Electronic properties

starting from all unknowns from tripos q's + key eqns

bravais lattice

• translational periodic array of lattice points

• all lattice points indisinguishable

• each lattice point has motif of atoms/molecules attached

unit cell

parallelogram defined by lattice vectors

tessellates plane - periodic translation covers plane

primitive unit cell

contains 1 lattice point

unit cell volume in 3D

$$a_1 \cdot (a_2 \times a_3)$$

how is the NaCl structure formed?

fcc

parameter a

motif: Na+ (0, 0, ½ a)

Cl- (0, 0, 0)

Hexagonal net: how to construct

motif: (0,0), (0, $\frac{a}{\sqrt{3}}$)

diamond structure

($\frac{1}{4} a$, $\frac{1}{4} a$, $\frac{1}{4} a$)

lattice energy

Energy to go from solid lattice $\rightarrow$ widely separated gaseous ions

see derivation

assumptions of 1D FEG

ignore all interactions

no boundaries

“free particle” = no PE

describe as ideal gas (collisions etc)

Wavefunction and energy for FEG.

What is this wavefunction form often called?

How can the real and imaginary parts of this wavefunction be plotted?

$\psi = e^{ikx}$ (PLANE WAVE)

Re = cos(kx)

Im = isin(kx)

$$E = \frac{\hbar^2 k^2}{2m_e}$$

$$k = \frac{2\pi}{\lambda}$$

momentum of a plane wave in 1D and 3D

$$-i\hbar\frac{\partial}{\partial x} e^{ikx} = k\hbar e^{ikx}$$

1D: $\hbar k$

3D: $\hbar(k_x, k_y, k_z)$

how can we easily construct a wavefront in real space?

split into $\lambda_x$ and $\lambda_y$

How do we quantise FEG?

BvK boundary conditions: CYCLIC

dispersion curve for FEG

energy NOT QUANTISED!!!

fermi-dirac distribution

$$f(E) = \frac{1}{e^{(E-E_F)/kT} +1}$$

Drude model drift velocity + conductivity

$$v_d = \frac{e\Epsilon\tau}{m_e}$$

$$E_f$$

$$E_f = \frac{\hbar²k_f^2}{2m_e}$$

$$e^{ikx} + e^{-ikx}$$

$$2\cos{(kx)}$$

$$e^{ikx} - e^{-ikx}$$

$$2\sin{(kx)}$$

1D band energy using Huckel approx. (derive)

$$\alpha + 2\beta \cos{(ka)}$$

for $-\pi/a \leq k \leq +\pi/a$

See supo q 20

2D band energy using Huckel approx

$$\alpha + 2\beta[\cos{(k_y a)} + \cos{(k_x a)}]$$

effective mass eqn

$$\frac{1}{m_e} = \frac{1}{\hbar^2}\frac{\text{d}^2 E_k}{\text{d} k^2}$$

CURVATURE!!!!

What does an infinite effective mass imply?

e^- not accelerated by applied e-field

$$R_{\text{eff}}$$

$$R_H \frac{m_e*}{m_e \epsilon_r^2}$$

$E_n$ for a donor/acceptor level

$$E_n = \frac{(-)R_{\text{eff}}}{n^2}$$

DONOR: E -ve (E = 0 @ CB bottom).

n = 1 most - ve E

ACCEPTOR: E +ve (E = 0 @ VB top).

n = 1 most + ve E

$$a_{0, \text{eff}}$$

$$a_{0,\text{eff}} = a_0 \frac{\epsilon_r m_e}{m_e*}$$