Phys 446 Solid State Physics Summary Last Lecture Lecture 8 ™ Free electron model – simplest way to describe electronic properties of (Ch. 4.9 – 4.10, 5.1-5.6) metals: the valence electrons of free atoms become conduction electrons in crystal and move freely throughout the crystal.

Last time: Discussed the free electron (Drude) model applied to electronic specific heat and electrical conductivity.

Today: Finish with Free electron model. Thermal conductivity. Motion in magnetic field: cyclotron resonance and Hall effect Start new chapter: energy bands in solids

Summary Last Lecture (continued)

™ Fermi energy - energy of the highest occupied electronic level at T = 0 K; 3D case: 2 3 1 3 1 3 =2 ⎛ 3π 2 N ⎞ ⎛ 3π 2 N ⎞ = ⎛ 3π 2 N ⎞ ⎜ ⎟ k = ⎜ ⎟ v = ⎜ ⎟ EF = ⎜ ⎟ F ⎜ ⎟ F ⎜ ⎟ 2m ⎝ V ⎠ ⎝ V ⎠ m ⎝ V ⎠ ™ Density of states of 3D free electron gas: ™Electrical conductivity: σ 3 2 ne2τ dN V ⎛ 2m ⎞ 1 2 3N = − D(E) = = 2 ⎜ 2 ⎟ E = m* dE 2π ⎝ = ⎠ 2E ρ = ρ + ρ (T) ™ Heat capacity of free electron gas i ph at low temperatures kBT << EF : ™Thermal conductivity: Wiedemann-Franz law 2 π 2 ⎛ k ⎞ K = LσT L = ⎜ B ⎟ 3 ⎝ e ⎠

Motion in a magnetic field: cyclotron resonance Applied magnetic field →the Lorentz force: F = −e[E+(v × B)] Perfect metal, no electric field - the equation of motion is:

Let the magnetic field to be along the z-direction. Then dv dv x = −ω v y eB c y = ωcvx where ω = - cyclotron frequency dt dt c m For moderate magnetic absorption fields (~ few kG),

ωc ~ few GHz. e.g. for B = 0.1 T,

fc = ωc/2π = 2.8 GHz

Cyclotron resonance – peak in absorption of electromagnetic waves at ωc Used to measure the effective mass in metals and semiconductors

Summary of free electron model Limitations of free electron model ™ Free electron model – simplest way to describe electronic properties of The free electron model gives a good insight into many properties of metals: the valence electrons of free atoms become conduction electrons in metals, such as the heat capacity, thermal conductivity and electrical crystal and move freely throughout the crystal. conductivity. However, it fails to explain a number of important ™ Fermi energy - the energy of the highest occupied electronic level at T = 0 properties and experimental facts, for example: K; 2 3 1 3 1 3 =2 ⎛ 3π 2 N ⎞ ⎛ 3π 2 N ⎞ = ⎛ 3π 2 N ⎞ ⎜ ⎟ k = ⎜ ⎟ v = ⎜ ⎟ • the difference between metals, semiconductors and insulators EF = ⎜ ⎟ F ⎜ ⎟ F ⎜ ⎟ 2m ⎝ V ⎠ ⎝ V ⎠ m ⎝ V ⎠ •It does not explain the occurrence of positive values of the Hall ™ Density of states of 3D free electron gas: ™Electrical conductivity: coefficient. σ 3 2 ne2τ dN V ⎛ 2m ⎞ 1 2 3N = − •Also the relation between conduction electrons in the metal and the D(E) = = 2 ⎜ 2 ⎟ E = m* dE 2π ⎝ = ⎠ 2E number of valence electrons in free atoms is not always correct. ρ = ρi + ρph(T) Bivalent and trivalent metals are consistently less conductive than the ™ Heat capacity of free electron gas monovalent metals (Cu, Ag, Au) ™Thermal conductivity: at low temperatures kBT << EF : Wiedemann-Franz law ⇒ need a more accurate theory, which would be able to answer these 2 π 2 ⎛ k ⎞ questions – the band theory K = LσT L = ⎜ B ⎟ 3 ⎝ e ⎠

