What is Solid State Physics?

Transcription

1 What is Solid Stat Physics? Branislav K. Nikolić Dpartmnt of Physics and Astronomy, Univrsity of Dlawar, U.S.A.

2 Quantum Hamiltonian of Solid Stat Physics Th gnral thory of quantum mchanics is now almost complt. Th undrlying physical laws ncssary for th mathmatical thory of a larg part of physics and th whol of chmistry ar thus compltly known, and th difficulty is only that th xact application of ths laws lads to quations much too complicatd to b solubl. Hˆ P. A. M. Dirac 1929 NN 2 2 NN Pn ZnZm = + n= 1 2Mn 2 n m= 1 Rn Rm T N : motion of nucli V : intraction btwn nucli N N N 2 2 N pi i= 1 2Mi 2 i j= 1 ri rj T : motion of lctrons V :int raction btwn lctrons p V iħ N 2 r N N, N P R Z iħ r n= 1 i= 1 n i :intraction btwn lctrons and nucli n R

3 Complxity in Solid Stat Hamiltonian Evn for chmist, th task of solving th Schrödingr quation for modst multilctron atoms provs insurmountabl without bold approximations. Th problm facing condnsd mattr physicist is qualitativly mor svr: N N N 10 CM Thorist is ntrappd in th thrmodynamic limit Enrgy scals: 10 V 10 V

4 Th Way Out: Sparat Lngth and Enrgy Scals Enrgy: Tim: Lngth: ω, T < 1 ħ τ 1V 10 V x i x j, q 1 Å ħ J s forgt about atom formation + forgt about crystal formation m + Born-Oppnhimr approximation for 1 mn H ˆ = T ˆ + V ˆ + V ˆ, V ˆ ( r+ r ) = V ˆ ( r), r = na + na + na lctronic I I n I n

5 Non-Intracting Elctrons in Solids: Band Structur Calculations In band structur calculations lctron-lctron intraction is approximatd in such a way that th rsulting problm bcoms an ffctiv singl-particl quantum mchanical problm ˆ ( ) IE,H,LDA ( ) IE,H,LDA (, ) IE,H,LDA ( ) IE,H,LDA r φkb r = ε k b φkb r H Hˆ = Tˆ + Vˆ + Vˆ lctronic I Hˆ = Tˆ + Vˆ = H ( r ) IE N I IE i i= 1 n( r ) H T V d n n ˆ 2 H 2 Hartr = ˆ ˆ + I + r, ( r) = φk b( r) = ( r + rn ) r r H ε ( k, b) E LDA ˆ ˆ 2 n( r ) δexc [ n ] LDA LDA LDA IE ( ) H = H r + dr +, Exc [ n ] = dr n ( r ) xc ( n ( r )) r r δn( r) F

6 From Many-Body Problm to Dnsity Functional Thory (and its LDA approximation) Classical Schrödingr quation approach: SE 1 Ψ... Ψ V ( r) Ψ( r,, r ) avrag of obsrvabls Exampl: Particl dnsity n N d d d N * ( r) = r2 r3 rn Ψ ( r,, rn ) Ψ( r,, rn ) DFT approach (Kohn-Hohnbrg): n ( r) Ψ ( r,, r ) V ( r) Ψ N ( r,, r ) Ψ [ n ( r)] 0 1 N 0 0

7 Many-Body Wav Function of Frmions It is with a havy hart, I hav dcidd that Frmi-Dirac, and not Einstin is th corrct statistics, and I hav dcidd to writ a short not on paramagntism. W. Pauli in a lttr to Schrödingr (1925). All lctrons in th Univrs ar idntical two physical situations that diffr only by intrchang of idntical particls ar indistinguishabl! ΨS P Ψ ( x,, x,, x,, x ) = P Ψ( x,, x,, x,, x ) ij 1 i j n ij 1 j i n ˆ ˆ ˆ ˆ ij = ij Ψ Ψ = Ψ ij Ψ ij = Pˆ Hˆ HP ˆ ˆ A Pˆ APˆ Pˆ A APˆ +, ˆ is vn Ψ ˆ ˆ ΨA Pα A Pα Ψ S = ΨS, Pα Ψ A =, ˆ Ψ A Pα is odd ˆ ˆ ˆ ˆ 1 P, ˆ α A Ψ = A Ψ A= εα Pα N! α

8 Pauli Exclusion Principl Thr is no on fact in th physical world which has gratr impact on th way things ar, than th Pauli xclusion principl. I. Duck and E. C. G. Sudarshan, Pauli and th Spin-Statistics Thorm (World Scintific, Singapor, 1998). Two idntical frmions cannot occupy th sam quantum-mchanical stat: 1 2 i i i i 1,2,, N 1,2,, N 1,2,, N 1,2,, N 1 2 1,2,, N kσ ( ε µ )/ k T 1 Hˆ = hˆ + hˆ + + hˆ hˆ φ = φ Hˆ Φ = E Φ E = φ1 (1) φn (1) 1 1 Φ = n = kσ N! φ ( N ) φ ( N ) N N N B + 1

9 Braking th Symmtry QUESTION: In QM w larn that th ground stat must hav th symmtry of th Hamiltonian - so thr can't b a dipol momnt (intractions btwn ions and lctrons hav no prfrrd dirction in spac). On th othr hand, ammonia molcul obviously has dipol momnt? RESOLUTION: Th ammonia molcul ground stat is a suprposition of stats, so as to rcovr th symmtry of th Hamiltonian. Howvr, at short tim-scal molcul can b trappd in on of th stats (du to larg potntial barrir for tunnling btwn th stats), and w masur non-zro dipol momnt. QUESTION: What about largr molculs ( > 10 atoms) which hav dfinit thr-dimnsional structurs which brak th symmtry of th Hamiltonian? RESOLUTION: W cannot undrstand th structur of molculs starting from Quantum Mchanics of lmntary particls - w nd additional thortical idas (mrgnt phnomna)!

