Equations Flashcards

1
Q

Momentum

A

p=hbark

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2
Q

Energy of free electron

A

E=hbar^2k^2/2m

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3
Q

Schrodinger eqn

A

H|ψ>=E|Ψ>, H|C>=E|C>

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4
Q

Momentum operator

A

P=-ihar*laplacian

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5
Q

Hamiltonian

A

H=1/2mP(operator)^2+V=-habr^2/2m*Δ^2+V

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6
Q

Hamiltonian matrix elements

A

=<Hi|H|Hj>

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7
Q

2nd order perturbation theory

A

E = E0+<ψn|H|ψn>+Σn’≠n<ψn|Hpert|ψn’>/εn-εn’

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8
Q

LCAO for diatomic molecule

A

(E1 0 0 E2)(C1 C2) = E (1 S S 1) (C1 C2)

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9
Q

Energy for LCAO when E1=E2=E0

A

E = Eo±|U|

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10
Q

(C1 C2) when E1=E2

A

1/root(2)*(1 ±1)

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11
Q

E for LCAO when E1≠E2

A

E=Eav±root(Ecov^2+Eionic^2)

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12
Q

Ecov

A

Ecov=2|U|

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13
Q

Eionic

A

=|E1-E2/2|

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14
Q

Eav

A

=E1+E2/2

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15
Q

Bloch Theorem

A

ψ(r+a) = exp(ikr)ψ(r)

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16
Q

Bloch function

A

ψn(r)=exp(ikr)un(r)

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17
Q

average p

A

<p>=<ψnk|P|ψnk> = hbark + <Un|p|Un> = Pcrystal ( + Pnn atomic part which =0 when definite parity)
</p>

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18
Q

Effective mass

A

1/m*=1/hbar^2d^2E/Dk^2=1/m(1+Ep/Eg)

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19
Q

Ep (energy of kane momentum)

A

Ep=2Pcv^2/m

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20
Q

Energy of dipole in magnetic field

A

Hso proportional σΔV*P

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21
Q

Vg Group velocity

A

Vg=dω/dk=1/hbarΔE

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22
Q

acceleration with group velocity

A

a = dVg/dt=M^-1*F

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23
Q

M^-1

A

M^-1=1/hbar^dE/dkidkj

24
Q

Important Lattice Points

A

L = 1/2[111]2π/a
Γ = [000]
X = [100]2π/α

25
Bowing
E(x) = EA(1-x)+EBx+bx(1-x)
26
State radius
an=a0n^2
27
binding energy
En=-Er/n^2
28
Number of states per unit volume
g(E)dE=g(k)dK
29
3D density of states
g(E) proportional m*^3/2E^1/2
30
2D density of states
g(E) proportional m*E^0
31
1D density of states
g(E) proportional m*^1/2E^-1/2
32
boltzmann occupation function
fB(E) proportional exp(-E/kT)
33
Fermi occupation function
f(E) = 1/(1+exp((E-Ef)/kt)
34
Number of electrons per unit volume
n=Nc*exp((Ef-Ec)/kt)
35
Nc
Nc=2(me*kT/2πhbar^2)^3/2
36
Number of holes per unit volume
p = Nv*exp((Ev-Ef)/kT)
37
Law of mass action
np=ni^2=NcNv*exp(-Eg/kT)
38
Fermis golden rule for number of excitations per unit vol
Rcv = 2π/hbar|/2|^2*gcv(e)
39
Reduced Mass
1/μ = 1/mc*+1/mv* (replace m* with μ in DOS formulas for joint density of states)
40
Intensity
I = F^2/2Z0 = hbarωφ
41
wavelength in terms of k
λ= 2π/k=2πc/ω=hc/E
42
current density
J = σε
43
Drift velocity
vd= με (μ = mobility)
44
mobility
μ = qτ/μ*
45
conductivity
σ= 1/ρ = neμ = nq^2τ/m*
46
Lorentz Force
F = q(ε+vg x B)
47
Hall Resistance
Rh=1/nq
48
cyclotron frequency
ωc = qB/m*
49
Landau level energy
Ε = hbar^2*kz^2/2m*+(l+1/2)hbarωψ
50
Lattice Scattering
μl proportional T^-3/2
51
impurity scattering
μi proportional T^3/2
52
1/total scattering
1/μ = 1/μl+1/μi
53
potential
dφ(x)/dx = ε(x)
54
Gauss' law for charge density in 1D
dε(x)/dx = 1/ε*ρ(x)
55
Electric field
ε = F/d = V/d (uniform) = Q/4πε0r^2 (radial)
56
Magnetic field
B = μ0I/2πr
57
Magnetic force
F = qvBsin(90) = m*v^2/r (centripetal force)