vacancy scattering contribution, an ex-cellent agreement is obtained between experimental andtheoretical values of κ . κ computed from RTA, is also found to retrieve thetrend of experimental κ ( T ) quite well, especially athigher temperatures. However, the over-resistive natureof RTA due to the treatment of umklapp and normalscattering in equal footing, causes an underestimationof κ compared to the experimental values. Neverthe-less, the trend κ ( T ) agrees well at higher temperatures.Cumulative lattice thermal conductivity is presented as6a function of phonon frequencies for different tempera-tures. The density of heat carrying phonons or ratherthe phonon density of states plays a crucial role in de-termining κ L . Acoustic phonons emerge as the domi-nant ( ∼ κ is attributed to the variation of phonon mean free pathand consequently the variation of phonon lifetime withtemperature.Phenomenological models are also found to be quite ef-fective in describing and identifying the underlying phys-ical mechanism of thermal transport. The experimentalvalues of κ in amorphous phase of GeTe are describedusing the minimal thermal conductivity model, proposed by Cahill et al. [20]. For crystalline phase data, Slackmodel [18] is also employed and it has been found to bestrikingly similar with RTA based solutions. Both RTAbased solutions as well as expression from phenomeno-logical Slack model for GeTe indicate that the umklappphonon phonon scattering is significant for the temper-ature regime studied in this work. Umklapp phonon re-laxation time is found to obey ω − dependence at higherfrequencies and therefore umklapp scattering parameterhas been obtained, which remains almost constant for thewhole temperature range studied. This complete experi-mental and theoretical exercise to elucidate the thermalconductivity of GeTe at high temperatures can furtherassist in improving the thermal management for otherTe based phase change materials by understanding heatdissipation, localization and transport with more clarity. ACKNOWLEDGMENTS
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