Solid state physics - v karpus

Solid State Physics • Kietųjų kūnų fizika Физика твердого тела • Festkörperphysik Vytas Karpus (PFI: 364 k.; tel.: 2619475 (of.), 2313217 (h.); karpus@pfi.lt) http://www.pfi.lt/studies/doctor_l.html • C Kittel, Introduction to Solid State Physics (John Willey & Sons, 1996) Ч Киттель, Введение в физику твердого тела (Москва: Наука, 1978) • N W Ashcroft and N D Mermin, Solid State Physics, (Holt, Rinehart, and Winston, 1976) Н Ашкрофт, Н Мермин, Физика твердого тела (Москва: Мир, 1979) • J M Ziman, Principles of the Theory of Solids (Cambridge University Press, 1972) Дж Займан, Принципы теории твердого тела (Москва: Мир, 1974) • А С Давыдов, Теория твердого тела (Москва: Наука, 1976) • В Л Бонч-Бруевич, С Г Калашников, Физика полупроводников (Москва: Наука, 1977) • A Juodviršis, M Mikalkevičius, S Vengris, Puslaidininkių fizikos pagrindai (Vilnius: Mokslas, 1985) • A Matulis, Kietojo kūno fizika (Vilnius: Petro ofsetas, 2000) • V Karpus, Kietųjų kūnų fizika Kristalinis būvis (Vilnius: Ciklonas, 2002) Physics: Lecture (2004) V Karpus, Solid State Crystal structures The crystalline state = the equilibrium state of solids Experimental fact or theorem? Crystals: • long-range order • periodicity Investigations of symmetry of crystals (mineralogy) Crystallography: Symmetry: • point symmetry • translation symmetry • 17–18th c Steno (1638–1686) Hẵy (1743–1822) Hessel, Bravais, Schưnflies, • 19th c Fedorov (Е С Федоров) classification of crystals V Karpus, Solid State Physics: Lecture (2004) Translation symmetry atoms & equivalent points primitive cell a2 Wigner–Seitz cell a1 crystal structure/lattice Bravais lattice lattice vectors l = n1a1 + n2a + n3a , n i = , ±1, ±2 , K Vcell = (a1[a 2a ]) vectors of primitive translations translation operator ˆ Tl f (r ) = f (r + l ) atomic positions ˆ Tl = exp(l∇) R (n0 ) = l n + ρα α V Karpus, Solid State Physics: Lecture (2004) Oh Point symmetry Schönflies O Herman-Mauguin mirror plane σ n m rotation-reflection axis Sn Cn i n inversion centre rotation axis rotation-inversion axis i Cn Th Td T Oh , Th Oh , T h O, Oh T, Td , Th Td O h , Td O h , Td V Karpus, Solid State Physics: Lecture (2004) Point groups A set of point symmetry operations of a body constitutes its point group Group A set of elements A, B, C, is called a group G if a law of their “multiplication” is defined, and the following conditions are satisfied: Examples of point groups: • Cn = rotations of Cn-axis • O = all rotations of a cube • if A and B ∈ G, then AB ∈ G • Oh = Ci ì O ã multiplication is associative, (AB)C = A(BC) • T = all rotations of a tetrahedron • the set contains an element E such that AE = EA = A ã Th = Ci ì T ã for all A ∈ G there exist B such that AB = BA = E • Td (4 C3 , S4 , and σ) • Cnh (C1h ≡ Cs ) , Cnv , Dn (D2 ≡ V ) V Karpus, Solid State Physics: Lecture (2004) Classification of crystals System Parameters of crystallographic cell Lattice type Point group Triclinic (T) a≠b≠c α≠β≠γ P C1 , Ci Monoclinic (M) a≠b≠c α = γ = ½π ≠ β P, C C2 , Cs , C2h Orthorhombic (O) a≠b≠c α = β = γ = ½π P, C, I, F D2 , C2v , D2h Tetragonal (Q) a=b≠c α = β = γ = ½π P, I C4 , S4 , C4h , D4 , C4v , D2d , D4h Trigonal a=b=c α = β = γ ≠ ½π R C3 , C3i , D3 , C3v , D3d Hexagonal (H) a=b≠c α = β = ½π, γ = 2π/3 P C6 , C3h , C6h , D6 , C6v , D3h , D6h Cubic (K) a=b=c α = β = γ = ½π P, I, F T , Th , Td , O , O h or rhombohedral (R) Hessel 1830 Bravais 1848 Schönflies and Fedorov 1890-1894 V Karpus, Solid State Physics: Lecture (2004) Miller indices wc c b a ua vb (hkl) [130] (110) [hkl] [[uvw]] c (110) 1 : : = h:k :l u v w [110] b 31 22 a b a (510) u :v: w = h:k :l V Karpus, Solid State Physics: Lecture (2004) Examples of crystal structures: Ionic crystals Na Cs - Cl + Cl Bravais lattice: fcc σ= (σ = 8) coordination number = a = 5.63 Å KCl, AgBr, MgO, MnO + - Bravais lattice: sc σ= coordination number = a = 4.11 Å CsBr, CsI, β´-CuZn V Karpus, Solid State Physics: Lecture (2004) Examples of crystal structures: Metallic crystals bcc c hcp 8 fcc b a c Bravais lattice: bcc σ= (σ = 2) coordination number = aNa = 4.23 Å Li, Na, K, Rb, Cs a 7 a1 = (− a + b + c ) a = (a − b + c ) a = (a + b − c ) 6 b Bravais lattice: hexagonal σ= (σ = 6) coordination number = 12 c/a = 1.633 Mg, Cd, Zn Bravais lattice: fcc σ= (σ = 8) coordination number = 12 Al, Cu, Ag, Ne a1 = (b + c ) a = (c + a ) a = (a + b ) V Karpus, Solid State Physics: Lecture (2004) Examples of crystal structures: Covalent crystals diamond structure Bravais lattice: fcc σ= (σ = 8) coordination number = aC = 3.56 Å Si, Ge, α-Sn zinc blende structure Bravais lattice: fcc σ= (σ = 8) coordination number = aZnS = 5.41 Å GaAs, AlAs, ΙnP, CdTe V Karpus, Solid State Physics: Lecture (2004) [A Mooradian, in: Laser Handbook (North Holland Publ., 1972)] Raman scattering C.V Raman, Indian J Phys 387 (1928) C.V Raman and K.S Krishnan, Nature 121 501 (1928) G Landsberg and L Mandelstam, Naturwiss 16 557 (1928) (1930 Nobel prize) ∆Ω = mω N qλ anti − Stokes = Stokes + N qλ = exp(−hω / k BT ) hω T = 33.7 meV hω L = 36.7 meV exp hω p = 16 meV theor hω p = 16.8 meV (n = 1.75 1017 cm −3 ) V Karpus, Solid State Physics: Lecture 16 (2005) Brillouin scattering A Hassine et al., J Appl Phys 77 6569 (1995) [L Brillouin, Ann Phys (Paris) 17 88 (1922)] Ω = Ω'±ω qλ k = k '±q ˆ ω qλ = sλ (q) q ∆Ω = m2Ω ˆ sλ (q) sin(ϑ / 2) c / nr The frequency shift depends • on the frequency of the incident light • on the scattering angle V Karpus, Solid State Physics: Lecture 16 (2005) Bragg diffraction 2Λ sin α = mλ Acousto-optics Relation with the Brillouin scattering m = 0,±1,±2, K k '−k = mq 2k sin α = mq Frequency shift ~ ~ Ω´= Ω comoving reference frame The Doppler shift: The first experimental observations: • P Debye and F W Sears, Proc Nat Acad Sci (USA) 18 409 (1932) • R Lucas and P Biquard, J Phys Rad 464 (1932) velocity of the frame ~ Ω = Ω + Qs lab frame wavevector comoving frame Ω´= Ω + mω qλ V Karpus, Solid State Physics: Lecture 16 (2005) Raman – Nath diffraction Λ (sin α − sin α i ) = mλ m = 0,±1,±2, K C.V Raman and N.C