unit 3 Flashcards
all imp formulas
μ=Vd/E, μ=eZ/m (mobility)
μ=1/pne, P=RA/L
Vd=eEz/m
z=λ/vth
E= Vd/μ
λ=vtz
λf=VfZ
Vf=√2Ef/m
J=σE, v=jpl, Vd=j/ne
time btw collision = 1/z
NOTE: take area as 1mm or 2mm, they expect to take dimension
J=neVd
V=Vde^-t/z => imp
t=z, V=Vde^-1
1/2mvth^2= 3/2kT
vth = √3KbT/m
Average velocity of e- due to thermal energy
random motion doesnt contribute to a net drift of electrons
CURRENT DENSITY
J = I/A=(ne^2τ/m)Ε = σΕ
σ=conductivity
σ= (ne^2t)/m = neμ
RESISTIVITY
J/E = σ= (ne^2t)/m = neμ
TOTAL
p= 1/σ= m/(ne^2t)/= 1/neμ
σ = ne^2λ/√3mKbT
electon drift m*dv/dt =
eE − kmv
Merits and demarks of CFET(Classical free electron theory)
CFET (Drude, Lorentz)
they treated electrons as if they were like molecules of an ideal gas
MERITS OF CFET
- explained conduction in terms of free e-
- showed that current is due to drift motion of e-
- showed that resistance is due to obstruction faced by e- due to vibrating atoms
-showed that as T increases J
- showed ohms law is valid
-> free electron gas
DEMARKS
->CFET could not give correct dependence of resistivity(σ) on T(Temp).
but experiment σ ∝ 1/√T
σ ∝ 1/T
2)experimentally values of specific heat of electrons in a metal is 1% of theoretical values
Ce ≈ 10^-4RT
3) Conductivity variations with electron concentrations.
σ= ne^2τ/m
nCu<nAl
8.4x1028/m^3 and Al is 18.1x1028/m^3
σcu>σAl
Copper with lesser electronic concentration has a higher electrical conductivity as compared to Aluminum.
CANNOT MAKE A STATEMENT LIKE THAT
expected linear dependency on the free
electron concentration
But, experimental results show no linear dependence
CFET COULDNT PROVE K/σ = LT
As per CFET resistivity arises due to scattering mechanism with
stationary ionic centers
how does current flow in a metal, explain in terms of drift velocity
the average velocity experienced by the electrons is called drift velocity
In a metal, free electrons move randomly in all directions due to thermal energy.
This random motion does not result in a net flow of charge
When an electric field is applied across the metal the electrons experience a force that causes them to drift in a direction opposite to the electric field (since electrons are negatively charged).
The electrons still move randomly due to thermal motion, but now there is a slight overall bias in their movement, thus net flow of charge. aka drift velocity
FORCE ON e-
Fe=-eE
a = Fe/m a=acceleration
drift velocity
Vd=etE/m= μE
μ=et/m = vd/E
drift velocity per unit electric field
particles (distinguishable, indistinguishable) using stats and Somerfield
exclusive : alone not together
non-exclusive: together like crows
distinguishable:
P(E) ∝ e^-E/Kbt (maxwell-Boltzmann)
indistinguishable:
(particles of light)=> non exclusive (BOSE-EIENSTIEN) BOSSON
exclusive: e- photons neutrons (FERMI-DIRAC) Fermious
IN THE FORMULA TAKE EF AS 0, and if E > EF its ‘+’, else E is ‘-‘ for ex -0.05, ot 0.05
ISA: if the energy levels are symmetrically located to the fermi level, probability will be one
sommerfield: the arrangement of particles together/not together in energy levels
Boltzman: no spin property
Bossons: spin integral values(0,1,2)
Fermions spin half integral(1/2,3/2,5/2…)
Fermi-Dirad Statistics/fermi energy
Occupation probabilities of valence electrons estimated using
Fermi Dirac statistics for Fermions
Valence electrons above the Fermi level contribute to conductivity
No contribution from valence electrons below the Fermi level
* Thus, all valence electrons are not conduction electrons
* The effective number of electrons above the Fermi level:
neff=n*KT/Ef
Thus a small fraction of valence electrons excited above the Fermi
level only contribute to conductivity
g(E) = π/2(8m/h^2)^3/2E^1/2
P(E) ∝ e^-E/Kbt
P(E) = 1/(1+e^E-Ef/KbT )
f(E) = Fermi factor
Ef => fermi level
Fermi energy- energy corresponding
to the highest occupied level at 0K
At 0K all the states below the Fermi energy are filled and all the states above are empty
highest energy level electrons can fill in 0K
No electrons can occupy energy levels beyond Ef
P(E)*g(E)=> number of electrons with a given E
=> total num of e-
Particles with spin ±
𝟏/𝟐 are classified as Fermions
for E<Ef T-0K
