Thermal Conductivity in Superlattices

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1 006, November Thermal Conductivity in Superlattices S. Tamura Department of pplied Physics Hokkaido University

2 Collaborators and references Principal Contributors: K. Imamura Y. Tanaka H. J. Maris B. Daly References. S. Tamura, Y. Tanaka, and H. J. Maris, Phys. Rev. B60, 67 (999). B. C. Daly, H. J. Maris, K. Imamura and S. Tamura, Phys. Rev. B66, 430 (00) 3. K. Imamura et al., J. Phys. Condensed Matter 5, 8679 (003) 4. B. C. Daly, H. J. Maris, Y. Tanaka and S. Tamura, Phys. Rev. B67, (00)

3 Outline of Talk 0. Introduction Phonons in superlattices. Out-of-plane conductivity - 0 Experimental results - Effect of Brillouin-zone folding - Effect of relaxation time (anharmonicity) - 3 Effect of interface roughness. In-plane conductivity

4 pplication of superlattice structures Operation of devices is greatly affected by thermal conductivity κ () Semiconductor laser large κ is preferred Lifetime is reduced by the heat produced () Thermoelectric devices low κ materials are preferable B B D Figure of merit : Z S = ( V= S T, S : Seebeck coefficient ) ρ κ Smaller thermal conductivity produces larger temperature difference

5 Phonon dispersion relations in a bulk crystal Zincblende structure Dispersion relations ω = ω ( k, j) in Gas (a cubic crystal) Ga s a optical lattice dynamics Brillouin zone k z acoustic elasticity theory k x k y ( π a)

6 Synthetic periodicity and zone-folding Bulk crystalline solids (D) a D Periodic superlattices (SLs) unit period a (atomic spacing) D ω SL mini (folded) zone bulk iii B B B iii synthetic periodicity D ( = n a ) Brillouin-zone folding 0 π D ω = ω ( q) π π D = nλ = n q = n (Bragg condition) min q D q π a ( λ = a)

7 How to calculate the phonon dispersion relations in SLs Elasticity theory (valid for sub-thz phonons) ρ i ( r, t) S ( r, t) S (,) r t = c ij u t k = l j ijkl ij x l j u (,) r t x k u S i ij : lattice displacement : stress tensor ρ :density c ijkl : elastic stiffnes tensor ρ ( r ) i c ijkl u, t uk (,) r t =, ( i =,,3) t x x j l : wave equations Plane wave solution (, ) ur Christoffel equation i( kr ωt) t = a ee ( ) ρω δ c k k e = 0, ( i =,,3) im ijmn j n m one longitudinal (L) two transverse (T) ω = ω ( k, j)

8 Wave fields and transfer matrices for superlattices Wave fields in layer d db ( ) W n z n Γ Φ ( z ) iii B B B iii z (, t) (, t) 6 ( j) Un x ( j) e ( j) = n ( j) z, S n x j= σ a exp( ik z) e i ( k x ωt) ( 3 pairs of counterpropagating waves ) = W n ( z) e i ( k x ωt) Definitions of transfer matrices n ( ) =Γ Φ( ) n W z z T =Γ Φ ( d ) Γ - T T T 6 6 B matrix

9 Transfer matrix T W Transfer matrix in periodic SLs ( z ) T W( z ) n+ = () Propagation normal to the layer interfaces (z direction) k 0 () Single mode, e.g., u = ( 0, 0, U( z, t) ) Matrix elements of T matrix Z = ρ v acoustic impedance dett = n = for L mode W ( z ) n z Dn + z n U n = Sn z x = ( x,0) B B B B Z T = cos( k d) cos( kbdb) sin( kd) sin( kbdb) ZB Z T = sin( kd) cos( kbdb) + cos( kd) sin( kbdb) ZB ZB T = sin( k d) cos( kbdb) cos( kd) sin( kbdb) Z T = T ( B) D = d + d : unit period B

10 Bloch theorem Perfect periodicity and phonon dispersion relation k = 0 W = exp( iqd) W = T W n n n q : Bloch wave number Dispersion relation in periodic SLs cos( q D) = Tr ( T) / ωd ωdb Z ZB ωd ωdb =cos cos + sin sin v vb ZB Z v vb S. M. Rytov, Sov. Phys. coust., 68 (956) Tr( T )/ < :frequency band Tr( T )/ > :frequency gap Tr( T )/ = v vb ω = ωm = mπ + d db π π < q < : mini-zone D D π q = 0, ± : zone center and boundaries D : m -th order Bragg freq.

