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Home , Amorphous solid, Condensed matter physics, State of matter

What is State ? Branislav K. Nikoli ć Department of Physics and , University of Delaware, U.S.A.

PHYS 624: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html Condensed Systems

Hard Matter

Colloidal Dispersions Melts and Crystalline Non-Crystalline (, Insulators, Solids ) Crystalline Solids + Amorphous Biomatter Defects Solids () (, (point, membranes, dislocations, Polymer nucleic acids) surfaces and Solids (Glass interfaces) and Rubber)

PHYS 624: Introduction to Solid State Physics Basic Notions of Condensed Matter

 excitations which look nearly as individual particles as possible - modern condensed matter theory asserts that a solid is actually a of weakly interacting quasiparticles . Scalar (=0)

Broken , long-range order, and order parameters → Phases of matter are characterized by the symmetry of their ground (lowest ) state (Landau, 1937).

 Transitions between different broken symmetry states: Classical (at T>0) and Quantum (at T=0).

PHYS 624: Introduction to Solid State Physics Quasiparticles vs Collective Excitations

− : This is a collective : a e excitation , corresponding to consisting of a real coherent motion of all the electron and the cloud of − in the solid. It is quantized effective charge of qe lattice vibration with a typical opposite sign due to energy scale ℏω ∼ 0.1 eV exchange an correlation effects arising interaction : bound state of electron with all other in and hole with binding energy10 eV the system. The electron is a with spin ½. The : longitudinal charge Fermi energy (highest oscillations of the entire electron occupied state) is of the gas relative to the lattice of order of 5 eV and the due to the long-range of the Fermi velocity is 10 8 cm/s , Coulomb interaction so it can be treated like a nonrelativistic particle. The of quasiparticle can : be substantially different : a quasiparticle , like electron, whose motion collective than that of the free in polar crystals distorts the lattice of positive and excitation of electron. negative ions around it – polaron has different mass the spin than the electron. degrees of freedom on the Hole: a quasiparticle , like the electron, but of opposite charge; it corresponds to the absence crystalline of electron for a single-particle state which lies below the Fermi level. The notion of a hole is lattice with an particularly convenient when the reference state consists of a quasiparticle states that are energy scale fully occupied and are separated by an from the unoccupied state. ℏω ∼ 0.001− 0.1 eV

PHYS 624: Introduction to Solid State Physics Experimental Probes of Condensed Matter Phases and Quasiparticle Dynamics

: Send or X-rays into the system with prescribed energy and ; measure the energy and momentum of the outgoing neutrons or X-rays. NMR: Apply static B and measure absorption and emission of magnetic radiation at frequencies of the order geB of ω = . c m : Measure the response of macroscopic variables (energy, , etc.) to variations of the , , etc. Transport: Set up a potential ∇ ϕ or thermal gradient ∇ T and measure the electrical or current. The gradients can be held constant or made to oscillate at finite frequency.

PHYS 624: Introduction to Solid State Physics Quantum Hamiltonian of “““”The general theory of quantum is now almost complete. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of 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 N2 2 N NP e N Z Z Hˆ =n + n m ∑ ∑ − ∂ ∂ n=12Mn 2| n ≠ m = 1 R n R m | p≡−iℏ, P ≡− i ℏ ∂r ∂ R TN: motion of nuclei V N− N : interaction between nuclei Nep2 e2 N e 1 NN N e Z +∑i + ∑ − e2 ∑∑ n =2M 2|| ≠ = r− r = = |R− r | i1i i j 1 i j n1 i 1 n i T : motion of electrons V − :interaction between electrons and nucle i e Ve− e :interaction between electrons e N

PHYS 624: Introduction to Solid State Physics Complexity in Solid State Hamiltonian

Even for , the task of solving the Schrödinger equation for modest multi- electron atoms proves insurmountable without bold approximations.