The problem of electrons in a solid – a many-electron problem Bloch theorem Write the Schrödinger equation the approximation of non-interacting The full Hamiltonian contains not only the one-electron potentials electrons: describing the interactions of the electrons with atomic nuclei, but also pair potentials describing the electron-electron interactions The many-electron problem is impossible to solve exactly ⇒ ψ(r) – wave function for one electron. simplified assumptions needed Independent electrons, which obey a one-electron Schrödinger The simplest approach we have already considered - a free electron equation a periodic potential U(r) = U(r+ T) - Bloch electrons model Bloch theorem: the solution has the form where The next step is an independent electron approximation: assume that all the interactions are described by an effective potential. uk(r)= uk(r+T) - a periodic function with the same period as the lattice One of the most important properties Bloch theorem introduces a wave vector k, which plays the same of this potential - its periodicity: fundamental role in the motion in a periodic potential that the free U(r) = U(r+ T) electron wave vector k plays in the free-electron theory. ħk is known as the crystal momentum or quasi-momentum Another conclusions following from the Bloch theorem: Number of states in a band the wave vector k can always be confined to the first Brillouin zone The number of states in a band within the first Brillouin zone is equal This is because any k' not in the first Brillouin zone can be written as to the number of primitive unit cells N in the crystal. k' = k + G Then, if the Bloch form holds for k', it will also hold for k Consider the one-dimensional case, periodic boundary conditions. Energy bands Allowed values of k form a uniform mesh whose unit spacing is 2π/L Substitute the solutions in the Bloch form into the ⇒ The number of states inside the first zone, whose length is 2π/a, is (2π/a)/(2π/L) = L/a = N, where N is the number of unit cells Schrodinger equation, obtain:

A similar argument may be applied in 2- and 3-dimensional cases. with periodic condition: uk(r)= uk(r+T) For any k, find an infinite number of solutions with discrete energies Taking into account two spin orientations, conclude that there are 2N En(k), labeled with the band index n independent states (orbitals) in each energy band.

For each n, the set of electronic levels specified by En(k) is called an energy band. The information contained in these functions for different n and k is referred to as the band structure of the solid.

Nearly free electron (weak binding) model Now, turn on a weak potential. Consider it as a weak periodic First step: empty-lattice model. When perturbation in Hamiltonian. the potential is zero the solutions of the From perturbation theory have: 2 Schrödinger equation are plane waves: ψ j,k' U i,k E (k) = E 0 (k) + 0 U ψ 0 + 2 2 1 i i i,k i,k ∑ 0 0 0 = k 0 ik⋅r k ', j Ei (k) − E j (k) E (k) = ψ k (r) = e V 2m0 c where index i refers to ith band; 0 refers to empty-lattice model. where the wave function is normalized The first term is the undisturbed free-electron to the volume of unit cell Vc value for the energy. The second term is the mean value of the potential in the state i, k – constant independent of k – can set to zero The third term – the 2nd order correction – vanishes except k' = k + G The third term can be rewritten as There are special k points for which the energy levels become degenerate and the relationship holds for any non-zero value of the potential:

UG - Fourier transform of the crystal potential U

Finally we obtain for the energy: This conduction implies that k must lie on a Bragg plane bisecting the line joining the origin of k space and the However, the perturbation theory cannot be applied when the reciprocal lattice point G potential cannot be considered as a small perturbation This happens when the magnitude of the potential becomes comparable with the energy separation between the bands, i.e. ⇒ a weak periodic potential has its major effect on those free electron levels whose wave vectors are close to ones at which the Bragg reflection can occur. In this case we have to solve the Schrödinger equation explicitly

In order to find the energy levels and the wave functions near these Illustrate this behavior using a one-dimensional lattice points we need to invoke the degenerate perturbation theory. The splitting of the bands at each Bragg plane in the extended-zone The result: scheme results in the splitting of the bands both at the boundaries and at the centre of the first Brillouin zone.