11 Perferct periodic SLs (normal propagation) k = 0 (00) (Gas) 5 (ls) 5 SL d = d = B.83 5 = 85 nm Dispersion relations Transmission rates Bragg reflections frequency gaps Frequency (THz) transmission dips for a finite periodic structure T ω L 0.5 T L ω T ω.0 L T N (# of periods) = qd/π Transmission rate

12 Perferct periodic SLs (oblique propagation) k 0 (00) (Gas) (ls) SL L phonon incidence from a Gas substrate θ L = 45 d = d = 40 B T gap Intermode Bragg reflection opening of frequency gaps reduction of group velocity

13 Thermal conductivity (κ) in semiconductor SL Measurements of thermal conductivity (κ) in semiconductor SL. T. Yao, ppl. Phys. Lett. 5, 798 (987) bulk Gas In plane κ in Gas/ls SLs Reduction of 0-70 % depending on bi-layer thickness. S. Lee, D. Cahill, and R. Venkatasubramanian, ppl. Phys. Lett. 5,798 (987) κ in Si/Ge SLs 3. W. S. Capinski, H. J. Maris et al., Phys. Rev. B59, 805 (999) Out of plane κ in Gas/ls SLs

14 Out of plane thermal conductivity (κ) in semiconductor SL W. S. Capinski, H. J. Maris et al., Phys. Rev. B59, 805 (999) pump probe (Gas) Gas/ls-SL n -(ls) n SL t = 0.5 ps 0 µm heat Gas substrate Picosecond Ultrasonics a pump-probe optical technique measure the change in optical reflectivity R due to phonons R 0. µm (0.5~) µm periods () T= K (B) n : # of monolayers (repeat distance of SL) T () t l

15 Thermalization time and increase in film temperature The time for T to become uniform inside the l film film thickness ~0. µm d τ = = π l Dl 3 ps (300 K) τ echoes thermal diffusivity ~0.45 cm / s The increase in the l-film temperature optical reflectivity ~0.9 energy of pump pulse.3 nj ( R) Q Tl = C d l l K at 300 K specific heat per volume area of the spot

16 Determination of the thermal conductivity (κ) The rate of heat flow across l-sl interface K [ () ( 0,) ] Q = σ T t T z = t l Kapitza conductance The rate of temperature change in the l film SL 0. mm T l l Q B B B z=0 T (,) z t SL z iii Tl () t Cl dl = σk[ Tl () t TSL ( z = 0,) t ] () t Heat flow within the SL (~D) heat capacity in the SL (average of Gas and ls) C TSL(,) z t TSL(,) z t SL = κ t z (B) σ () ssume K and κ (adjustable parameters) () With () and (B), determine l () T t T () t = β R () t l thermoreflectance

17 Experimental out-of-plane thermal conductivity (κ) in SLs 0.0 bulk Gas (300K) K 00K / K TEMPERTURE (K) # of monolayers n () κ decreases as T increases () κ decreases as the number of monolayer n decreases How we understand these experimental results (3) κ is much smaller than κ Gas (~/0 for x-sl)

18 Lattice thermal conductivity (κ) Fourier s law J = κ T Expression for thermal conductivity lattice specific heat relaxation time group velocity in the direction of T κ = κ κ = ω τ λ λ, λ Cph ( λ) λvλ, z λ = ( k, j) C ( ω) exp( ω/ kt ) ph ( ω) = kt [ exp( ω / kt ) ] wave vector phonon mode

19 Effect of Brillouin-zone-folding in SLs Opening of frequency gaps and the associated groupvelocity reduction near the zone center and boundaries () P. Hyldgaard and G. D. Mahan, Phys. Rev. B56, 0754 (997) simple cubic lattice () S. Tamura, Y. Tanaka, and H. J. Maris, Phys. Rev. B60, 67 (999) FCC lattice 3D FCC lattice model (harmonic approximation) Periodic B. C. M K d M B B d B B a () M MB () Force constant K is the same for the atomic pairs -, -B, B-B

20 Equations of motion for the lattice displacement a = Gas or Si B = ls or Ge K B d d B = n a = n a B B x K x x x x Mn u lmn = ( ul+, m+, n + ul+, m, n+ + ul+, m, n + ul+, m+, n + u + u + u + u 8u x x x x x l, m+, n l, m, n+ l, m, n l, m, n l, m, n + u u + u u y y y y l+, m+, n l+, m, n l, m, n l, m+, n z z z z ) l+, m, n+ l+, m, n l, m, n u l, m, n + + u u + u D= d + d B r = ( x, z ) lmn lm n = (, l m, n) a u = u ( z)exp[ i( k x + qnd ωt)] ( N ) lmn n lm u + + ( z) = u ( z) ( n n + nb ) n n n n B Bloch function Periodic function