Nagendra Nath, Proc Indian Acad Sci 406, 413 (1935) Proc Indian Acad Sci 75, 119, 459 (1936) Schaeffer – Bergmann experiment (1938) V Karpus, Solid State Physics: Lecture 16 (2005) Resumé Neutron spectroscopy NB Umklapp and normal processes • Raman scattering: Combination scattering Raman scattering Brillouin scattering Ω' = Ω m ω Acousto-optics Bragg diffraction Raman – Nath diffraction V Karpus, Solid State Physics: Lecture 16 (2005) Anharmonic effects Thermal expansion Phonon-phonon interaction • phonon liftime • thermal conductivity V Karpus, Solid State Physics: Lecture 17 (2005) Thermal expansion ∂L α= L ∂T linear expansivity ∂V β= V ∂T va ( R) = va ( R ( 0) ) + ΦU − Φ 3U Φ k BT U = 2Φ α=1β α = const T > ΘD Grüneisen theory (quasi-harmonic approximation) [E Grüneisen, Ann Physik 26 393 (1908); Handb Phys 10 (1926)] F = F (V , T ) = const + ⎛ ∆V ⎞ BV ⎜ ⎟ + Fph ⎝ V ⎠ ω qλ ∆V = −γ V Grüneisen parameter ∆ω qλ the free energy of the static lattice ω qλ hω qλ ⎞ ⎛ ⎟ Fph = k BT ∑ ln⎜ 2sh ⎜ 2k BT ⎟ q ,λ ⎝ ⎠ α = γ K cv ∆ω qλ = −γ qλ γ= K =− ⎛ ∂V ⎜ V ⎜ ∂p ⎝ ⎞ ⎟ ⎟ ⎠T cv cv,qλ = ∆V V ∑γ λ c qλ v,qλ q, ∂N qλ hω qλ V ∂T V Karpus, Solid State Physics: Lecture 17 (2005) Thermal expansion (contraction) of solids α = γ K cv guitar string effect [G.K White, Contemp Phys 34 193 (1993)] V Karpus, Solid State Physics: Lecture 17 (2005) Phonon-phonon interaction hω q1λ1 = hω q 2λ2 + hω q3λ3 q1 = q + q hω q1λ1 + hω q 2λ2 = hω q3λ3 q1 + q = q Selection rules (Peierls 1955): no solutions for T1 + T1 → T2 T1 + T1 → T1 T1 + L → T2 L + T→ L T1 + L → L T1 + T1 → L T2 + T2 → T2 T1 + T2→ T L+L →L 1,2 T2 + L → L V Karpus, Solid State Physics: Lecture 17 (2005) Thermal conductivity Fourier law jQ = −κ ∇T thermal current density thermal conductivity [κ ] = W cm K cal W = 4.186 s cm K cm K Phonon contribution jQ = Electron contribution ph dT dΩ v x uT ( x ) = − cv s 2τ ph dx 4π ∫ ph κ ph = cv s 2τ ph el κ el = cv v 2τ el V Karpus, Solid State Physics: Lecture 17 (2005) Thermal conductivity: Metals Fermi Wiedemann – Franz law (1853) κ el mv c = σ 3e n el v = π kB T 3e 3k B T = 2e εF mv el cv Boltzmann π2 nk B k BT / ε F ⎧2.44 10 −8 WΩ / K ⎪ Lorenz number: ⎨1.11 10 −8 WΩ / K ⎪2.22 10 −8 WΩ / K Boltzmann ⎩ Fermi k BT nk B Fermi Boltzmann Drude mistake experiment Ag Al κ/σT (10-8 WΩ/K2) el κ el cv v 2τ el k T = ph ~ B >> κ ph cv s τ ph ms Li 2.22 2.31 2.14 ⎞ m⎛ s ⎜ ⎟ ms ≈ 1.6 K × m0 ⎜ 10 cm/s ⎟ ⎝ ⎠ κexperiment (W cm-1K-1) ph κ ph = cv Li2τ ph Cu s 2 Au Ag Al T = 373 K 0.73 3.82 3.1 4.17 2.3 T = 273 K 0.71 3.85 3.1 4.18 2.38 el κ el =κ1 ≠ κ v 2τ elin metals cv (T ) el cv ∝ T τ el ∝ T −1 V Karpus, Solid State Physics: Lecture 17 (2005) Phonon-phonon scattering Thermal conductivity Dielectrics ph v ? Vanh(4) τ ph = τ ph (T ) q1 = q + q Peierls law only the Umklapp processes contribute to the thermal resistivity -1 τ ph = exp(−Θ D / 3.5 T ) [H.E Jackson and C.T Walker, PRB 1428 (1971)] κ = c s τ ph Vanh(3) High temperatures, T >> ΘD τ ph ⎧ N ∝T ∝⎨ N ∝T2 ⎩ κ ∝ 1/ T n anh (3) anh (4) n = 1− Low temperatures, T lph -ph (N) / τ ph