f(E) = 1/e-∞+1 =1 E-EF=’-‘
at 0k all e- states below fermi level are filled
for E>Ef
f(E) = 1/e∞+1 =0; T is 0 on denominator thus its ∞
g(E)*f(E) = N(E) (energy states density)
for T>0K and E=Ef
1/e^0+1=1/2; E-EF=0
fermi level: Prob of a level getting occupied is 50% at Ef,
n=π/3(8m/h^2)^3/2Ef^3/2 (electon concentration)
Ef=(3/π)^2/3(h^2/8m)n^2/3
n is the number of electrons per unit volume
Ef=1/8(3/π)(h^2/8m)^3/2n^2/3
0.121h^2/mn^2/3
Ep(Cu)=7ev
vp=√2Fp/m
vf=10^6
vp=10^-4
concept of fermi temperature
fermi mean free path
fermi temperature L
KbTf = Ef
this temp = fermi energy
QFET: λf = vf2
CFET:λ = vth
vf>vth
when a wave moves through orderly obstacles, it will move without collisions, when impurity is present the electron waves collide, thus resistance is formed
As per quantum free electron model, valence electrons near
Fermi energy only are excited into conduction band
* At temperature T, thermal energy available is kBT
* Thermal energy required to excite the last electron at the bottom
of the energy band at temperature Tf
is kBTf
Then,
𝑬𝒇 = 𝒌𝑩𝑻𝒇
* This temperature is termed Fermi temperature
Fermi velocity is greater than the thermal velocity and drift
velocity (from CFET) of electrons
does vf depend on T’
does Ef depend on T’
fermi energy temp difference
only small change thus it doesnt depened
Ef(T) = EF(0)[1-π^2/12*(KbT/Ef(0)^2]
0 K has the highest energy of electrons
At T=0, the system is in its ground state, and all electrons occupy the lowest possible energy states. The energy levels are filled up to Ef
which becomes the maximum energy for electrons.
There is no thermal excitation at 0 K, so electrons cannot jump to higher energy states.
This creates a sharp cutoff at EF, meaning no electron has energy greater than EF
QFET valence e- near
Fermi energy only are excited into conduction band
At temperature T, thermal energy available is kBT
* Thermal energy required to excite the last electron at the bottom of the energy band at temp Tf
is kBTf =
Tf= ISA!!(fermi energy)
SIGNIFICANCE OF THIS TEMP= BOTTOM e- to conduction band
Then, Ef = kBTf
- This temp is termed Fermi temp
- An example - For Copper (Ef = 7 eV)
Fermi temperature ≈ 81000K
all valence electrons cannot be conduction electrons
.
what does an e- travelling through a metal detect when it approaches an obstacle
KBT ∝ R^2; KBT ∝ πR^2
No of collisions ∝ πR^2
No of collisions of ∝ KbT Mean Free Path
1/No of collisions
projectile is a circle
which QFET assumption was wrong,
and rectangular an array, of an 1D finite potential
Bloch function
Free e- are not free from electrostatic potentials
* Move in a periodic P.E approximated by rectangular P.E due to the regular arrangement of the ionic centers
If V(x) is the P.E at x then, then 𝑽 𝒙 + 𝒂 = 𝑽 𝒙 , the periodic P.E is invariant under translation through lattice parameter (a)
The P.E is lower close to ‘+’ ions in lattice
* The valence e- of diff atoms experience
similar P.E
ktan(ka/2) =a
kcot(ka/2)=-a
(Q^2K^2/2QK)sinhθbsinka + coshθbcoska+
coshθbcoska=cosk(a+b)
QFET could not explain the behavior of semi-conductors and insulators which have energy gap btw conduction band and valance band,
E(k)–k diagram for the system show discontinuity in the energy
at zone boundary of 𝒌 = ±𝒏(𝝅/a)
the assumption which is wrong is
e- in a metal are treated as if they are in an infinity 3D potential well
look at potential well using the V formula diagram: e^ikx(from diagram)
the graph made results in prob density of electrons graph, and will prob be same across adjacent points >
Energy of electron and KP model and simplified boundary condition in terms of k and q
KP FAILURE
an array of 1D finite potential adjacent boundaries can have similar properties ex ψII=ψII’ but not the same
a periodic pattern but the b(barrier gap, is stretched very thin, this its 0) and Vo goes to infinite
approximated the periodic potential as a long chain of coupled finite rectangular wells,
calculated by solving time-independent 1D
Schrödinger’s wave equations for the two regions I and II
ψx = C1e^ikx+C2e^-ikx
ψII=C3e^Qx+C4e^-Qx
Using Bloch theorem and the boundary conditions for continuity of the
wave function, can be obtained.