21 Determination of dispersion relations x SL ( n = n = ) B iqd iqd M ω + CC SS 0 Cx( + e ) 0 Sx( e ) iqd iqd SS M ω + CC 0 0 Cy( + e ) S y( e ) iqd iqd iqd 0 0 M ω 4 K Sx( e ) Sy( e ) C+ ( + e ) det iqd iqd Cx( e + ) 0 Sx( e ) MBω + CC SS 0 = iqd iqd 0 Cy( e + ) Sy( e ) SS MBω + CC 0 iqd iqd iqd Sx( e ) Sy( e ) C+ ( e + ) 0 0 MB ω 4K 0 ω = ω ( k, q) C = Kcos k a, S = iksin k a x x x x ( C, S ) = ( k k ) y y x y C = K(cos k a+ cos k a) + CC = K( cosk a cos k a ) SS = Ksink a sink a x x x y y y

22 Dispersion relations for k =0 8 7 [ν ls ] max (5 5)-Gas/ls 0 [ ν Si ] max (5 5)-Si/Ge 6 [ν Gas ] max 8 [ ν Ge ] max not contribute to κ qd/π qd/π

23 Effect of group velocity and weighted density of states To see the effect of group velocity, we have calculated κ = κ τ ω λ λ / λ Cph ( λ)vλ, z λ Phonon density of states weighted by v λ, z < v( ω) >= δ( ω ω )v z λ λ, z V λ = ds λ 3 ( π ) j vλ v λ, z ω = ω λ

24 DOS weighted by (v z ) and calculated κ ~ κ / τ ls Gas n= n=5 n=0 Gas/ls (Gas) n (ls) n ls Si Ge n= n=5 n=0 Si/Ge (Si) n (Ge) n Si Frequency (THz) Frequency (THz) (Gas) n (ls) n bulk ls ~/ (Si) n /(Ge) n bulk Si ~/ ls Gas n= n= n=5 n= Si Ge n= n= n=5 n= Temperature (K) 300 K K Temperature (K)

25 κ ~ κ / τ vs number of monolayers ls (T=300K) Gas (T=300K) Si (T=300K) Ge (T=300K) T=50K T=300K T=00K T 00K 300K # of monolayers n bulk Gas (300K) n 00K expt # of monolayers n Gas/ls n 0. T=50K T=300K T=00K # of monolayers n κ ~ κ / τ Si/Ge Calculated exhibits T and n dependences opposite to the experimental results Effect of τ is important!! n

26 How to include the effect of relaxation time Molecular dynamics (MD) calculations () B. Daly et al., Phys. Rev. B66, 430 (00) () ssume an initial temperature distribution of the system T(x,t = 0), e. g., ( T 0. T ) 0 0 computer simulated temperature decay SL n () MD is used to determine the rate at which T (t) is changed by heat diffusion L x Tt () = Txt (,)cos( π x/ L) 0 x dx L x (3) thermal diffusion constant D is deduced from Tt = T Dt L x ( ) 0 exp( 4 π / ) (4) thermal conductivity is given by κ = D C ph

27 note on the MD calculation of Daly et al (3) thermal diffusion constant D is deduced from Tt = T Dt L x ( ) 0 exp( 4 π / ) computer simulated temperature decay [ no good fit to the simulated T (t) for a small t ] / v (4 D/ t) ( t 0) n n ad hoc modification for the solution for T (t) x 4 Dt 4Dt v 4Dt + v t t v t ( t 0) 4 Dt ( t ) π Tt () = T exp 4Dt v 0 L x 4Dt + v t t the MD results for Tt ()/ T 0 are fit to this form using D and v as adjustable parameters

28 How to include the effect of relaxation time- Molecular dynamics (MD) calculations (B) K. Imamura et al., J. Phys. Condensed Matter 5, 8679 (003) heat-reservoir method J () Heat current () Temperature profile T( x) Fourier s law J = κ T κ hot heat bath T H buffer layer cold heat bath T C harmonic spring anharmonic spring

29 Hamiltonian and interatomic potential Hamiltonian H p = + Φ m ' ' interaction with + heat reservoirs = (, l m, n) a r lmn ( m = m or m ) Gas ls Interatomic potential 3 rd order anharmonicity β β ' Φ =Φ ( u ) = u + u 6 u = u u 3 ' ' ' ' ' ' relative displacement 3Ba C + C β =, B = Bulk modulus 3 9Bγ β ' = Gruneisen parameter (γ = ) lgorithm: Symplectic Integrator (SI) method

30 Heat reservoirs and energy exchange with atoms Heat reservoirs n x = nx = Nx () consist of particles with energy (E H and E L ) given by Boltzmann distribution at T H and T L () particle collides elastically with an atom at the end of the SL with a given probability w (~0.) at each time step hot heat bath T H cold heat bath T C (3) Energy given to (removed from) the atom at the site ( n =, n, n ) [( n = N, n, n )] x x y z x y z Q p ( s) ( s) (, ny, nz ) H,( n, n ) = E H, y z m Q ( s) ( s) ( N x, ny, nz) L,( n, n ) = EL y z p m (with collision) Q ( s) ( s),( n, n ) Q L,( n, n ) 0 = = (without collision) H y z y z

31 Heat current and steady state The heat flowing into (from) the SL at a time t (averaging over time steps s to reduce fluctuations) Q t Q s ' ( s) H() = H,( ny, nz ) s' t s= n, n y z Q t Q s ' ( s) L() = L,( ny, nz ) s' t s= n, n y z t = 4 ps : an interval of time step (from energy conservation).0 heat flowing out from the SL Steady state for t > τ 0 Q () t = Q () t H Q () t : t indep. H Heat flux (defined in the steady state) J = QH () t dt τ 0 ( τ τ ) 0 cross-sectional area of SL L τ Heat flux (0 6 J cm - s - ) < Q L (t) > 0.5 < Q H (t) > heat flowing into the SL τ 0.5 Time (ns) steady state τ 0 =. 4 ns.0.5

32 Temperature profiles ( n, n, n ) 4 x y z t T( x = nxa) p total # of lattice points in 3k N N m the lateral plane =N y N z / B y z n, n n y z x buffer layers superlattice T L a good conductor a poor conductor

33 note on the temperature profiles Temperature (K) T H T H (Gas) (ls) without disorder N x (monolayers) (a) T L 00 T L () No disorder is present T decreases exponentially with distance T( x) ' x / L x ' T L = TH ' TH κ T anharmonicity resists the heat flow /T-law Temperature (K) T H (Gas) (ls) with interfacial roughness (f = 0.5) () Disorder is present T decreases linearly with distance TL TH T( x) = TH + x 0 κ T L X 00 0 (b) N x (monolayers) T L 00 scattering from disorder is T-indep.

34 Size independence and thermal conductivity For a small system size some phonons travel ballistically between the reservoirs Fourier s law is inapplicable ( κ depends on the system size) Longitudinal direction Lateral direction N y =N z =8 0.8 Bulk Gas N y =N z = N x = # of lattice points N x N ( =N ) y z

35 Thermal conductivity in ideal SL The simulated κ includes the effect of lattice anhamonicity effects of group-velocity reduction due to zone-folding simulated result for ideal superlattice Expt. 0. T=300K # of monolayers (n) () For large # of monolayers, κ approaches to the experimental values () For small # of monolayers, κ does not explain the expeimental results nharmonicity is not the main origin of the reduction of κ for small n

36 Thermal conductivity in SL with interface roughness- How to include the interface roughness For the last atomic monolayer of each SL layer, we assign to each atom the mass either M Gas or M ls randomly with a probability f layer B layer interface layer ideal SL with roughness f =0.5 ( f max =0.5)

37 Thermal conductivity in SL with interface roughness- MD results of Daly et al. f = ideal SL f = 0. f = 0.0 f = 0.5 expt T=300K Thermal conductivity (W/cm K) # of monolayers (n) () κ in SLs with n < 0 is strongly affected by the interface roughness () κ in SLs with f = coincides well with κ exp

38 Thermal conductivity in SL with interface roughness- Temperature variations κ SL (W cm - K - ) = 0 Ideal SL 300 K 400 K Bilayer thickness (monolayers) Gas 300K Gas 400K κ SL (W cm - K - ) f = 0.5 Disordered SL 300 K 400 K Bilayer thickness (monolayers) () κ in ideal SLs κ min with respect to n () κ in strongly disordered SLs T-indep.

39 . In-plane thermal conductivity Expt. (by Yao) Zone-folding effect for the in-plane bulk Gas in plane out of plane reduction ~5 % unable to explain Yao s experiment Weighted density of states < > v( z ω) = δω ( ωλ )v V λ λ, z in plane out of plane

40 Comparison between the SLs without and with roughness Ideal SL κ / τ T=300 K MD calculation SL with interface roughness Expt. MD calculation MD calculation (f=0.5)

41 Summary I. MD calculation with a classical fcc lattice model for the thermal conductivity (κ) of semiconductor SLs -) For perfect, periodic SLs, the reduction of κ is in good agreement with the calculation based on the effect of Brillouin-zone folding -) With the addition of interfacial roughness, the dependence of κ on SL period similar to that seen experimentally is obtained II. These results are obtained with () conventional MD simulations which assumes the hot and cold thermal reservoirs () a MD simulation for the relaxation of an inhomogeneous temperature distribution III. The MD calculation have proven to be a powerful method for studying the thermal properties in SLs.

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