Solid State Physics
Full text
Solid State Physics: Part 1 Structure, symmetry, mechanical and thermal properties Kamil Postava and Tibor Fördös
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Solid State Physis: Part 1 Struture, symmetry, mehanial and thermal properties Fyzika p evnýh látek: 1. £ást Struktura, symetrie, mehaniké a termální vlastnosti Kamil Postava and Tib or Fördös kamil.postavavsb.z, tibor.fordosvsb.z Faulty of Materials Siene and Tehnology, VB Tehniká univerzita Ostrava (tel. 4545, 5170) Novemb er 1, 2024 1 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Summary Part 1: Struture, symmetry, mehanial and thermal prop erties of solids 1 Intro dution, rystal symmetry Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution 2 Bonding in solids and mehanial prop erties Chemial b onding in solids Mehanial prop erties 3 Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion 2 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Summary Part 2: Eletroni prop erties of solids and appliations 4 Eletroni p erop erties of solids Condutivity, eletrial transp ort, and eets on interfaes Energy bands and eletrons in rystal Miro eletroni appliations 5 Optial prop erties of solids Free eletrons Drude theory Lo alized eletrons Lorentz damp ed harmoni osillator 6 Magneti prop erties of solids Intro dution to magnetism Magneti moment, magnetism in solids Magneti domains Appliations of magnetism 3 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Literature 1 Ch. Kittel, Intro dution to Solid State Physis, John Wiley & Sons, 8th ed., 2022. 2 P. Hofmann, Solid State Physis: An Intro dution, 2nd ed., Wiley-VCH, 2015. 3 I. Kraus, J. Fiala, Elemntární fyzika p evnýh látek, VUT Praha 2016. 4 R. E. Hummel, Eletroni Prop erties of Materials, Springer-Verlag, 2nd ed., 1993. 5 C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mehanis (Méhanique quantique) Wiley, 1991. 6 M. A. Wahab, Solid state physis: Struture and properties of materials, Narosa, 3rd ed., 2015. 7 H. Ibah and H. Luth, Solid-State Physis: An Intro dution to Priiples and Material Sienes, Springer, 4th ed., 2009. 8 S. H. Simon, The Oxford Solid State Basis (Oxford University Press: Oxford, 2013). 9 N. W. Ashroft and N. David Mermin, Solid State Physis (Harourt: Orlando, 1976). 10 J. F. Nee, Physial properties of rystals, Oxford Clarendon Press 1992. 4 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Intro dution to Solid state physis Subjet of Solid state physis: solid (rigid) state of matter relate atomi (mirosopi) struture with marosopi physial prop erties (mehanial, thermal, eletrial, optial, magneti) theoretial basis of material siene wide range of appliations mehanial engineering, material siene, eletronis, photonis, magnetism and spintronis 5 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Crystalline and amorphous solids 6 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Material prop erties of rystalline and amorphous solids Crystalline solids higher density, hardness, sharp eletron band struture, b etter optial prop erties, anisotropy p ossible theoretial mo dels (band struture theory, ab-initio, rst priniples quantum mehanial mo dels) Amorphous and rystalline Silion optial prop erties: 0 1 2 3 4 5 6 7 8 −20 −10 0 10 20 30 40 50 Photon energy (eV) Dielectric functions: a−Si, c−Si ε1 ε2 7 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Crystal lattie Bravais lattie p erio di p ositions of the enters of atoms (bases) r=n1a1+n2a2+n3a2 , lattie onstants |a1| , |a2| , |a3| Primitive unit ell volume of spae, ontains only one lattie p oint, translated ll the whole spae Wigner-Seitz ell spae loser to one lattie p oints 8 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Cubi b , f, hexagonal hp 15 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Miller indexes of rystal planes (hkl) desription of planes {hkl} set of planes equivalent to (hkl) [hkl] denotes diretions in the lattie basis 16 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution X-ray diration determination of rystal struture X-rays (Ro entgen rays) high energy eletromagneti radiation (wavelength λ omparable with lattie onstant) Bragg ondition X-ray diration 2dsin θ=n λ, where d is distane b etween atomi planes, θ is the angle of inidene and diration (measured from surfae) and n is integer. 