CMP Technical Report No. 4 Department of Computational Nanomaterials Design ISIR, Osaka University

2 2................................. 2.2......................... 2 3 3 3................................ 3 3.2............................... 4 3.3.................................. 6 3.4............................... 8 4

MDS MDS u i (t) MDS Si 64 2 2. u i (t) u i (t) q u i (t) = q u i (q)e iω qt () u i (t) u i (q) u i (q) q u ( q) 2 = i u i (q) u i (q) (2) ū 2 ū 2 g(ω) ω ω + ω { 2 Mω2 ū(ω) 2 = n(ω) + } ωg(ω) (3) 2 3 n(ω) ū(ω) 3 ω

2.2 3 u i (t)u j () (4) J ij (ω) = u i (t)u j () e iωt dt (5) G(u i (t), u j ()) = i T [u i (t), u j ()] (6) E vib = D ij u i u j (7) 2 i,j { } E vib = 2 lim I D i,j G(u i (t), u j ()) (8) t + i,j 8 6 G(i, j; ω) ω G(ω) E vib = dω ω 2 I {G(ω)} (9) 2 π I {G(ω)} = {(n(ω) + ) + n(ω)} g(ω) () 2ω 9 3 4 5 G R (ω) J ij (ω) = 2[n(ω) + ]I { G R (ω) } () 2

ω k 3 Si 64 ==================== K-Space Setup ============================== k sampling point set Nkpts = No NM index in p in c A/gmin i/o star WTK GM / 2 / 2.. sum of wtk =. ==================== PW_Expansion ============================== Cutoff in the reciprocal space am =.74 (rel. units) kcut = 3.28776 (ab^-) with 2Pi Ecut =.893 (Ry) UNIT of K.362 (a.u.) Planewave expansion with NHDIM = 585 Name: GM / 2 Nstr= WTK=. NPW= 585 INV= Sum over WTK. 3. MDS MDS parameters ============================ time step :. (Rysec) = 4.8378 (fsec) max iterations : 25 Simulation time :.29 (psec) mds resume : NO way to give v : random initial T_atom : 2. (K) =.76 (Ry) Thermo control : OFF T at () = 2 [K] (2) T 6 K T span =.29 [ps] 3

MDS noctrl dt = 25 Etot.4.2 -.2-7.2-7.4-7.6-7.8-8 -8.2 Log E -.4 2 4 6 8-8.4.5.5 Epot.4.3.4.3 Ekin.2 2 4 6 8 ITER.2 : Time evolution of energies of Si 64. t=25 Rysec. ν = 27.28 [cm ] (3) t t =25 Rysec 2 t = Rysec 3 3.2 MDS T span t = 25 Rysec f = [/Rysec] 25 α/2 = 25 = [a B 25 274.74 ] = 27.28 [cm ] (4) 4

MDS noctrl dt = -6.4 Etot -. -6.6-6.8-7 Log E -7.2 -.2-7.4 5 5 2 25-55..5 Epot -55.2-55.3.4.3 Ekin -55.4.2 5 5 2 25 ITER 2: Time evolution of energies of Si 64. t= Rysec..2 (ps). u x -. -.2 5 5 2 25 STEP 3: Displacement of atom, u x in relative units. The black line shows data obained by setting t= Rysec, while the red line 25 Rysec. 5

<u(ω) - 2 > [a.u.^2] x -3 5 4 3 2 Frequency t=25 5 5 2 25 t= - ( x 27.28 [cm ]) 4: Fourier spectra of atomic displacements. The black line shows data of t=25, while the red line of t= Rysec. 4 27.28 cm 4 2 u i (ω) 4 MDS t 3.3 g(ω) 3 g(ω) T T at (t) T = 522 [K] (5) T span 2.4 ps ū 2 (ω) 5 6

<u(ω) - 2 > [a.u.^2] x -3 5 4 3 2 T=522 K 274 2 3 4 5 - Frequency ( x 3.64 [cm ]) 5: Power spectra at different temperatures. [a.u.]. T=522 K 274 g(ω) 2 3 4 5 6 ω 6: Phonon density of states g(ω) [cm -] 7

3 /ω g(ω) 6 g(ω) 52 cm TO 2 cm A 5 ū 2 g(ω) g(ω) 3.4 k i i i = lκα l κ α Si κ =, 2 u κα (kω) = l exp[ik R l ]u lκα (ω) (6) κ S α (kω) = κ u κα(kω)u κα (kω) (7) T = 522 K 2.4 ps 8

<u(ω) - 2 > [a.u.^2] x -4 k=(,,) (/2,,) (,,) 2 3 4 5 - Frequency ( x 3.64 [cm ]) 7: k dependent phonon peaks. Shown is the longitudinal branch along the line. k is expressed in 2π/a. x -4 <u(ω) - 2 > [a.u.^2] (,,) k=(,,) (/2,,) 2 3 4 5 - Frequency ( x 3.64 [cm ]) 8: The transverse branch along the line. k is expressed in 2π/a. 9

7 k = (,, ) x L X (u x (ω), u 2x (ω)) Γ x 39 {-.553, -.66386} {-.683262 +.5988 i,.685896 -.4392 i} LA LO X x 3 {.462358+.246 i, -.23872 +.29449 i} 8 k = (,, ) z T 7 8 k = (,, ) k /2 (2π/a ) 4 ω k