• (a) a solid with delocalized electron states,
• (b) e.g., conducting polymer, benzene pi electrons, Li metal
• (c) obtain the one electron delocalized electron states involving N atomic orbital basis AO(x-R) (where R represents the atom coordinates and x those of the electron) i.e., the so called "Tight Binding Approximation":

 II. What the students do know at this stage of the course:

• wave-particle duality for a free electron moving in a given space, say the macroscopic dimensions of a ring of radius r, requires that the wave associated with the particle fit an integer number (m) of times in the space of motion, the circle. The deBroglie wave length gives the momentum p in the direction of motion,
• the wave function associated with the free electron moving on the circumference (call this the x-direction) gives the coefficients C(R) where the distance on the arc x(R)= r * angle identifying the position R in the macroscopic ring:

(c1) the real TB orbitals for pi states in benzene and butadiene using the C(2p_z) AO as basis,

(c2) show that the energy for these states become continuous to within k_BT in a band as the number of atoms in the polyene increases to say 100. Use the relation:

<TB_m|H|TB_m> = E₀(AO) + Coulomb Term (U)

+ Resonance Term t(a) Sum_R{C_m(R) C_m(R-1)+C_m(R) C_m(R+1)}/N

  The resonance term is the most important term, it can be evaluated semi-empirically by spectroscopy (e.g., benzene). It depends on the bond distance a as:

(c3) use trigonometry to show that:

Sum_R{C_m(R) C_m(R-1)+ C_m(R) C_m(R+1)}/N = (SQC4)

i.e., the energy of the TB state (say N=100) depends on finite m as a parabola:

(c4) Show that the number of states filled (N/2) each with two electrons in a state (m) is related to the volume occu pied by the system V. In 1D the space of motion is the circumference and the highest filled orbital is identified by the Fermi momentum k_F =1/2 1/a:

In 3D the highest filled state identifies the Fermi momentum k_F:

2 (dp_xdx)(dp_ydy)(dp_zdz)/ h³ > # e in space (dp_xdx)(dp_ydy)(dp_zdz).

1. Given

• (a) a solid has symmetry,
• (b) e.g., conducting polymer, benzene pi electrons, Li metal
• (c) obtain the reciprocal space

 III. We obtain the reciprocal space b₁,b₂,b₃ by the definition:

  and the others are obtained by rotation 1=>2=>3=>1 in the relation (SQC7).

The reciprocal lattice point (h,k,l) is then built with the unit vectors b₁,b₂,b₃. The Brillouin Zone BZ is obtained by the space bound by b₁/2,b₂/2,b₃/2 as an analogy of the one dimensional metal that has the highest filled state at k_F=1/2 1/a.

the fcc lattice with translation vectors:

a₁ =1/2,1/2,0; a₂ =0,1/2,1/2; a₃ =1/2,0,1/2

and the hcp lattice with translation vectors:

a₁ =1/2,3^1/2/2,0; a₂ =1/2,-3^1/2/2,0; a₃ =0,0,1

I.Given

• (a) a solid with a given symmetry,
• (b) e.g., conducting polymer, benzene pi electrons, Li metal
• (c) obtain the Bragg Condition in reciprocal space

II. Every lattice has unit vectors that define the periodicity in 3D in units of the principal axes a,b,c:

 a₁,a₂,a₃ with a reciprocal lattice b₁,b₂,b₃.

 This means that:

or that the scattering vector:

is a vector of the reciprocal lattice.

The reflection condition is extremely simple. All that a student needs to do is:

then take their difference. If the sphere passes through two reciprocal lattice points then the Bragg Condition is satisfied, i.e., the magnitude of the scattering vector is:

Figure 1. Unit vectors

I.Given

• (a) a solid with a given symmetry,
• (b) e.g., conducting polymer, benzene pi electrons, Li metal
• (c) obtain the expansion for the potential energy in the reciprocal space

II. Every lattice has unit vectors that define the periodicity in 3D in units of the principal axes a,b,c:

a₁,a₂,a₃ with a reciprocal lattice b₁,b₂,b₃.