17 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution X-ray diration X-rays diration instrumentation single-rystal diration three-axis goniometer to adjust angles or p olyhromati soure p owder randomly oriented grains, single angle-of-inidene san 18 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Laue desription of X-ray diration in rystal Sattering of X-rays by eletrons (lose to atoms) Desription by plane waves (soure and detetor are far from sample) r p osition of satterer k,k′ wavevetors of soure and detetor |k|=|k′|= 2π/λ , λ is wavelength Eletri eld from satterer (from volume dV= d2r ) E(R′)∝e−ik′rρ(r) eikr E(R), where ρ(r) relates to eletron density (sattering probability). Result eletri eld from whole sample (interferene inluded) and dirated intensity: E(K) = Zρ(r) e−iKrd2r, I(K) = Zρ(r) e−iKrd2r 2 , where K=k′−k is the sattering vetor. 19 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Laue desription of X-ray diration in rystal Sattering of X-rays by eletrons (lose to atoms) Desription by plane waves (soure and detetor are far from sample) r p osition of satterer k,k′ wavevetors of soure and detetor |k|=|k′|= 2π/λ , λ is wavelength Eletri eld from satterer (from volume dV= d2r ) E(R′)∝e−ik′rρ(r) eikr E(R), where ρ(r) relates to eletron density (sattering probability). Result eletri eld from whole sample (interferene inluded) and dirated intensity: E(K) = Zρ(r) e−iKrd2r, I(K) = Zρ(r) e−iKrd2r 2 , where K=k′−k is the sattering vetor. 19 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Laue desription of X-ray diration in rystal Sattering of X-rays by eletrons (lose to atoms) Desription by plane waves (soure and detetor are far from sample) r p osition of satterer k,k′ wavevetors of soure and detetor |k|=|k′|= 2π/λ , λ is wavelength Eletri eld from satterer (from volume dV= d2r ) E(R′)∝e−ik′rρ(r) eikr E(R), where ρ(r) relates to eletron density (sattering probability). Result eletri eld from whole sample (interferene inluded) and dirated intensity: E(K) = Zρ(r) e−iKrd2r, I(K) = Zρ(r) e−iKrd2r 2 , where K=k′−k is the sattering vetor. 19 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Reipro al spae, reipro al lattie dirated eld relates to Fourier transform of the lattie Reipro al latie reipro al lattie vetor K=kb1+lb2+lb3 relates with the real lattie R=n1a1+n2a2+n3a3 by the ondition: eiR·K= 1 or R·K= 2π m, m is integer. Relation b etween lattie vetors: ai·bj= 2πδij , b1= 2πa2×a3 a1·(a2×a3),b2= 2πa3×a1 a1·(a2×a3),b3= 2πa1×a2 a1·(a2×a3). 20 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Real and reipro al spae Real lattie Reipro al lattie Lattie p erio diity basis R=n1a1+n2a2+n3a3 p ositions of atoms (m,Å) K=kb1+lb2+lb3 wavevetors (m −1 ) eiR·K= 1 or R·K= 2π m paralel planes desrib ed by Miller indexes (klm) p oints in reipro al spae [k, l, m] Exp erimental observation mirosopy (eletron, STM) diration (X-ray, eletron, neutron) Laue ondition k′−k=K p oint in reipro al spae (generalized Bragg ondition) Elementary primitive Bravais ell Wigner-Seitz ell Brillouin zone 21 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Basi rystal symmetries Real lattie Reipro al lattie simple ubi (s) simple ubi (s) b o dy-entered ubi (b ) fae-entered ubi (f) fae-entered ubi (f) b o dy-entered ubi (b ) hexagonal hexagonal 22 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Eletron diration wave harater of moving eletrons Low energy eletron diration (LEED) Davisson, Germer, 1925 29 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Wave partile duality for photons in Young exp eriment Interferene of single photons on double-slit z θ d x a xx 30 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Wave-partile duality Eletromagneti waves Photons Partile waves eletrons Wave partile duality Relation b etween Partile energy and wave frequeny E=hf =~ω, where h= 6.626 10−34 J s is the Plak onstant and ~=h 2π Relation b etween Partile momentum and wavelength (wave vetor) p=h λp=~k The realation for momentum an b e derived for photon using the equations E=mc2 , p=mc , f=c/λ . 31 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Eletron mirosopy Resolution of optial mirosop e: y=0.61 λ An , where An=nsin σ is the numerial ap erture. 32 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Eletron mirosopy Resolution of optial mirosop e: y=0.61 λ An , where An=nsin σ is the numerial ap erture. 33 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Eletron lithography Same limits for resolution of optial observation (mirosopy) and writing (lithography). 34 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Wavelength of eletron Velo ity of eletron by voltage U (valid for v≪c ): e U =1 2mv2−→ v=r2eU m Wavelength of eletron aelerated by voltage U: p=h λ−→ λ=h p=h √2meU For example for U= 50 kV, m= 9.11 ·10−31 kg, e= 1.602 ·10−19 C, λ= 5.5·10−12 m. 35 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Wavelength of eletron (relativisti approah) Relativisti equation for energy: E=m c2=m0c2 q1−v2 c2 , p =mv =m0v q1−v2 c2 , E2= (m0c2)2+(pc)2. Total energy of eletron E=m0c2+eU Therefore 2m0c2eU +e2U2=p2c2 and p=h/λ Wavelength of eletron (relativisti approah): λ=h √2m0eU 1 q1 + eU 2m0e2 36 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Wavelength of eletron (relativisti approah) Relativisti equation for energy: E=m c2=m0c2 q1−v2 c2 , p =mv =m0v q1−v2 c2 , E2= (m0c2)2+(pc)2. Total energy of eletron E=m0c2+eU Therefore 2m0c2eU +e2U2=p2c2 and p=h/λ Wavelength of eletron (relativisti approah): λ=h √2m0eU 1 q1 + eU 2m0e2 36 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Wave funtion of a partile wave funtion ψ(r, t) desribing quantum state of a partile probability of nding a partile at the lo ation r |ψ(r, t)|2=ψ(r, t)ψ∗(r, t)Rψ(r, t)ψ∗(r, t)d3r= 1 time dep endene of wave funtion Time-dep endent Shrö dinger equation i~∂ ∂t ψ(r, t) = −~2 2m∆ψ(r, t) + V(r, t)ψ(r, t) oresp ondene E=p2 2m+V E→i~∂ ∂t p→~ i∇ 37 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Eletron in retangular p otential stationary Shrö dinger equation: ∆ϕ(~r) + 2m ~2(E−V)ϕ(~r) = 0 solution of Shrö dinger equation in one dimension d2ϕ(x) dx2+2m ~2(E−V)ϕ(x) = 0 E > V : E−V=~2k2 2m , ϕ(x) = Aeikx +A′e−ikx (harmoni) E < V : V−E=~2ρ2 2m , ϕ(x) = Beρx +B′e−ρx (exp onenial) b oundary onditions: ϕ(x) , dϕ(x) dx ontinuous 43 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Examples: Quantum p otenial well Potenial well: simple approximation of eletron in eletrostati p otenial eld of atomi ore eletron states in thin lm −→ semiondutor quantum well laser 44 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Innite quantum well Solution inside well ( E > V ): ϕ(x) = Aeikx+Be−ikx,2mE ~2=k2 Boundary onditions: ϕ(0) = ϕ(L) = 0 Condition of eletron existene R|ϕ(x)|2dx= 1 Solution: A+B= 0 , AeikL +Be−ikL = 0 B=−A , ei2kL = 1 45 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Innite quantum well Solution inside well ( E > V ): ϕ(x) = Aeikx+Be−ikx,2mE ~2=k2 Boundary onditions: ϕ(0) = ϕ(L) = 0 Condition of eletron existene R|ϕ(x)|2dx= 1 Solution: A+B= 0 , AeikL +Be−ikL = 0 B=−A , ei2kL = 1 45 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Innite quantum well Solution inside well ( E > V ): ϕ(x) = Aeikx+Be−ikx,2mE ~2=k2 Boundary onditions: ϕ(0) = ϕ(L) = 0 Condition of eletron existene R|ϕ(x)|2dx= 1 Solution: A+B= 0 , AeikL +Be−ikL = 0 B=−A , ei2kL = 1 45 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Innite quantum well Energy levels are disrete E=~2π2 2mL2n2 , where n is thequantum numb er Wave funtions ϕ(x) = q2 Lsin kx =q2 Lsin nπ Lx. 46 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Innite quantum well qualitatively similar solution as eletron in Hydrogen atom eletrostati Coulomb p otenial V(r) = −e2 4πε0r solution of Shrö dinger equation in spherial o ordinates disrete (quantized) energy: En=−m0e4 32π2ε2 0~2 1 n2=−13.61 n2(eV ) 47 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Finite quantum well Solution in three regions separately: I.(E < V =V0)ϕ1(x) = B1eρx,2m(V0−E) ~2=ρ2 II.(E > V = 0) ϕ2(x) = Aeikx +A′e−ikx,2mE ~2=k2 III.(E < V =V0)ϕ3(x) = B3e−ρx. Boundary onditions: ϕ1(0) = ϕ2(0) , ϕ2(L) = ϕ3(L) , dϕ1(x) dx=dϕ2(x) dx x=0 , dϕ2(x) dx=dϕ3(x) dx x=L 48 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Finite quantum well Solution in three regions separately: I.(E < V =V0)ϕ1(x) = B1eρx,2m(V0−E) ~2=ρ2 II.(E > V = 0) ϕ2(x) = Aeikx +A′e−ikx,2mE ~2=k2 III.(E < V =V0)ϕ3(x) = B3e−ρx. Boundary onditions: ϕ1(0) = ϕ2(0) , ϕ2(L) = ϕ3(L) , dϕ1(x) dx=dϕ2(x) dx x=0 , dϕ2(x) dx=dϕ3(x) dx x=L 48 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Finite quantum well 49 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Tunnel eet 55 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Tunnel eet 56 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Tunnel eet for E < V0 : T= C A2 2 =T0e−2 ~√2m(V0−E)L for E > V0 : R= A1 A2 2 = 1−p1−V0/E 1 + p1−V0/E !2 57 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Sannig tunelling mirosopy (STM) advaned version of Atomi fore mirosopy (AFM) surfae observation at atomi sale 58 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Perio di p otenial in rystal (one dimensional ase) Mo del of eletrons in solid-state rystal: eletrostati Coulomb p otenial V(r) = −Z e2 4πε0r p erio di system of atoms in rystal mo del in one dimension (explains fundamental phenomena) V(x) approximized by retangular p otential 59 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Kronig-Penney mo del Approximation of p otential eld in rystal using p erio di array of quantum wells: Potenial energy desib ed by (I.) p otenial well a separated by (I I.) p otenial barrier b . Lattie onstant =a+b 60 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Kronig-Penney mo del Solution of Shrödinger equation in two regions: I. p otential wells a , V= 0 d2ϕ(x) dx2+α2ϕ(x) = 0 α2=2m ~2E I I. p otential barriers a , V=V0 d2ϕ(x) dx2−γ2ϕ(x) = 0 γ2=2m ~2(V0−E) solution in p erio di p otential is p erio di Blo h theorem ϕ(x) = u(x) eiKx, where u(x) = u(x+a+b) Blo h funtion (p erio di with the p erio diity of the lattie) and K is the Bloh wavevetor. 61 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Kronig-Penney mo del Substituing ϕ(x) = u(x) eiKx into dierential equations: I.d2u1 dx2+ 2iK du1 dx−(K2−α2)u1= 0, II.d2u2 dx2+ 2iK du2 dx−(K2+γ2)u2= 0. Bondary onditions: u1(0) = u2(0) , u1(a) = u2(−b) , du1(x) dx=du2(x) dxx=0 , du1(x) dxx=a=du2(x) dxx=−b Solution: I. u1= e−iKx (Aeiαx +Be−iαx), II. u2= e−iKx (Ce−γx +Deγx). 62 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution Kronig-Penney mo del Substituing ϕ(x) = u(x) eiKx into dierential equations: I.d2u1 dx2+ 2iK du1 dx−(K2−α2)u1= 0, II.d2u2 dx2+ 2iK du2 dx−(K2+γ2)u2= 0. Bondary onditions: u1(0) = u2(0) , u1(a) = u2(−b) , du1(x) dx=du2(x) dxx=0 , du1(x) dxx=a=du2(x) dxx=−b Solution: I. u1= e−iKx (Aeiαx +Be−iαx), II. u2= e−iKx (Ce−γx +Deγx). 