This results is particularly simple for point lying on the Bragg plane:

Obtain

The magnitude of the band gap is equal to twice the Fourier component of the crystal potential. Intermediate Summary Origin of the energy gaps ™ The Bloch theorem: the wave function for an electron in periodic potential can be written in the form: where

uk(r) = uk(r+T) - a periodic function with the period of the lattice

™ The energy spectrum of electrons consists of a set of continuous energy bands, separated by regions with no allowed states - gaps

™ Function E(k) satisfies the symmetry properties of a crystal, in particular, the translational invariance: E(k) = E(k +G) This allows considering the first Brillouin zone only. We focused on the energy values Also, inversion symmetry: E(k) = E(-k) got away from the zone edges ™ Nearly free electron model – weak crystal potential. Electron behaves essentially as a free particle, except the wave vectors close and to the boundaries of the zone. near the zone edges. In these regions, energy gaps appear: Eg = 2|UG|

Now, let's see how the wave functionsψ are modified by (weak) crystal For simplicity we consider a one-dimensional lattice, for which the potential. From the perturbation theory, have for the first band zone edges are k=½G = π/a (Bragg reflection occurs) ψ (away from the zone edge): π The result: π 0 U 0 G 1 0 ψ 0 1 i()a x −i()π a x ψ 1(k) = 1 (k) + ψ 2 (k) 0 0 ψ ± = 1/ 2 []1 ( a) ± 2 (π a) = 1/ 2 ()e ± e E1 (k) − E2 (k) L L Here, again, we leave only the nearest-band (2nd) term, as we did for at the zone edge, the scattering is so strong that the reflected wave the energy levels – because of the large denominator for higher bands. has the same amplitude as the incident wave ⇒ the electron is 0 0 ikx represented there by a standing wave, unlike a free particle Functions ψ (k) are those of free electrons: ψ 2 (k) ~ e 0 2 If k is not close to the zone edge, the coefficient of ψ 2 (k) is very small The distribution of the charge density is proportional to |ψ| , so that

So, 2 π 0 1 ikx ψ ~ cos x - higher ψ (k) ≈ψ (k) = e - behaves like free-electron + 1 1 L1/ 2 a energy However, near the edge, the denominator becomes very small – 2 π - lower must use the degenerate perturbation theory – the functions ψ − ~ sin x 0 0 are treated equally a energy ψ 1 (k) and ψ 2 (k) Tight binding model Assumptions: – atomic potential is strong, electrons are tightly bound to the ions

– the problem for isolated atoms is solved: know wave functions φn and energies En of atomic orbitals – weak overlapping of atomic orbitals

Start with 1D case N 1 ikX Bloch function in the form: (k, x) e j (x X ) ψ = 1/ 2 ∑ φn − j N j=1 th where Xj = ja – position of the j atom, a – lattice constant; th ψn(x- Xj) – atomic orbital centered around the j atom – large near Xj , but decays rapidly avay from it. Small overlap exists only between the neighboring atoms φ

(N −1)/ 2 φ The function chosen satisfies the Bloch theorem: got ikX j E(k) = e n (x) H φn (x − X j ) N N φ ∑ 1 ikX 1 −ik (x− X ) j=− N / 2 ψ (k, x) = e j (x − X ) = eikx e j φ (x − X ) φ N 1/ 2 ∑ n j N 1/ 2 ∑ n j j=1 j=1 separate the term j = 0 from the others: periodic function φ E(k) = (x) H (x) + ' eikX j (x) H φ (x − X ) near X , ψ(k,x) ≈ eikXj φ (x-X) ~ φ (x-X) - behaves like atomic orbital n n ∑ n n j j n j n j  j energy of electron  includes tunneling effects The energy of the electron described by ψ(k)isE(k) = 〈ψ(k)|H|ψ(k)〉 in an isolated atom φ significant only for j = ±1

1 ik ( X − X ) Obtain j j ' 2 2 E(k) = e n (x − X j' ) H φn (x − X j ) = d ∑ Write the Hamiltonian H = − +V (x) N j, j' 2 2m0 dx Summation over j, j' covers all the atoms in the lattice. with crystal potential as a sum of atomic potentials: For each j', the sum over j gives the same result ⇒ get N equal terms (N −1)/ 2 φ v(x) – potential due to the ikX V (x) = v(x − X ) = v(x) +V '(x) E(k) = e j (x) H φ (x − X ) (put j' = 0) ∑ j atom in the origin; ∑ n n j j j=−N / 2 V'(x) – due to all the others