III. The Fourier expansion of the lattice potential in the reciprocal lattice basis b₁,b₂,b₃ is:

where

g = h b₁+k b₂+l b₃

and

V_g= V(x) e^{-2 i x.g} d³x, (SQC13)

 and if x = (x₁,x₂,x₃) are the direct lattice positions of the atoms in the unit cell (in units of (a₁,a₂,a₃) then in (SQC13):

x.g= x₁ h + x₂ k + x₃ l

  is independent of the magnitude of a,b,c. The physical significance is the Bloch Theorem. The wave function is given in a Fourier expansion:

and

 The Schrödinger relation follows from the potential expansion (SQC12)

 0= {-/2 + V(x)-E} PSI_k =

SUM_g {1/2(2g+k)² + V₀₀₀ - E} C_g e ^{ix.( 2 g+k)}

+ SUM_g",g' {V_g' } C_g" e ^{ix.( 2 (g"+g')+k)} = 0. (SQC14)

 The sum of terms with equal exponents must vanish, i.e., when g=g"+g':

is the "so called" Secular Equation SE from which the coefficients C_g are obtained.

For free electrons in (SQC14) in atomic units:

 E⁰_k = V₀₀₀ + k²/2

then

 defines the planes confining the space where the electrons move in reciprocal space called the Brillouin Zone. For nearly free electrons:

 |V_g|<<<V₀₀₀,

the coefficients are obtained from:

 {1/2(2g+k)² + V₀₀₀ - E⁰_k} C_{g = -} {V_g } C₀₀₀

or

2C_g = V_g/{|g|² + g.k}

 except when the denominator {|g|² + g.k} vanishes, otherwise:

At the edge of the BZ the degeneracy (say E₁ = E₂) indicated by (SQC15) is resolved by solving a simple SE for non-vanishing coefficients C₁, C₂, i.e.,

(E-E₁) C₁ + V_g C₂ = 0

V_g C₁ +(E-E₂) C₂ = 0, (SQC17)

E =(E₁+E₂)/2 {((E₁-E₂)/2)² + |V_g|²}^1/2, (SQC17)

 E = E₀ -U -

4 t{cos(1/2 k_ya) cos(1/2 k_xa)+cos(1/2 k_za) cos(1/2 k_xa)+cos(1/2 k_ya) cos(1/2 k_za)}

(c2) obtain the approximation for small k.

(c3) For what values of k is E at a minimum if AO is s-type, p_z type along the bond?

(c4) What would E be for a body centered cubic lattice?

I.Given

• (a) a solid with a given symmetry,
• (b) e.g., conducting polymer, benzene pi electrons, Li metal
• (c) obtain the macroscopic properties

II. Every lattice has unit vectors that define the periodicity in 3D in units of the principal axes a,b,c: a₁,a₂,a₃ with a reciprocal lattice b₁,b₂,b₃ and corresponding energy bands in the first BZ.

III. All thermodynamic and transport properties must be considered:

v =1/ (E(k)/ k). (SQC19)

ev.F = e/ (E(k)/ k).F,

or,

eF/ = dk/dt. (SQC20)

The loss per unit time of electrons in state k due to collisions a time apart is:

n(k)/

the gain from transitions into the state are n_av/, so the net loss is:

-dn(k)/dt = (n(k) - n_av)/

and in the absence of an external field:

n(k,t) = n_av + {n(k) - n_av}exp(-t/).

In the presence of an electric field in the x-direction the electrons move with a rate of change:

-eF_x/ (n(k)/ k_x)

giving the Boltzmann Equation:

{n(k) - n_av}/+ eF_x/ (n(k)/ k_x)=0. (SQC20')

  By analogy a magnetic field H changes k at the rate:

dk/dt =e/c v ^ H. (SQC21)

It should be noted however that in a magnetic field H the electron moves in closed orbits normal to the direction of the field and the energy levels will be discrete.

2 f(E) = 2/{1+exp(E-)/k_BT}, (SQC22)

 where is the chemical potential. The total free energy is then:

and the Fermi energy E= is determined such that:

N_e = 2 SUM_i f(E_i)

E(T) = 2 SUM_i E_i f(E_i)= E f(E) dZ/dE dE

= E₀ + (k_BT)² /3 (dZ/dE)_{(evaluated at )}, (SQC24)

where is the Fermi energy, Z(E) is the number of electrons with energy less than E.

dE(T)/dT = 2/3 k_B {k_BT (dZ/dE)_{(evaluated at )}}. (SQC25)

E = E₀ - k²/2m_eff

N(E) dE = (2m_eff)^3/2 (E-E₀)^1/2 /4 dE, (SQC26)

or,

N(E) = 1/|GRAD E| dS/(2)³ (SQC26')

where the integral is over a surface where the energy is a constant E.