62 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Subjet of solid state physis Crystal lattie and symmetry Intro dution to quantum mehanis, wavepartile duality Shrö dinger equation, examples of solution *Kronig-Penney mo del Disp ersion relation for eletron in p erio di rystal set of 4 b oundary ondition has nontrivial solution A+B=C+D (−iK +iα)A+ (−iK −iα)B= (−iK −γ)C+ (−iK +γ)D e(−iK+iα)aA+ e(−iK−iα)aB= e−(−iK−γ)bC+ e−(−iK+γ)bD (−iK +iα) e(−iK+iα)aA+ (−iK −iα) e(−iK−iα)aB= = (−iK −γ) e−(−iK−γ)bC+ (−iK +γ) e−(−iK+γ)bD 63 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Attrative and repulsive fores Bonding in solids realized usualy using valene eletrons Interatomi p otenial φ(r) = A rn−B rm, n > m attrative fore repulsive fore (Pauli exlusion priniple) Lennard-Jones p otenial n= 12 , m= 6 70 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Various mehanism of b ounding in solids Dierent typ es of b onding in solds have dierent origin dep end on band struture and walene eletrons inuene physial prop erties Cohesive energy = Energy of isolated atoms Energy of solid ioni Coulomb attration b etween ions, stron ovalent sharing eletrons b etween dierent atoms, strong metalli valene eletrons = ondution eletrons Hydrogen eletrostati fore through hydrogen van der Waals quantum utuation often mixed b onding 71 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Ioni b onding Coulomb attration b etween eletrop ositive and eletronegative atoms E=−Md e2 4πε0a, where Md is Madelung onstant (dep end on rystal struture) Alali metal + Halogenid: NaCl, LiF, KBr, et. 72 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Covalent b onding sharing of eletrons (with antiparalel spins), b etween dierent atoms b etween atoms with half-lled outer shells semiondutors, high εr , high refrative index H 2 , N 2 , Silion, Ge, Diamond, gephene, C 60 , GaAs, SiC 73 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Metalli b onding valene eletrons are weekly b ounded to atoms onduting eletrons high eletrial and thermal ondutivity, high reetivity metals 74 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Hydrogen b onding and Van der Waals b onding Hydrogen b ond with very eletronegative atom (F, O) itramoleular interation H 2 O, DNA, rystal saharides Van der Waals b onding purely quantum mehanial origin quantum utuations, dip ole moment graphite (b etween graphite planes), ombination with other b onds (water ie) 75 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Mehanial prop erties of solids Eet of mehanial stress σ=F S (N.m −2= Pa) where F is the fore and S is area solids resp onse by deformation strain. ε=∆l l Desription (piture): marosopi (phenomenologial) desrib es resp onse (b ehaviour) of solids on stess. mirosopi based on interatomi fores (b onds) 76 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Mehanial prop erties of solids exp erimental tehniques Measurement of mehanial prop erties of solids 77 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Mehanial stress and strain in solids 1 elasti deformation reversible, linear relation b etween stress and strain 2 plasti deformation ireversible 3 frature 78 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Elasti deformation marosopi piture Ho oke's law tensile ompressive stress σ=E ε, where E is the Young's mo dulus, σ=F S , ε=∆l l . shear stress τ=G α, where G is the mo dulus of rigidity, τ=F S , α=x l . isotropi (hydrostati) pressure p=−K∆V V where K is the bulk modulus Poisson ratio ν : ∆l2 l2=∆l3 l3=−ν∆l1 l1 79 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Ho ok's law tensor of elasti stiness o eients Tensor of elasti stiness o eients Cijkl 81 elements σij , εij symmetri −→ indep endent 36 elements additional energy onsideretation (isotermal, reversible) Cijkl =Cklij −→ indep endent 21 elements Tensor notation 11 22 33 23,32 31,13 12,21 Matrix notation 1 2 3 4 5 6 σ11 σ22 σ33 σ23 σ31 σ12 = C11 C12 C13 C14 C15 C16 C12 C22 C23 C24 C25 C26 C13 C23 C33 C34 C35 C36 C14 C24 C34 C44 C45 C46 C15 C25 C35 C45 C55 C56 C16 C26 C36 C46 C56 C66 ε11 ε22 ε33 2ε23 2ε31 2ε12 86 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Ho ok's law tensor of elasti stiness o eients Sp eial ase of tensor of elasti stiness o eients for isotropi matrial: σ11 σ22 σ33 σ23 σ31 σ12 = C11 C12 C12 000 C11 C12 000 C11 000 1 2(C11 −C12) 0 0 1 2(C11 −C12) 0 1 2(C11 −C12) ε11 ε22 ε33 2ε23 2ε31 2ε12 Relation b etween the tensor omp onenets and Young mo dulus: C11 =E(1 −ν) (1 + ν)(1 −2ν), C12 =E ν (1 + ν)(1 −2ν), G =E 2(1 + ν)K=E 3(1 −2ν) 87 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties *Other examples of using tensor material desription Thermal expansion o eients αij εij =αij ∆T. Piezo eletriity (piezeletri mo dulus dijk ) εij =dijkEk, where E is the eletri els vetor. photo elastiity magneto-eletriity magneto-optis eletro-opti magnetostrition (magneto elastiity) et. 88 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Mehanial prop erties mirosopi desription Elasti deformation in solids hange of interatomi distane fore prop ortional to derivative of interatomi potential φ . Cohesive energy (how deep φ -minimum is), Young's mo dulus (urvature near minimum). Taylor series of interatomi p otential: φ(x) = φ(a) |{z} equilibrium +φ′(a) 1! (x−a) |{z } zero near equilibrium +φ′′(a) 2! (x−a)2 |{z } elasti b ehaviour Ho ok's law +φ′′′ (a) 3! (x−a)3+· · · |{z } negligible 89 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Young's mo dulus of dierent materials Metals (metalli bound) value of G≈15 −300 GPa W, Mo strong b onding, high melting p oint Covalent b onding strong diamond, graphite||planes, graphene Van der Waals, Hydrogen b onding weak 90 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Plasti deformation σY yield stress limit of appliability of Ho ok's law mirosopi mehanism of plasti deformation 91 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Chemial b onding in solids Mehanial prop erties Plasti deformation shear deformation of rystal overestimate σY p oint defets (vaany, interstitial) and dislo ations essenial for explanation of mehanial prop erties moving of edge dislo ations (easy-glide region), dislo ation pinning on impurities (hardening region, arb on in steel; Be in Cu) 92 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Thermal prop erties of solids heat apaity C=∂E ∂T sp ei heat apaity c=C m , Q=m c ∆T thermal ondutivity transfer of thermal energy from hot to old part of material thermal expansion ∆l=l α ∆T , where α is the thermal expansion o eient Origin of thermal prop erties lattie vibrations, phonons free eletrons 93 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Lattie vibration one-dimensional ase Thermal vibration (displaement) of atom −→ repulsion fore F(x−a) = −γ(x−a) Estimation of γ from Young mo dulus E and Ho ok's low σ=E ε . σ=F S=γ∆a a2 , ε=∆a a E=γ a 94 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Lattie vibration one-dimensional ase Equation of motion: Md2un dt2=−γ(un−un−1) + γ(un+1 −un) = −γ(2un−un−1−un+1) solution in the form of wave un(t) = uei(kan−ωt) , un±1(t) = une±ika ω=r2γ(1 −cos ka) M = 2rγ M sin ka 2 95 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Lattie vibration one-dimensional ase Lattie with two atoms p er unit ell: ω2 1,2=γ1 M1 +1 M2±γs1 M1 +1 M22 −4sin2kb 2 M1M2 for k≈0 : ω1≈0 aousti branh ω2= 2γ(1 M1+1 M2) optial branh oupling