The problem facing condensed matter is qualitatively more severe:

23 NN∼ N e ∼ 10 Energy scales: 10−2 eV− 10 4 eV CM Theorist is entrapped in the “thermodynamic limit”

PHYS 624: Introduction to Solid State Physics The Way Out: Separate Length and Energy Scales

Energy: ω ,T< 1 e V ∆τ ℏ ℏ −15  ∼ ∼− ∼ 10 s Time: 1eV 10 19 J − − 1 Length: |xi x j |, q ≫ 1Å forget about formation + forget abo ut crystal formation m + Born-Oppenheimer approximation for e ≪ 1 m N ⇓ ˆ=++ ˆˆˆˆ += ˆ =++ Helectronic TVVVe eI−−− ee, eI (rr n ) V eI − ( rr ), n nnn11 a 22 a 33 a

PHYS 624: Introduction to Solid State Physics “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 … ˆ φ ε φ IE,H,LDA= IE,H,LDA IE,H,LDA HIE,H,LDA ()rkb () r (,) k b k b () r ˆ= ˆˆˆ + + Helectronic TVe eI− V ee − Ne ˆ= ˆ + ˆ = HIE TVeeI− ∑ H IE (r i ) i=1 ′ ˆ=++ ˆ ˆ 2 ′ n(r ) =φ H 2 =+ HTVedHartree e e− I ∫ r,() n r∑ |()|kb rrr n ( n ) |r− r ′ | ε H ≤ (k ,b ) E F n(r′ ) δELDA [ n ] HHedˆ=+ ˆ ()rr2 ′ +xc e , EndnenLDA [] = rrr ()(()) LDA LDA IE ∫|r− r′ |δn () r xce ∫ xc

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 ( r ,… , r )⇒ average of observables 1 Ne Example: Particle density n()r= Ndd rr⋯ d rrr Ψ* (,,)(,,) … Ψ rr … ∫2 ∫ 3 ∫ N Ne N e

DFT approach (Kohn-Hohenberg): n()r⇒ Ψ (,,) rr… ⇒ V () r 0 0 1 Ne Ψ(,r… , r ) ≡ Ψ [()]n r 01Ne 00

PHYS 624: Introduction to Solid State Physics Pauli Exclusion Principle for Atoms

Without the exclusion principle all electrons would occupy the same . There would be no chemistry, no life!

PHYS 624: Introduction to Solid State Physics Many-Body Wave Function of

“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 .” W. Pauli in a letter to Schrödinger (1925).

All electrons in the are identical → two physical situations that differ only by interchange of are indistinguishable! Ψ = Ψ Pxxxij(,,,,,,)1 ……………… i jnij xPxx (,,,,,,)1 jin xx PHHPˆˆˆˆ= ←ΨΨ=Ψ Aˆ PAP ˆˆ† ˆ Ψ⇔ PAAP ˆ ˆˆ = ˆ ij ij ij ij ↑ + Ψ ,Pˆ is even Ψ Ψ ˆ ˆ  A α S A α Pα Ψ=Ψ, P Ψ=  S S A − Ψ ˆ  A ,Pα is odd ˆˆΨ= ˆ Ψ ˆ = 1 ε ˆ PAα A, A∑ α α P N! α

PHYS 624: Introduction to Solid State Physics 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: ˆ =ˆ + ˆ + + ˆ Hhh1 2 … h N heˆ φφ = ⇒ Hˆ Φ = E Φ iiii1,2,,……… Ne 1,2,, N e 1,2,, N e E= ee + +… + e 1,2,,… Ne 1 2 N e φ φ (1)… (1) 1 Ne Φ = 1= 1 1,2,… , N ⋮ ⋱ ⋮ ⇒ nkσ µε − e N! e(kσ )/ kB T +1 φ φ ()N⋯ () N 1 e Ne e

PHYS 624: Introduction to Solid State Physics Energy Bands and Rigid Band Filling

E

Wide-gap Intrinsic Two-band One-band

PHYS 624: Introduction to Solid State Physics Electron-Electron Interactions in Metals

Bloch theory of electrons in metals: completely independent particles (e.g., even more sophisticated Hartree-Fock fails badly because dynamic correlations cancel miraculously exchange effects). Why is long-range strong (of the order of ) Coulomb interaction marginal in metals? 1. In system with itinerant electrons, Coulomb interaction is very −1 effectively screened on the length scale of k F . 2. In the presence of the scattering rate between + ω ω 2 electrons with energy E F ℏ vanishes proportional to since the Pauli principle strongly reduces the number of scattering channels that are compatible with energy and momentum conservations – Landau “quasielectrons” live very long! DFT (and LDA) only describes an equivalent one-electron substitute system with the same ground- state energy and electron density. Altough we may expect that the ground-state properties are reasonably described, the wave functions and excitation (band structures) may be used only if the prerequisites of the Landau are fulfilled.

