Advanced Solid state Physics lecture notes

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Solid State Physics Chetan Nayak Physics 140a Franz 1354; M, W 11:00-12:15 Oce Hour: TBA; Knudsen 6-130J Section: MS 7608; F 11:00-11:50 TA: Sumanta Tewari University of California, Los Angeles September 2000Contents 1 What is Condensed Matter Physics? 1 1.1 Length, time, energy scales . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Microscopic Equations vs. States of Matter, Phase Transitions, Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Broken Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Experimental probes: X-ray scattering, neutron scattering, NMR, ther- modynamic, transport . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 The Solid State: metals, insulators, magnets, superconductors . . . . 4 1.6 Other phases: liquid crystals, quasicrystals, polymers, glasses . . . . . 5 2 Review of Quantum Mechanics 7 2.1 States and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Density and Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 -function scatterer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Double Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Many-Particle Hilbert Spaces: Bosons, Fermions . . . . . . . . . . . . 15 3 Review of Statistical Mechanics 18 iiChapter 1 What is Condensed Matter Physics? 1.1 Length, time, energy scales We will be concerned with:  ;T 1eV 1  jx xj;  1A i j q as compared to energies in the MeV for nuclear matter, and GeV or even TeV , in particle physics. The properties of matter at these scales is determined by the behavior of collections 23 of many ( 10 ) atoms. In general, we will be concerned with scales much smaller than those at which gravity becomes very important, which is the domain of astrophysics and cosmology. 1Chapter 1: What is Condensed Matter Physics? 2 1.2 Microscopic Equations vs. States of Matter, Phase Transitions, Critical Points Systems containing many particles exhibit properties which are special to such sys- tems. Many of these properties are fairly insensitive to the details at length scales  shorter than 1A and energy scales higher than 1eV which are quite adequately described by the equations of non-relativistic quantum mechanics. Such properties are emergent. For example, precisely the same microscopic equations of motion 23 Newton's equations can describe two di erent systems of 10 H O molecules. 2 2 X d x i m = rV (x x ) (1.1) i i j 2 dt j6=i Or, perhaps, the Schr odinger equation: 0 1 2 X X h  2 A r + V (x x ) (x ;:::;x ) =E (x ;:::;x ) (1.2) i j 1 N 1 N i 2m i i;j However, one of these systems might be water and the other ice, in which case the properties of the two systems are completely di erent, and the similarity between their microscopic descriptions is of no practical consequence. As this example shows, many-particle systems exhibit various phases such as ice and water which are not, for the most part, usefully described by the microscopic equations. Instead, new low-energy, long-wavelength physics emerges as a result of the interactions among large numbers of particles. Di erent phases are separated by phase transitions, at which the low-energy, long-wavelength description becomes non-analytic and exhibits singularities. In the above example, this occurs at the freezing point of water, where its entropy jumps discontinuously.Chapter 1: What is Condensed Matter Physics? 3 1.3 Broken Symmetries As we will see, di erent phases of matter are distinguished on the basis of symmetry. The microscopic equations are often highly symmetrical for instance, Newton's laws are translationally and rotationally invariant but a given phase may exhibit much less symmetry. Water exhibits the full translational and rotational symmetry of of Newton's laws; ice, however, is only invariant under the discrete translational and rotational group of its crystalline lattice. We say that the translational and rotational symmetries of the microscopic equations have been spontaneously broken. 1.4 Experimental probes: X-ray scattering, neu- tron scattering, NMR, thermodynamic, trans- port There are various experimental probes which can allow an experimentalist to deter- mine in what phase a system is and to determine its quantitative properties:  Scattering: send neutrons or X-rays into the system with prescribed energy, momentum and measure the energy, momentum of the outgoing neutrons or X-rays.  NMR: apply a static magnetic eld,B, and measure the absorption and emission by the system of magnetic radiation at frequencies of the order of =geB=m. c Essentially the scattering of magnetic radiation at low frequency by a system in a uniform B eld.  Thermodynamics: measure the response of macroscopic variables such as the energy and volume to variations of the temperature, pressure, etc.Chapter 1: What is Condensed Matter Physics? 4  Transport: set up a potential or thermal gradient,r',rT and measure the electrical or heat currentj,j . The gradientsr',rT can be held constant or Q made to oscillate at nite frequency. 1.5 The Solid State: metals, insulators, magnets, superconductors In the solid state, translational and rotational symmetries are broken by the arrange- ment of the positive ions. It is precisely as a result of these broken symmetries that solids are solid, i.e. that they are rigid. It is energetically favorable to break the symmetry in the same way in di erent parts of the system. Hence, the system resists attempts to create regions where the residual translational and rotational symmetry groups are di erent from those in the bulk of the system. The broken symmetry can be detected using X-ray or neutron scattering: the X-rays or neutrons are scattered by the ions; if the ions form a lattice, the X-rays or neutrons are scattered coherently, forming a di raction pattern with peaks. In a crystalline solid, discrete subgroups of the translational and rotational groups are preserved. For instance, in a cubic lattice, rotations by =2 about any of the crystal axes are symmetries of the lattice (as well as all rotations generated by products of these). Translations by one lattice spacing along a crystal axis generate the discrete group of translations. In this course, we will be focussing on crystalline solids. Some examples of non- crystalline solids, such as plastics and glasses will be discussed below. Crystalline solids fall into three general categories: metals, insulators, and superconductors. In addition, all three of these phases can be further subdivided into various magnetic phases. Metals are characterized by a non-zero conductivity at T = 0. Insulators have vanishing conductivity atT = 0. Superconductors have in nite conductivity forChapter 1: What is Condensed Matter Physics? 5 T T and, furthermore, exhibit the Meissner e ect: they expel magnetic elds. c In a magnetic material, the electron spins can order, thereby breaking the spin- rotational invariance. In a ferromagnet, all of the spins line up in the same direction, thereby breaking the spin-rotational invariance to the subgroup of rotations about this direction while preserving the discrete translational symmetry of the lattice. (This can occur in a metal, an insulator, or a superconductor.) In an antiferromagnet, neighboring spins are oppositely directed, thereby breaking spin-rotational invariance to the subgroup of rotations about the preferred direction and breaking the lattice translational symmetry to the subgroup of translations by an even number of lattice sites. Recently, new states of matter the fractional quantum Hall states have been discovered in e ectively two-dimensional systems in a strong magnetic eld at very low T. Tomorrow's experiments will undoubtedly uncover new phases of matter. 1.6 Other phases: liquid crystals, quasicrystals, polymers, glasses The liquid with full translational and rotational symmetry and the solid which only preserves a discrete subgroup are but two examples of possible realizations of translational symmetry. In a liquid crystalline phase, translational and rotational symmetry is broken to a combination of discrete and continuous subgroups. For instance, a nematic liquid crystal is made up of elementary units which are line seg- ments. In the nematic phase, these line segments point, on average, in the same direction, but their positional distribution is as in a liquid. Hence, a nematic phase breaks rotational invariance to the subgroup of rotations about the preferred direction and preserves the full translational invariance. Nematics are used in LCD displays.Chapter 1: What is Condensed Matter Physics? 6 In a smectic phase, on the other hand, the line segments arrange themselves into layers, thereby partially breaking the translational symmetry so that discrete transla- tions perpendicular to the layers and continuous translations along the layers remain unbroken. In the smectic-A phase, the preferred orientational direction is the same as the direction perpendicular to the layers; in the smectic-C phase, these directions are di erent. In a hexatic phase, a two-dimensional system has broken orientational order, but unbroken translational order; locally, it looks like a triangular lattice. A quasicrystal has rotational symmetry which is broken to a 5-fold discrete subgroup. Translational order is completely broken (locally, it has discrete translational order). Polymers are extremely long molecules. They can exist in