with eletri eld in infrared region 99 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Lattie vibration 1D, nite length of rystal Finite length of rystal perio di (Born-Karman) b oundary onditions eikan = eika(N+n)−→ eikNa = 1, k =2π aN m, where N is numb er of atoms (numb er of p oints in 1st BZ) Corresp ond to quantized energy levels El(k) = (l+1 2)~ω(k) quantum of lattie vibration phonon , ω(k) phonon disp ersion, phonons are b osons and fulll Bose-Einstein statistis 100 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Lattie vibration generalization to 3D 3-dimmensional solution with two atoms p er unit ell: 3·2 equation of motion, wavevetor k→k , solution of determinant of 6×6 matrix, 6 solutions of ω(k) Al only 3 aousti branhes, Diamond (two atoms in unit ell) 3 aousti + 3 optial branhes (1 longitudinal, 2 transverse) 101 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity of materials Clasial thermo dynamis approah: the rst low of thermo dynamis (onservation of energy) dQ= dE+pdV Heat apaity CV=∂Q ∂T V−→ Q=CV∆T CV→0 for T→0 Sp ei heat apaity c=C m−→ Q=m cV∆T Molar heat apaity CM=C n , where n is the amount of substane (mol) 102 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity of materials Clasial statistial-physis approah (Dulong-Petit): lattie atoms osillators, mean energy of osillator in 1D: < E >=kBT , where kB is the Boltzman onstant mean energy of osillator in 3D: < E >= 3kBT Heat apaity C=∂ < E > ∂T = 3 kB , molar heat apaity C= 3kBNA= 3R= 24.9 J.K −1 103 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity of materials Theory for heat apaity of solids: 1 lassial atomisti theory Dulong-Petit 2 Einstein mo del (quantization of phonon energy) 3 Debye mo del (inludes density of phonon states) 104 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity (diamont) 105 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity Einstein mo del Einstein mo del for heat apaity lattie vibration is quantizied (phonons), energy of osillators: En=n+1 2~ωE numb er of phonons Bose-Einstein statistial distribution < n >=1 e~ωE/kBT−1 < E >= 3NA< n > +1 2~ωE Heat apaity: C=∂ < E > ∂T = 3R~ωE kBT2e~ωE/kBT (e~ωE/kBT−1)2 106 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity Einstein mo del Einstein mo del for heat apaity Heat apaity: C=∂ < E > ∂T = 3R~ωE kBT2e~ωE/kBT (e~ωE/kBT−1)2 disagreement for low T (ep onenial instead T3 dep endene) −→ solved by Debye mo del 107 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Heat apaity Debye mo del Einstein mo del Debye mo del Energy levels En=n+1 2~ωEE=~ω e~ω/kBT−1 quantum osillators lattie vibrations 108 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Phononis Thermal and mehanial prop erties (sound) Aousti prop erties (sound) relate to mehanial prop erties of solids Thermal prop erties heat apaity, heat ondution, heat expansition 115 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Phononis Relation b etween thermal and mehanial properties (sound) 116 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Phononis Mehanial and thermal prop erties are inuened by struture p erio diity at atomi level (natural material prop erties) nanometer sale (artiial materials) −→ phononis (analogy with photonis) phononi materials and phononi metamaterials 117 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
Intro dution, rystal symmetry Bonding in solids and mehanial properties Lattie vibrations, phonons, thermal prop erties Lattie vibrations, phonons Heat apaity Thermal ondutivity and thermal expansion Phononis 118 K. Postava, T. Fördös, FMT, TUO Solid State Physis 1 (Fyzika p evnýh látek 1)
NPO_VŠB-TUO_MSMT-16605/2022 Thank you for your attention kamil.postav[email protected] tibor.f[email protected] This work is licensed under CC BY 4.0