φ φ φ φ φ φ Let's consider the interaction term – has two terms in the sum: X = ± a V'(x) is smallφ compared to v(x) near the origin. Return to E(k): φ φ φ φ φ For X = a have n (x) H n (x − a) = E(k) = (x) H (x) + ' eikX j (x) H φ (x − X ) φ n n ∑ n n j 2 2 j = d φ = (x) − + v(x − a) (x − a) + (x) V '(x − a) (x − a) n 2 n n φ n The first term: 2m0 dx 2 2 φ  = d −γ (x) H (x) = (x) − + v(x) (x) + (x) V '(x) (x) n n n 2 n n φ n E n (x) n (x − a) - negligible γ 2m0 dx  n  − β φ overlap integral E − energy of atomic level because of small overlap β n φ = −∫ n *(x)V '(x − a)φn (x − a)dx sign is chosen so that β is positive, β = − *(x)V '(x)φ (x)dx the atomic functions are symmetric ⇒ get the same result for X = -a ∫ n n since V' is negative β φ γ ' ikX j Return to E(k) = En − + ∑ e n (x) H φn (x − X j ) β is small: φn are large onlyβ near the origin, where V' is small j

ika −ika β φ Now have E(k) = En − − (e + e )= En − − 2γ cos ka Have ' ikX j E(k) = En − + ∑ e n (x) H φn (x − X j ) j 2 ⎛ ka ⎞ LetE0 = En − β − 2γ then E(k) = E0 + 4γ sin ⎜ ⎟ ⎝ 2 ⎠ 2 ⎛ ka ⎞ The results obtained can be extended to 3D case. E(k) = E0 + 4γ sin ⎜ ⎟ ⎝ 2 ⎠ For simple cubic lattice, get Original energy level En has broadened into an energy band. ⎡ k a ⎛ k a ⎞ k a ⎤ 2 ⎛ x ⎞ 2 ⎜ y ⎟ 2 ⎛ z ⎞ E(k) = E0 + 4γ ⎢sin ⎜ ⎟ + sin ⎜ ⎟ + sin ⎜ ⎟⎥ The bottom of the band is E0 ⎣⎢ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠⎦⎥ - located at k = 0 −1 ⎛ d 2 E ⎞ Effective mass is The band width = 4γ – proportional to the overlap integral m* = =2 ⎜ ⎟ ⎜ 2 ⎟ determined by the ⎝ dk ⎠ For small k, ka/2 << 1 (near the zone center) curvature of 2 2 dispersion E(k) − E0 = γa k - quadratic dispersion, same as for free electron 2 Generally, anisotropic: =2k 2 = 1 E(k) − E = where m* = - effective mass 0 2m* 2a2 γ ⎛ 1 ⎞ 1 ∂ 2 E = i,j = x, y, z ⎜ ⎟ 2 - inverse effective mass tensor For k = π/2, you will find that −1 ⎝ m* ⎠ij = ∂ki∂k j ⎛ d 2 E ⎞ 2 Generally, m* = =2 ⎜ ⎟ 1 = 1 ⎜ 2 ⎟ 1 ∂E m* = − ⎝ dk ⎠ Velocity of the Bloch electron: v = 3D: v = ∇k E(k) 2a2 γ = ∂k =

Density of states Metals, Insulators, semimetals, semiconductors Number of electronic states per unit energy range (E, E+dE): D(E)dE D(E) – density of states. insulator (semi) metal metal =2k 2 In the effective mass approximation, E(k) = 2m*

Number of states dNstates in the shell (E, E+dE):

π π 3 2 1 2 1 ⎛ 2m* ⎞ 1 2 dN states = 4 k dk = ⎜ ⎟ E dE ()2 3 4π 2 ⎝ =2 ⎠

Taking into account spin, multiply by 2.

Get 3 2 1 ⎛ 2m* ⎞ D(E) = ⎜ ⎟ E1 2 2π 2 ⎝ =2 ⎠ Summary Summary 1 ™ Tight binding model – strong crystal potential, weak overlap. ™Velocity of the Bloch electron: v = ∇ E(k) The band width increases and electrons become more mobile = k (smaller effective mass) as the overlap between atomic wave ™In the presence of an electric field the electron functions increases moves in k-space according to the relation: ™ Concept of effective mass: in a periodic potential electron moves as This is equivalent to the Newton’s second law if we assume that the electron in free space, but with different mass: momentum is equal to ħk −1 2 ⎛ d 2 E ⎞ ⎛ 1 ⎞ 1 ∂ E m* = =2 ⎜ ⎟ ⎜ ⎟ = i,j = x, y, z 2 2 ™Dynamical effective mass: ⎜ dk ⎟ ⎝ m* ⎠ij = ∂ki∂k j ⎝ ⎠ ™ Metals: partially filled bands; insulators – at 0 K the valence band is m* is inversely proportional to the curvature of the dispersion. 2 full, conductance band is empty. In a general case the ⎛ 1 ⎞ 1 ∂ E ⎜ ⎟ = i,j = x, y, z Semiconductors and semimetals. effective mass is a tensor: 2 ⎝ m* ⎠ij = ∂ki∂k j 1 ™p = ħk is called the crystal momentum or quasi-momentum. ™ Velocity of the Bloch electron: v = ∇ E(k) c ψ = k The actual momentum is given by k − i=∇ ψ k remains constant in perfectly periodic lattice 3 2 Can show that p = m v, where m is the free electron mass, 1 ⎛ 2m* ⎞ 1 2 0 0 ™ Density of states. Simple case: D(E) = ⎜ ⎟ E v is given by the above expression 2π 2 ⎝ =2 ⎠

Physical origin of the effective mass Current density dv Free electron model: j = −env ; n - the number of valence electrons Since p = m0v - true momentum, one can write: m = F = F + F 0 dt tot ext L per unit volume, and v - the velocity of electrons. The total force is the sum of the external and lattice forces. Generalize this expression to the case of Bloch electrons.

dv F In this case the velocity depends of the wave vector ⇒ But m = m ext 0 dt 0 m* need to sum up over k vectors for which there are occupied states available: So, we can write Fext m* = m0 Convenient to replace the summation by the integration. Fext + FL The volume of k-space per allowed k value is ∆k = 8π3/V ⇒ we can write the sum over k as The difference between m* and m0 lies in the presence of the lattice force FL

obtain for the current density: Holes Electrical conductivity σ ne2τ Already know that completely filled bands do not contribute to the Free electron model: = − F current m* Therefore, can write: Let's obtain the corresponding expression within the band theory. Same idea: we had for the current density: No electric field - the Fermi sphere is cantered at the origin. The total can equally well write this in the form: current of the system is zero. ⇒ the current produced by electrons occupying a specified set of levels in a band is precisely the same as the current that would be produced if Applied field → the whole Fermiδ sphere is displaced the specified levels were unoccupied and all other levels in the band eE The average displacement is kx = − τ ε were occupied but with particles of charge +e - holes. = ⎛ ∂E ⎞ -ev g( )⎜ ⎟ δk The current density: jx = −evF,xnuncomp. = F ,x F ⎜ ⎟ x Convenient to consider transport of the holes for the bands which are ∂kx σ⎝ ⎠ε F almost occupied, so that only a few electrons are missing. ⎛ ∂E ⎞ τ τ ⎜ ⎟ = =v j = e2 v g(ε )E 1 2 2 ⎜ ⎟ F,x ⇒ x F ,x F F ⇒ = e vF F g(ε F ) ∂kx 3 Can introduce the effective mass for the holes. It has a negative sign. ⎝ ⎠EF

Cyclotron resonance Hall effect Lorentz force: F = −e(v × B) dk We had 1 Equation of motion: Re = − = = −e(v(k)×B) n e dt δ e 1 e analogously, if only holes are present, have Rh = Change in k in a time interval δt: k = − (v(k)×B)δt nhe = δ if both electrons and holes are present = δk Period of cyclic motion: T = t = − ∫ eB ∫ v(k) σ R 2 + R 2 R = e e σh h ω π σ 2 −1 ()e +σ h Generalized cyclotron frequency for 2 eB ⎛ δk ⎞ = ⎜ ⎟ Bloch electron: c = ⎜∫ v(k) ⎟ ⎝ ⎠ You will show this in your next homework When effective mass approximation eB ω = is applicable, v(k) = ħk/m* ⇒ c m* Summary ™ Physical origin of effective mass: crystal field ™ Concept of the hole: consider transport of the holes for the bands, which are nearly occupied. σ

™ Electrical conductivity: 1 2 2τ = e vF F g(ε F ) 3 ω π −1 2 eB ⎛ δk ⎞ ™ Generalized cyclotron frequency: c = ⎜ ⎟ = ⎝ ∫ v(k ) ⎠

™ Hall coefficient for metals with both electrons and holes: σ R 2 + R 2 R = e e σh h σ 2 ()e +σ h