solution or a chemical re- action can take place which cross-links them, thereby forming a gel. A glass is a rigid, `random' arrangement of atoms. Glasses are somewhat like `snapshots' of liquids, and are probably non-equilibrium phases, in a sense.Chapter 2 Review of Quantum Mechanics 2.1 States and Operators E A quantum mechanical system is de ned by a Hilbert space,H, whose vectors, are associated with the states of the system. A state of the system is represented by E i the set of vectors e . There are linear operators,O which act on this Hilbert i space. These operators correspond to physical observables. Finally, there is an inner D E E E product, which assigns a complex number,  , to any pair of states, ,  . A E state vector, gives a complete description of a system through the expectation D E E D E values, O (assuming that is normalized so that = 1), which would i be the average values of the corresponding physical observables if we could measure E them on an in nite collection of identical systems each in the state . y The adjoint,O , of an operator is de ned according to D  E D  E y  O =  O (2.1) E E In other words, the inner product between  andO is the same as that between E E y O  and . An Hermitian operator satis es y O =O (2.2) 7Chapter 2: Review of Quantum Mechanics 8 while a unitary operator satis es y y OO =OO = 1 (2.3) IfO is Hermitian, then iO e (2.4) is unitary. Given an Hermitian operator,O, its eigenstates are orthogonal, D E D E D E 0 0 0 0  O  =   =   (2.5) 0 For 6= , D E 0   = 0 (2.6) If there aren states with the same eigenvalue, then, within the subspace spanned by these states, we can pick a set of n mutually orthogonal states. Hence, we can use E E the eigenstates  as a basis for Hilbert space. Any state can be expanded in the basis given by the eigenstates ofO: E E X = c  (2.7)   with D E c =  (2.8)  A particularly important operator is the Hamiltonian, or the total energy, which we will denote by H. Schr odinger's equation tells us that H determines how a state of the system will evolve in time. E E ih  =H (2.9) t If the Hamiltonian is independent of time, then we can de ne energy eigenstates, E E H E =E E (2.10)Chapter 2: Review of Quantum Mechanics 9 which evolve in time according to: E E Et i h  E(t) =e E(0) (2.11) An arbitrary state can be expanded in the basis of energy eigenstates: E E X = c E (2.12) i i i It will evolve according to: E E E t X j i h  (t) = c e E (2.13) j j j For example, consider a particle in 1D. The Hilbert space consists of all continuous complex-valued functions, (x). The position operator, x , and momentum operator, p are de ned by: x  (x)  x (x) p  (x)  ih  (x) (2.14) x The position eigenfunctions, x(xa) =a(xa) (2.15) are Dirac delta functions, which are not continuous functions, but can be de ned as the limit of continuous functions: 2 x 1 2 a (x) = lim p e (2.16) a0 a  The momentum eigenfunctions are plane waves: ikx ikx ih  e =hk  e (2.17) x Expanding a state in the basis of momentum eigenstates is the same as taking its Fourier transform: Z 1 1 ikx p (x) = dk (k) e (2.18) 1 2Chapter 2: Review of Quantum Mechanics 10 where the Fourier coecients are given by: Z 1 1 ikx p (k) = dx (x)e (2.19) 1 2 If the particle is free, 2 2 h  H = (2.20) 2 2m x then momentum eigenstates are also energy eigenstates: 2 2 h  k ikx ikx He = e (2.21) 2m If a particle is in a Gaussian wavepacket at the origin at time t = 0, 2 x 1 2 a (x; 0) = p e (2.22) a  Then, at time t, it will be in the state: Z 1 2 1 a hk  t 1 2 2 i k a ikx 2m 2 p p (x;t) = dk e e e (2.23) 1  2 2.2 Density and Current  Multiplying the free-particle Schr odinger equation by , 2 2 h    ih  = (2.24) 2 t 2m r and subtracting the complex conjugate of this equation, we nd     ih     ( ) = r r r (2.25) t 2m This is in the form of a continuity equation,  =rj (2.26) t The density and current are given by:   = Chapter 2: Review of Quantum Mechanics 11     ih    j = r r (2.27) 2m The current carried by a plane-wave state is: h  1 j = k (2.28) 3 2m (2) 2.3 -function scatterer 2 2 h  H = +V (x) (2.29) 2 2m x 8 ikx ikx e +Re if x 0 (x) = (2.30) ikx : Te if x 0 1 T = mV 1 i 2 h  k mV i 2 h k R = (2.31) mV 1 i 2 h  k There is a bound state at: mV ik = (2.32) 2 h  2.4 Particle in a Box Particle in a 1D region of length L: 2 2 h  H = (2.33) 2 2m x ikx ikx (x) =Ae +Be (2.34) 2 2 has energy E =h  k =2m. (0) = (L) = 0. Therefore,   n (x) =A sin x (2.35) LChapter 2: Review of Quantum Mechanics 12 for integer n. Allowed energies 2 2 2 h   n E = (2.36) n 2 2mL In a 3D box of side L, the energy eigenfunctions are:       n  n  n  x y z (x) =A sin x sin y sin z (2.37) L L L and the allowed energies are: 2 2   h   2 2 2 E = n +n +n (2.38) n x y z 2 2mL 2.5 Harmonic Oscillator 2 2 h  1 2 H = + kx (2.39) 2 2m x 2 q 1=4 1=4 Writing = k=m, p =p=(km) , x =x(km) ,   1 2 2 H = p +x (2.40) 2 p; x =ih  (2.41) Raising and lowering operators: p a = (x +ip )= 2h p y a = (x ip )= 2h (2.42) Hamiltonian and commutation relations:   1 y H = h  a a + 2 y a;a = 1 (2.43) The commutation relations, y y H;a =h  aChapter 2: Review of Quantum Mechanics 13 H;a =h  a (2.44) imply that there is a ladder of states, y y HajEi = (E +h  )ajEi HajEi = (Eh  )ajEi (2.45) This ladder will continue down to negative energies (which it can't since the Hamil- tonian is manifestly positive de nite) unless there is an E  0 such that 0 ajEi = 0 (2.46) 0 Such a state has E =h  =2. 0 y We label the states by their a a eigenvalues. We have a complete set of H eigen- states,jni, such that   1 Hjni =h  n + jni (2.47) 2 y n y and (a )j0i/jni. To get the normalization, we write ajni =cjn + 1i. Then, n 2 y jcj = hnjaajni n = n + 1 (2.48) Hence, p y ajni = n + 1jn + 1i p ajni = njn 1i (2.49) 2.6 Double Well 2 2 h  H = +V (x) (2.50) 2 2m xChapter 2: Review of Quantum Mechanics 14 where 8 1 ifjxj 2a + 2b V (x) = 0 if bjxja +b : V ifjxjb 0 Symmetrical solutions: 8 0 A coskx ifjxjb (x) = (2.51) : cos(kjxj) if bjxja +b with s 2mV 0 0 2 k = k (2.52) 2 h  The allowed k's are determined by the condition that (a +b) = 0:   1  = n + k(a +b) (2.53) 2 the continuity of (x) atjxj =b: cos (kb) A = (2.54) 0 coskb 0 and the continuity of (x) atjxj =b:    1 0 0 k tan n + ka =k tankb (2.55) 2 0 If k is imaginary, cos cosh and tani tanh in the above equations. Antisymmetrical solutions: 8 0 A sinkx ifjxjb (x) = (2.56) : sgn(x) cos(kjxj) if bjxja +b The allowed k's are now determined by   1  = n + k(a +b) (2.57) 2 cos (kb) A = (2.58) 0 sinkbChapter 2: Review of Quantum Mechanics 15    1 0 0 k tan n + ka =k cotkb (2.59) 2 Suppose we have n wells? Sequences of eigenstates, classi ed according to their eigenvalues under translations between the wells. 2.7 Spin The electron carries spin-1=2. The spin is described by a state in the Hilbert space: j+i + ji (2.60) spanned by the basis vectorsji. Spin operators: 0 1 0 1 1 B C s = A x 2 1 0 0 1 0 i 1 B C s = A y 2 i 0 0 1 1 0 1 B C s = (2.61) A z 2 0 1 Coupling to an external magnetic eld: H =g sB (2.62) int B States of a spin in a magnetic eld in the z direction: g Hj+i =  j+i B 2 g Hji =  ji (2.63) B 2 2.8 Many-Particle Hilbert Spaces: Bosons, Fermions When we have a system with many particles, we must now specify the states of all of the particles. If we have two distinguishable particles whose Hilbert spaces areChapter 2: Review of Quantum Mechanics 16 spanned by the bases E i; 1 (2.64) and E ; 2 (2.65) Then the two-particle Hilbert space is spanned by the set: E E E i; 1; ; 2  i; 1 ; 2 (2.66) Suppose that the two single-particle Hilbert spaces are identical, e.g. the two particles are in the same box. Then the two-particle Hilbert space is: E E E i;j  i; 1 j; 2 (2.67) E E If the particles are identical, however, we must be more careful. i;j and j;i must be physically the same state, i.e. E E i i;j =e j;i (2.68) Applying this relation twice implies that E E 2i i;j =e i;j (2.69) i so e =1. The former corresponds to bosons, while the latter corresponds to fermions. The two-particle Hilbert spaces of bosons and fermions are respectively spanned by: E E i;j + j;i (2.70) and E E i;j j;i (2.71) The n-particle Hilbert spaces of bosons and fermions are respectively spanned by: E X i ;:::;i (2.72) (1) (n) Chapter 2: Review of Quantum Mechanics 17 and E X  (1) i ;:::;i (2.73) (1) (n)  In position space, this means that a bosonic wavefunction must be completely sym- metric: (x ;:::;x;:::;x ;:::;x ) = (x ;:::;x ;:::;x;:::;x ) (2.74) 1 i j n 1 j i n while a fermionic wavefunction must be completely antisymmetric: (x ;:::;x;:::;x ;:::;x ) = (x ;:::;x ;:::;x;:::;x ) (2.75) 1 i j n 1 j i nChapter 3 Review of Statistical Mechanics 3.1 Microcanonical, Canonical, Grand Canonical Ensembles In statistical mechanics, we deal with a situation in which even the quantum state of the system is unknown. The expectation value of an observable must be averaged over: X hOi = whijOjii (3.1) i i where the statesjii form an orthonormal basis ofH andw is the probability of being i P in statejii. Thew 's must satisfy w = 1. The expectation value can be written in i i a basis-independent form: hOi =TrfOg (3.2) P where  is the density matrix. In the above example,  = wjiihij. The condition, i i P w = 1, i.e. that the probabilities add to 1, is: i Trfg = 1 (3.3) We usually deal with one of three ensembles: the microcanonical emsemble, the canonical ensemble, or the grand canonical ensemble. In the microcanonical ensemble, 18

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