V= SUM_l 1/2 K (u_l - u_l+1), (SQC27)

which requires that:

M ü_l = - SUM_l 1/2 K (u_l - u_l+1), (SQC28)

 where K is the force constant in the harmonic oscillator problem.

Let q=2/ when is the periodicity of the distortion. The general solution to (SQC28) is:

u_l =U_q exp(iq.l), (SQC29)

- /a<q</a and U_q is the amplitude of the motion. Hooke's Law is applied to the entire chain:

M Ü_q exp(iq.l)= - SUM_l 1/2 K (u_l - u_l+1)

M Ü_q = - 2K (1-cos(q.a))U_q, (SQC29')

which is the simple harmonic oscillator relation with a dispersion of frequencies:

2v_q = (K/M)^1/2 2 sin(q.a/2). (SQC30)

Lattice waves with different masses are easily written:

M₁ Ü₁ = - 2K U₁+ 2 K U₂ cos(q.a),

M₂ Ü₂ = 2 K U₁ cos(q.a)- 2K U₂. (SQC31)

The SE for the non-vanishing amplitudes has two roots:

2v = K{(1/M₁+1/M₂) {(1/M₁+1/M₂)²-4 sin² (q.a).M₁M₂}^1/2. (SQC30')

obtains:

M = (N₊ - N_-)= -(A/H)_M = 2 H (dZ/dE), (SQC32)

and if we neglect the interaction between electrons, the magnetic susceptibility:

X = (M/H)

In classical statistics:

where H is the Hamiltonian and the kinetic energy includes the vector potential, i.e.,

For a field H_z in the z direction, the motion along x obtains:

which is used to solve the Schrödinger equation. PSI is simple along the y, z directions, i.e.,

PSI_k = exp(i(k_yy+k_zz)) u(x)

-u"(x)/2 -{E₁ -(k_y-exH_z/c)²}u(x)= 0, (SQC35)

E₁=E₀ - (k_z)²/2m_e

is the eneargy of motion in the transverse plane. The solution to (SQC35) obtains:

E(n, k_z) = (2n + 1) H_z + (k_z)²/2m_e. (SQC35")

The number of states of energy less than E is then:

this then gives the Helmholtz free energy:

A = N_e -2 k_BT SUM_i ln{1+exp(E_i-)/k_BT}=

N_e -2 k_BT dE dZ/dE ln{1+exp(E_i-)/k_BT}

= N_e -2 dE Z(E) f(E). (SQC23')

 The Helmholtz free energy when E=₀>> k_BT and V = L₁ L₂ L₃ is:

A = N_e -128/3 V m_e^3/2/h³ E^1/2{0.4 E² - 1/16 (H_z )²}. (SQC23")

The effect of a periodic field on the electron in the nth cell is to change TB(x) to:

The most important result is that in the presence of a magnetic field H_z, the motion along x is described by p_x but along y it is described by p_y -i e H_z/c which does not commute with p_x, i.e., the particle velocity in a magnetic field must include the vector potential contribution and is written:

The particle flux is obtained from the number of particles moving through the unit volume per unit time:

where is the phase for a free particle (PSI = |PSI| exp (i)), the current density is:

Among the important properties observed in metals in a magnetic field are:

The effect of the field on the current then is to add a term to the Boltzmann relation:

to give:

Layer compounds such as the superconducting organic metals and cuprates with different conductivity in each of the different layers (e.g., anion layers CuO₂^-2, CuO, Ta₂F₁₁^-, and cation layers M⁺, M⁺³ and M⁺²) are very likely to exhibit magneto-resistance effects. They can be detected in an esr spectrometer (see Acrivos, 1996).

{e^-(s_z, k)+e^-(s_z', k')+Coupled Via Lattice Phonons}e₂⁼(S=s_z+s_z', K=k+k'), (SQC37)

with an equilibrium constant:

The chemical equilibrium takes into account the lattice phonons in the activity a of the respective particles. The change in the Gibbs free enthalpy in (SQC37') may be evaluated: