Condensed Matter Systems Hard Matter Soft Matter Colloidal Dispersions Liquid Polymer Melts and Solutions Crystalline Solids (Metals, Insulators, Semiconductors) Non-Crystalline Solids Crystals Crystalline Solids + Defects (point, Quasicrystals Amorphous Solids (Glass) Biomatter (proteins, membranes, dislocations, surfaces and interfaces) Polymer Solids (Glass and Rubber) nucleic acids) PHYS 624: Introduction to Solid State Physics

∑ ∑=i 1 2M ∑ji≠ =1 | r − r | Quantum Hamiltonian of Condensed Matter Physics “”The general theory of quantum mechanics is now almost complete. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” P. A. M. Dirac 1929 NN n=1 2Mn TN : motion of nuclei Pn2 e2 NN Zn Zm 2 n≠m=1 | Rn − Rm | VN −N : interaction between nuclei Hˆ = ∑ ∑ Quantization ∂ ∂ ∂r ∂R + p ≡ −iℏ , P ≡ −iℏ Ne pi2 e2 Ne 1 2 i j NN Ne ∑=i 1 2M ∑ji≠ =1 | r − r | e ∑∑ Zn + + − 2 n=1 i=1 | Rn − r | i i Te : motion of electrons Ve−e :int eraction between electrons Ve−N :interaction between electrons and nuclei PHYS 624: Introduction to Solid State Physics

N N ∼ Ne ∼ 10 Complexity in Solid State Hamiltonian Even for chemist, the task of solving the Schrödinger equation for modest multi- electron atoms proves insurmountable without bold approximations. The problem facing condensed matter physicist is qualitatively more severe: N N ∼ Ne ∼ 10 23 −2 4 CM Theorist is entrapped in the “thermodynamic limit” PHYS 624: Introduction to Solid State Physics

Time: Separate Length and Energy Scales Energy: ω , T < 1 e V The Way Out: Separate Length and Energy Scales Energy: ω , T < 1 e V ℏ ℏ 1eV 1 0 − 19 J ∆ τ ∼ ∼ ∼ 1 0 − 15 s Time: Length: | x i − x j |, q − 1 ≫ 1 Å forget about atom formation + forget about crystal formation me mN + Born-Oppenheimer approximation for ≪ 1 ⇓ Helectronic =T +V −I +V −e, V −I (r +rn ) =V −I (r), r = na1 +n2a2 +n3a3 PHYS 624: Introduction to Solid State Physics ˆ ˆe ˆe ˆe ˆe ˆe n 1

∑ “Non-Interacting” Electrons in Solids: Band Structure Calculations In band structure calculations electron-electron interaction is approximated in such a way that the resulting problem becomes an effective single-particle quantum mechanical problem … H IE,H,LDA (r )φk E,H,LDA (r ) = ε IE,H,LDA (k , b )φ IE,H,LDA (r ) ˆ Ib kb H electronic = Te + Ve − I + Ve − e N ˆ ˆ ˆ ˆ IE IEˆ ˆ ˆ ( )e e I iH T H−= = ∑ r e V+ i =1 2 n(r′) Hartree e e IH T e d− ′= + ∫ r ˆ ˆ ˆV+ , ( )n =r ∑ H| )b nφk 2( ) | (n= +r r r | r − r′ | ε H (k ,b)≤ EF LDA[ ]( ) eE nn δ′r LDA LDALDA IE ( ) , [ ] ( ) ( ( ))eH H e d E n d n e n′= + + =∫ ∫r r r r r ˆ 2 | r −r′ | δ n(r) PHYS 624: Introduction to Solid State Physics

From Many-Body Problem to Density Functional Theory (and its LDA approximation) “Classical” Schrödinger equation approach: SE Ψ ... Ψ V (r) ⇒ Ψ(r1 ,…, rNe ) ⇒ average of observables Example: Particle density n(r) = N ∫ dr2 ∫ dr3 ⋯∫ drN Ψ* (r,… , rNe )Ψ (r,… , rNe ) DFT approach (Kohn-Hohenberg): n0 (r) ⇒ Ψ 0 (r1 ,… , rN e ) ⇒ V (r ) Ψ 0 (r1 ,… , rN e ) ≡ Ψ 0 [n0 (r )] PHYS 624: Introduction to Solid State Physics

Pauli Exclusion Principle for Atoms Without the exclusion principle all electrons would occupy the same atomic orbital. There would be no chemistry, no life! PHYS 624: Introduction to Solid State Physics

Many-Body Wave Function of Fermions “It is with a heavy heart, I have decided that Fermi-Dirac, and not Einstein is the correct statistics, and I have decided to write a short note on paramagnetism.” W. Pauli in a letter to Schrödinger (1925). All electrons in the Universe are identical two physical situations that differ only by interchange of identical particles are indistinguishable! Pij Ψ ( x1 ,… , xi ,… , x j ,… , xn ) = Pij Ψ ( x1 ,… , x j ,… , xi ,… , xn ) ˆ ˆ † ˆ ˆ ↑ Ψ P ΨS =ΨS , P ΨA =  P A Ψ = A Ψ , A = 1 ∑εα P N ! α PHYS 624: Introduction to Solid State Physics + ΨA , Pˆα is even ΨS A ˆα ˆα − ΨA , Pˆα is odd ˆα ˆ ˆ ˆ ˆα

Pauli Exclusion Principle for Hartree- Fock or Kohn-Scham Quasiparticles “There is no one fact in the physical world which has greater impact on the way things are, than the Pauli exclusion principle.” I. Duck and E. C. G. Sudarshan, “Pauli and the Spin-Statistics Theorem” (World Scientific, Singapore, 1998). Two identical fermions cannot occupy the same quantum-mechanical state: H = h1 + h2 +… + hN h φi = ei φi ⇒ H Φ1,2,…, Ne = E1,2,…, Ne Φ1,2,…, Ne E1,2,…, Ne = e1 + e2 +… + eNe ˆ ˆ ˆ ˆ ˆi ˆ φ1 (1) … φNe (1) ⋮ ⋱ ⋮ φ1 ( Ne ) ⋯ φNe ( Ne ) 1 N ! 1 e(εkσ −µ )/ kBT +1 Φ1,2,…, Ne = ⇒ nkσ = PHYS 624: Introduction to Solid State Physics