10 Brokn Symmtris and Phass of Mattr Phass of mattr oftn xhibit much lss symmtry than undrlying microscopic quations. Exampl: Watr xhibits full translational and rotational symmtry of Nwton s or Schrödingr s quations; Ic, howvr, is only invariant undr th discrt translational and rotational group of its crystal lattic translational and rotational symmtry of th microscopic quations hav bn spontanously brokn! Ordr Paramtr Paradigm (L. D. Landau, 1940s): Dvlopmnt of phass in a matrial can b dscribd by th mrgnc of an "ordr paramtr (which fluctuats strongly at classical critical points): i ϕ s M, Ψ ( r) = ρ, ρ ( ω, r) typical

11 Hard Condnsd Mattr Phass Positiv ions arrang to brak translational and rotational symmtry it is nrgtically favorabl to brak th symmtry in th sam way in diffrnt parts of th systm bcaus of brokn symmtris th solids ar rigid (i.., solid). In crystallin solids discrt subgroups of th translational and rotational group ar prsrvd: mtals T 0 σ 0 insulators T 0 σ = 0 suprconductors T < T σ + Missnr ffct c All of ths thr phass can b furthr subdividd into frromagnts or antifrromagnts, which brak th spin rotational invarianc. Othr hard mattr: quasicrystals (translational ordr compltly brokn; rotational symmtry brokn to 5-fold discrt subgroup) and glasss (rigid but random arrangmnt of atoms; in fact, thy ar non-quilibrium phas a snapshot of liquids).

12 Soft Condnsd Mattr Phass Liquids (full translational and rotational group prsrvd) vs. Solids (prsrv only a discrt subgroups). Liquid crystallin phas translational and rotational symmtry is brokn to a combination of discrt and continuous subgroups: Polymrs xtrmly long molculs that can xist in solution or a chmical raction can tak plac which cross-links thm, thrby forming a gl.

13 Complxity and Divrsity of Crystallin Phass

14 Brokn Symmtris and Rigidity Phas of a Coopr pair dvlops a rigidity it costs nrgy to bnd phas suprflow of particls is dirctly proportional to gradint of phas: U ( x ) 1 ρ [ ( )] 2 s φ x j s = ρ s φ s 2 Phas Brokn Symmtry Rigidity/Suprflow crystal suprfluid suprconductivity F- and AF-magntism nmatic liquid crystal? translation gaug EM gaug spin rotation rotation Tim translation Momntum (shr strss) mattr charg spin (x-y magnts only) angular momntum Enrgy?

15 High Enrgy Physics: Lssons from Low Enrgy Exprimnts Andrson Higgs Mchanism Asymptotic Frdom in th Physics of Quark Confinmnt

16 Exprimntal Probs of Condnsd Mattr Phass Scattring: Snd nutrons or X-rays into th systm with prscribd nrgy and momntum; masur th nrgy and momntum of th outgoing nutrons or X-rays. NMR: Apply static magntic fild and masur absorption and mission of magntic radiation at frquncis of th ordr gb of ω. c = m Thrmodynamics: Masur th rspons of macroscopic variabls (nrgy, volum, tc.) to variations of th tmpratur, prssur, tc. ϕ Transport: St up a potntial or thrmal gradint and masur th lctrical or hat currnt. Th gradints can b hld constant or mad to oscillat at finit frquncy. B T

17 Wakly vs. Strongly Corrlatd Elctrons Exchang and Corrlation in 3D gas of nonintracting frmions: 3n sin x x cos x gσ, σ ( r, r ) = 3 2 x g ( r, r ) = 0, x = k r r σ, σ Corrlatd Elctron Systm: Strongly Corrlatd Elctron Systm: F 2 g σ Exampl of nw strongly corrlatd mattr: Fractional Quantum Hall Effct, σ ( r, r ) 0 g ( r, r ) g ( r, r ) σ, σ σ, σ

18 Elctron-Elctron Intractions in Mtals Bloch thory of lctrons in mtals: compltly indpndnt particls (.g., vn mor sophisticatd Hartr-Fock fails badly bcaus dynamic corrlations cancl miraculously xchang ffcts). Why is long-rang strong Coulomb intraction marginal in mtals? 1. In systm with itinrant lctrons, Coulomb intraction is vry ffctivly scrnd on th 1 lngth scal of. k F 2. In th prsnc of Frmi surfac th scattring rat btwn lctrons with nrgy EF + ω 2 vanishd proportional to ω sinc th Pauli principl strongly rducs th numbr of scattring channls that ar compatibl with nrgy and momntum consrvations Landau quasilctrons liv vry long!

19 Survival of Coulomb Intraction: Collctiv Excitations ħω p 6V k k + Q Bohm and Pins (1953): sparat strongly intracting gas into two indpndnt sts of xcitations (prognitor of th ida of rnormalization!) Q high nrgy plasmon low nrgy lctron-hol pairs

20 Quantum Criticality in Your Car Bumpr

21 Quantum Criticality: Exampl of High Tc Suprconductors