PHYS 624: Introduction to Solid State Physics Real Free Electrons vs. Landau Quasielectrons

Fermi Gas Fermi Liquid

A quasiparticle state is in fact − − − τ made of very large number of Ψ(t ) ∼ e iE( k ) t ℏ Ψ(t ) ∼ eiEkt( ) ℏ e t exact eigenstatates of the interacting system. The separation of these states is exponentially small in the system size and thus irrelevant physically for reasonable systems. The cluster of all these states form the quasiparticle with its average enery and lifetime (= inverse of the energy broadening).

PHYS 624: Introduction to Solid State Physics Exchange-Correlation Hole

Surrounding every electron in a solid there's an exclusion zone, called the exchange-correlation hole, into which other electrons rarely venture . This is the hole around an electron near the centre of a bond in silicon.

PHYS 624: Introduction to Solid State Physics From Weakly to Strongly Correlated Electrons

Exchange and Correlation in 3D gas of noninteracting fermions: − 2 ′ = − 3n sin xx cos x  gσ σ, (r , r ) 3  2 x  ′= = − ′ gσ σ,− (,)0,rr x k F | rr | ′ ≠ Correlated Electron System: gσ σ,− (,r r ) 0 ′ ′ Strongly Correlated Electron System: g σ σσ σ ,(,)rr∼ g , − (,) rr Example of novel strongly correlated matter:

Fractional Quantum

PHYS 624: Introduction to Solid State Physics Survival of Long-Range Coulomb Interaction: Collective Excitations

π 2 ω 4 ne − ℏ≃ℏp ∼ 5 20eV me

k k+ Q Q

Bohm and Pines (1953): separate strongly interacting gas into two independent sets of excitations (progenitor of the

idea of !) high energy → plasmon low energy → electron-hole pairs

PHYS 624: Introduction to Solid State Physics Breaking the Symmetry

QUESTION: In QM we learn that the must have the symmetry of the Hamiltonian - so there can't be a dipole moment (interactions between ions and electrons have no preferred direction in space). On the other hand, ammonia obviously has dipole moment? RESOLUTION: The ammonia molecule ground state is a superposition of states, so as to recover the symmetry of the Hamiltonian. However, at short time-scale molecule can be trapped in one of the states (due to large potential barrier for tunneling between the states), and we measure non-zero dipole moment. QUESTION: What about larger ( > 10 atoms) which have definite three-dimensional structures which break the symmetry of the Hamiltonian? RESOLUTION: We cannot understand the structure of molecules starting from of elementary particles - we need additional theoretical ideas ( emergent phenomena )! PHYS 624: Introduction to Solid State Physics Broken Symmetries and Phases of Matter

Phases of matter often exhibit much less symmetry than underlying microscopic equations. Example: exhibits full translational and rotational symmetry of Newton’s or Schrödinger’s equations; , however, is only invariant under the discrete translational and rotational group of its crystal lattice → translational and rotational symmetry of the microscopic equations have been spontaneously broken! Order Parameter Paradigm (L. D. Landau, 1940s): Development of phases in a material can be described by the of an "order parameter“ (which fluctuates strongly at the critical point): Ψ = ω ρ ρ iϕ M,()rs e ,typical (,) r

PHYS 624: Introduction to Solid State Physics Crystalline Hard Condensed Matter Phases

Positive ions arrange to break translational and rotational symmetry – it is energetically favorable to break the symmetry in the same way in different parts of the system → because of broken symmetries the solids are rigid (i.e., solid ) and exhibit long-range order . In crystalline solids discrete subgroups of the translational and rotational group are preserved metals insulators T → 0⇒ σ ≠ 0 T → 0⇒ σ → 0 ρ ρ

T T

PHYS 624: Introduction to Solid State Physics Broken Symmetry States of Metals

) n U(1 Broke Highly Anisotropic Superconductors T< T ⇒ σ → ∞ c Charge-Density Wave + Meissner effect

Spin-Density Wave

PHYS 624: Introduction to Solid State Physics Broken Spin Rotational Symmetry of

Metals and Insulators

n A

o s

r t n

t e t

c n i

e g f l a e F

E e

m M r r s

t r

i r i

n g m

t o a r a

e m g

r e n

n e

e b g a t

n n s a i e g

t s n

M I i

e e

r t

e H s

n

o

t S

Ground state symmetry determines the properties of collective excitations – Spin waves ()

PHYS 624: Introduction to Solid State Physics Non-Crystalline Hard CM Phases: Quasicrystals

Among the most well known consequences of periodicity is the fact that the only rotational symmetries that are possible are 2-, 3-, 4-, and 6-fold rotations ⇒ Five-fold rotations (and any n-fold rotation for n>6) are incompatible with periodicity . Quasicrystals =quasiperiodic crystals [D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn, "Metalic phase with with long-range orientational order and no ," Phys. Rev. Lett. 53 , 1951 (1984) ]: Translational order completely broken; rotational symmetry broken to 5-fold discrete subgroup. Penrose tile is a 2D example of a : There is perfect long-range order and no periodicity in the normal sense.

Quasicrystals posses unique physical properties: even though they are all alloys of two or three metals they are very poor conductors of electricity and of heat.

PHYS 624: Introduction to Solid State Physics Imaging Quasicrystals via STM

Scanning tunneling microscope image of a 10 nm 2 quasicrystal of AlPdMn with a Penrose tiling overlaid [Ledieu et al., Phys. Rev. B 66 , 184207 (2002)]

PHYS 624: Introduction to Solid State Physics Non-Crystalline Hard CM Phases:

Glasses : rigid but random arrangement of atoms; in fact, they are “non-equilibrium phase” – effectively they look more like a frozen “snapshot” of a liquid. The structural glassy materials arise when a liquid is cooled to an fast enough to prevent , which would otherwise occur if the time had been sufficient for the sample to reach true equilibrium at each temperature. Thus, “frozen” supercooled (which have enormous that prevents any large-scale flow on human time scales ) form a glassy that is substantially different from crystals: it is disordered and only metastable thermodynamically. Moreover, the essential physics of nonequilibrium glassy state (leading to rapid increase in response time, and phenomena like aging, rejuvenation, memory effects, …) is the prototype of a slow nonexponential found in diverse physical systems: spin glasses, disordered insulators (Coulomb glass), , superconductors, proteins, colloidal suspensions, Example: . granular assemblies , …

PHYS 624: Introduction to Solid State Physics Soft Condensed Matter Phases

 Liquids (full translational and rotational group preserved) vs. Solids (preserve only a discrete subgroups).  Liquid crystalline phase – translational and rotational symmetry is broken to a combination of discrete and continuous subgroups:

 – extremely long molecules that can exist in or a can take place which cross-links them, thereby forming a .

PHYS 624: Introduction to Solid State Physics Broken Symmetries and Rigidity

Phase of a develops a rigidity → it costs energy to bend phase → superflow of particles is directly proportional to gradient of phase: 1 2 Ux()∼ ρ φ φ ρ []∇ () x⇒ j = ∇ 2 s sss Phase Broken Rigidity/Superflow Symmetry crystal translation Momentum (sheer stress) superfluid gauge matter EM gauge charge F- and AF- spin rotation spin (x-y magnets only) nematic rotation angular momentum ? Time Energy? translation

PHYS 624: Introduction to Solid State Physics High Energy Physics: Lessons from Low Energy Experiments

Anderson Higgs Mechanism

Asymptotic Freedom in the Physics of Confinement

PHYS 624: Introduction to Solid State Physics Quantum Critical Matter

PHYS 624: Introduction to Solid State Physics Complexity and Quantum Criticality in Oxides Strongly Correlated Materials

PHYS 624: Introduction to Solid State Physics Complexity and Diversity of Crystalline Phases

PHYS 624: Introduction to Solid State Physics