物性物理学I_2.pptx

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1 phonon U r U = nαi U ( r nαi + u nαi ) = U ( r nαi ) + () nαi,β j := nαi β j U r nαi r β j > U r nαi r u nαiuβ j + β j β j u β j n α i () nαi,β juβj

2 調和振動子近似の復習 極 小 値近傍で Tylor展開すると U ( x) = U ( x ) + ( x x )U ( x ) + ( x x ) U ( x )! n ( x x ) U (n ) ( x ) + n! U ( x ) U ( x ) + ( x x ) U ( x )! + 極値近傍の振る舞い ( U ( x ) = ) 調和振動子型ポテンシャル U ( x) ポテンシャル中の質点の運動 E = x + U ( x ) + ( x x ) U ( x )! E = E U ( x ) d = ( x x ) + ( x x ) U ( x ) dt! U ( x ) = x x 原子の位置 変位ベクトル n ( N ) フックの法則=線形近似 他の原子からの寄与 n N rα F = U = ( x x )U ( x ) unα i un β i n N 個 rnα = t n + rα t n = n + n b + nc O n n + un β j n unα i n + u n β j n n() α i,β j n n + U := > rnα i rβ j

3 原子の運動方程式 dunα i () Mα + nα i,β juβ j = dt β j iq r iω t unα i = uα i ( q ) e + c.c. Mα 平面波解 n (格子点上) ω unα i F変換! ( NN 本) + Mα Mβ βj iq r iq rn () nα i,β je uβ j () n() α i,β j = α i,( n )β j 並進対称性! ω uα i + βj = (特性方程式)! 分散関係 iq r iq rn α()i,(n )β j e Mα Mβ Dα()i, β j= 動力学行列! Dα()i, β juβ j = ( ω = ω (q) N 本) ) 結合振動子モデルと周期性 N自由度の結合振動子 単原子種一次元格子 x = x + ( x x ) 連立の運動方程式 x = x N+ = (固定端条件) x = ( x x ) + ( x x ) x N = x N + ( x N x N ) 特性方程式 ω + ω + ω + x x x これらを行列表示して x x = xn から基準振動を解析する x xn xn xn+

4 x = x n ( x n n ) + ( x n+ x n ) (n ) x = x N+ = x n = Asin (qn ωt +θ) q q = λ ω sin (qn ωt +θ) = [ cos q ]sin (qn ωt +θ) ω = ( cosq ) = 4 sin q ω = sin q x x x x x N x N x N+ q x = x N+ = Asin iωt +θ ( ) = Asin{ iq(n +) iωt +θ} = sin( iωt +θ ) = sinq(n +)cos( iωt +θ ) + cos q(n +) sin( iωt +θ ) = sinq(n +)cos( iωt +θ ) = ( t) t sinq(n +) = q j = j j =,,, ( N +) ( ) N N x = Asin ( ωt +θ) =, x N+ = Asin ( n N + ωt +θ) = N + x i = Asin ( n i i ωt +θ) = Asin n N + N + cos( ωt +θ) = x i( ) x N+l( ) = Asin (n N + l N + l ωt +θ) = Asin n N + cos( ωt +θ)( ) n = ( ) n x l

5 ω = sin q q j = ( N +) j ( j =,,,, N, N +) q = ( N +) q = ( N +) = li n n N+ N N + dω dq = = d dq sin q = Hz, 8 c = ω ln D( ω ) D( ω ) q = ω = q = 4 /s D( ω ) = dq dω = = q cos c ( ) ω

6 固定端の場合の振動モードと定在波 包絡線 のイメージ q = q = ( N + ) q = ( N + ) q = ( N + ) N ( N + ) 隣どうしが逆位相で振動 自由端条件下の振動解 周期的境界条件 Floquet-Bloch解 xn = Aeiqn iω t q= を仮定 q は波数 ( λ ) ω x = e iq + eiq x 運動方程式に代入すると ω = ( cos q ) = 4 sin ω = q sin q 音響フォノンの 分散関係 周期(的)境界条件 x = x N より N N Aeiq iω t = AeiqN iω t q = ( N ) l l =, ±,, ±, x x x xn xn

7 N自由度の結合振動子 二原子種からなる無限一次元格子 運動方程式 ( = c (u v j uj (q) = Aq Floquet-Bloch解 j + u j + M i q j+ ω t 4 e ( + e 光学分枝! T! L! = N ( =, ±, ±, ±,...) N モード 光学モード 音響モード q! T! ω= ω un (q) Aq ω = + = Bq M M Aq ω = = Bq Forbidden gp! 音響分枝! M vn (q) ( + e iq ) ) q= M 4sin ( q ) ω = + ± + M M M フォノンの分散関係 + M A q = Bq v j (q) M ω iq i q j+ ω t 4 + Aq*e i q j+ ω t 4 + Bq*e uj (q) v j (q) 特性方程式 j M ω ( + e iq ) ( + eiq ) ω 特性行列 q= (M > ) i q j+ ω t 4 e vj (q) = Bq uj (q) ) v ) j = c v j + v j u j Mu For N tos in -D crystl! q (M + ) q= coustic odes (xt + L)! N- optic odes! (TO>LO, LA>TA in generl)!

8 M u j (q) v j (q) ω = sin q ω = M + ± M + 4sin ( q ) M M q = q = q = q = ω = sin q q = q = q = q =

9 = λ=? ゾーン端の振動モード = λ= + M / 光学分枝! M / / Forbidden gp! T! L! 音響分枝! q! = / T! (λ = ) q= q= = フォノンと光の結合 c ω= n 光の分散関係 光散乱振幅 光の分散関係 A (t ) = dr n(r ) exp( iδ r ) e iω t Δ 交点で相互作用 (結合)が発生 光学分枝 (λ = ) 散乱中心 n(r ) = n δ ( r r (t)) r (t) = r + u (t) ベクトル分解 音響分枝 iδ (r + u (t)) iω t A (t ) = A e ω= q (M + ) q= q= iδ r iδ u (t) iω t = A e e e

10 iδ r iω t = A e iδ u (t) e iδ r iω t iδ r = A e e ia Δ e u (t)e iω t (*) ( iq r iω ( q)t iq r +iω ( q)t u (t) = u ( q ) e +e M iδ r iω t (*) = A e e ia 弾性散乱項 n n i(δ q) r i(ω ±ω ( q))t Δ e e M 非弾性(Rn)散乱項 Δ q = G 散乱(位相整合)条件 ω = ω ± ω ( q) q = G 保存則との対応 光 ω = ω ± ω ( q) 光 フォノン 結晶? 光 光 フォノン 仮想準位! 格子振動 フォノン による分極率変調 ) P = ε χ() + χ() cosωt E cos ω t 弾性(Ryleigh)散乱項 = ε E χ() cos ω t + ε E χ() cos (ω Ω ) t Intensity (.u.) + ε E χ() cos Si-Ge Si-Si Ω 基底状態! g Si-Si Ge-Ge ωs 振動励 起状態! 非弾性(Rn)散乱項 (ω + Ω) t Pup: 5 n 55 ω P ストークス光! 中間状態)! Rn散乱の古典論 ( ) 反ストークス光! 平面波解 SiGe (x=.64) lloy Wvelength (n)

11 ) 結晶の熱的性質 フォノンの状態密度 ω +d ω ω +d ω V V d q = dsω dq ( ) ω ( ) ω v = dω dsω dsω V V = dω = dω g dq v ( ) ω qω ( ) g 群速度 D (ω ) dω = 各モードのDOS 4 q V V ω Dα (ω ) dω = = dω dω c ( ) cα α ω qω = 全振動モードの和 dω = cl,t,t dq D (ω ) dω = D (ω ) dω α = V + ω dω cl ct Siのフォノンの分散曲線と状態密度 de d de d = = = = DOS $ dq ω (q) ω + dω

12 Intensity (rb. units) TO Si-Si TO Si-Ge TO Ge-Ge TA NP Si TO+O Γ e-h Si TO Si TO K Si TA Photon energy (ev) $F.$A.$Johnson$Proc.$Phys.$Soc.$7,$65$(959$)$ E n = ω n + ( n =,, ) n P n = Aexp E n B T P n = n A n= e E n B T = Ae ω B T n= = Ae ω B T e nω B T e ω B T = P n = ( e ω T B )e nω B T

13 E = P n E n = ( e ω n B T )ω e nω T B n +/ n ( ) x n = n= x, nxn = n= x ( x) = ( e ω B T )ω n + E ω, T ( ) = ω e nω B T e ω / BT + n f BE = n = e ω / BT f MB = e E n B T U ( T ) = N A ω = N A B T C V = U T C V = R C V = R V = N A B CV/R T/T

14 ( ) = N A U T e ω / BT + ω C V = U T = R θ E T lic V = li T li T V = N A B C V = li T e θ E /T ω B T e ω / BT (e ω / BT ) (e θ E /T ) θ E = ω R θ E T T R θ E T B e θ E /T (e θ E /T ) = R exp θ E T C E = U ω V = N T A B V B T = R θ E T e θ E /T (e θ E /T ) ω e ω / BT (e ω / BT ) ω CV/R..5 C D = 9N T V B /T e z z 4 (e z ) dz. T/T h;p://hyperphysics.phyastr.gsu.edu/hbse/solids/phonon.htl

15 ( ) = dωd( ω ) U T (q = ) ( ) E ω,t ω = cq C V = V c + L c T ω D ω T E( ω,t )dω ω D N A = V c + ω D L c T ω dω C V = 9N A ω D = 9N A ω D ω D T = 9N A B T B / e ω / BT ω dω e ω / BT ω (e ω / BT ) B T ω dω /T e z z 4 (e z ) dz = ω B # (K) # C Ge 7 Si 64 Au 65 Al 8 Fe 467 Cs 8 T 9N A B e z + z /T T (+ z)z dz 9N A B lic V = 9N A B T T T + 4 T e z z 4 (e z ) dz 5 4 T N A B 4 R T

16 T ω T = ω( T ) n = N A T D T = N A ω T = T c = B T ω D = D c = B T U = n B T = N A B T D C V = U T = n B T = N A B T T J = κ T κ = Cv λ κ : T t = κ ρc P T x ρ : C p : ΔT = T x l x = T x v xτ T ( x) x n v x x + l x n v x T ( x + Δx) ΔT J = cδt nv x nv x = cnv xτ T x = Cv xτ T x

17 J = C v x τ T x = C v τ T x = T T Cvl κ x x κ = Cvl l < : l κ = Cv (l > ) ω + ω = ω q + q = q q + q = q + G T q q q q exp T q, B q q G q + q = q q + q = q + G

18 Glssbrenner, C. J. nd G. A. Slc, Phys. Rev. 4, 4A (964) A58-A69. h;p:// U(x) = U() +U () () x! +U() () x! +U(4) () x4 4! U () () =!c >, U () () =!g <, U (4) () = 4! f < x = xe U (x)/ BT dx e U (x)/ BT dx = xe βu (x) dx e βu (x) dx xe βu (x) dx (x + βgx 4 + β fx 5 )e βcx dx = 4 g c e βu (x) dx e βcx dx = / / β c x = g 4c β = g 4c B T 5/ β /

19 h;p://science4.co/pper/4579 Z = n= ( ) e ω B T n+ F T = U ( x) B T log Z = U ( x) + ω + BT ln( e ω BT ) F = B T log Z Od, Y. nd Y. Touru,! J. Appl. Phys. 56, (984) 4- F = F ( x ) + ( x x ) F x x=x U ( x) = U ( x ) + ( x x ) p = = F V ( x x ) = ω T ω x E( ω,t ) α = x dx dt = x α V = V dv dt = βv γ = lnω ( q, j) lnv lnω ln x q, j ( ) T E ω,t ( ) lnω ( q, j) lnv T E ω ( q, j),t β = V ( ) dp dv Si

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

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