* A transcendental equation gives variation of E with propagation k, has discontinuities (forbidden gap)
* Allowed regions, e- are free to move with 𝑬 =ℏ^2K^2/2m
E of electron will appear in Band, graphs proves here E(Free e-) and E-k diagram, break in energy and k=+-nπ/a
BOUNDARY Condition
x=0
ψI = ψII
dψI/dx = dψII/dx
x=a
ψI= ψII’
dψI/dx = dψII’/dx
PsinKa/Ka + cosKa=cosKa
some K values allowed, some k values not allowed
Bands of k=> sinK and E are same Bands of E
A transcendental equation with solutions gives variation of E with
propagation k, has discontinuities (forbidden gap)
* Allowed regions, electrons are free to move with energy 𝑬 =ℏ^𝟐𝑲𝟐
Using Bloch theorem and the boundary conditions for continuity of the
wave function, the solution can be obtained.
Completely filled lowest inner band followed by a band of
forbidden energy states
* The highest band of allowed states represents valence band
* The upper most occupied states form conduction band
Thus, there are allowed and forbidden energy states for the
electrons in solids
Materials are then classified as metals, semiconductors or
insulators on the basis of energy band structure
Conduction and valence bands in reduced k space (Brillouin zone)
look at the graph made here, the graph says, some K values are allowed, some arent allowed
Bands of K -> sinK and E are same Bands of E
KP failure
cant give right value of band gap
P vs T resistivity vs temp
p=p(1+aΔT)
ISA: explain the the curve
P=Pi+Pph
Pi= intrinsic => temp independent
Pph => Matthiessen’s rule
ph=> phonon(temp based) Quantized elastic energy, on collision based opposite of photon
Resistance is because of scattering of matter ways, very small temp scatter is minimal
consequence of electron interacting with a periodic potential
names of zones ISA!!!!
CONCEPT OF EFFECTIVE MASS
-π/a<k<+π/a 1st (Brillouin Zone)
-2π/a<k<-π/a => 2nd Zone
Completely filled lowest inner band followed by a band of
forbidden energy states
* The highest band of allowed states represents valence band
* The upper most occupied states form conduction band
are allowed and forbidden energy states for the
e- in solids
explains why insulators/semi conductors see valence band and conduction band
CONCEPT OF EFFECTIVE MASS
Motion of electrons in the crystal is governed by the E=ħ^2k^2/2m* m*=>
shows E is nonlinearly dependent on the propagation constant k
m* is not a constant and depends on the non
linearity of E
effective mass
charge carriers have an effective mass which depends on the
curvature of E-k
“Free” e- => apply electric field
F=ma=eE, a=eE/m how to measure a?, change in velocity/motion of a body
“Bound Electron e-“ => Bound to metal, forces holding it
eE+?=ma ?=Fc (forces due to surroundings on the electron)
eE=m(variable mass=> mass is not ∞)*a(push it mass wont budge)
you push it in one direction, it goes into an other diagram, so is mass (‘-‘), no!!
ISA: effective mass concept, just to capture effects of the electron, and mass aint negative and not ∞ heavy
Effective mass of electrons can be higher or lower than the rest mass
of the electrons and depends on the position of electron in the
particular band
Curvature of the E-k is ‘+’ in the conduction band the
effective mass is ‘+’
* Curvature in the valence band is ‘-‘ with a ‘-‘ m, and indicates the concept of hole conduction in the valence band
* Effective mass of e- can be higher or lower than the rest mass of the e-and depends on the position of
e- in the particular band
* Concept of effective mass helps to explains mobility of charge carriers
* Expression for electrical conductivity, 𝛔 =𝐧𝒆𝒇𝒇𝒆^𝟐𝝉/𝒎∗
why a break in graph effective mass
Matthien’s rule, super conductivity and Meissiens effect
why a break in the εk graph at k= +-=nπ/a
ISA: shape and of graph breaks because of standing waves are results of superposition as it obeys Bragg’s law
similar anology
2dsinθ = nλ
θ=90, 2d=nλ
Matthien’s rule, super conductivity and Meissiens effect
An unusual property of certain metals,
alloys, and ceramics in which electrical
resistance drops to zero when the
temperature is reduced below a critical
value (Tc - known as the transition
temperature)
tot temp: intrinsic temp+phonon temp based
p=m/ne^2z => p=m/ne^2z
=m/ne^2zi+m/ne^2zPh
1/z=1/zi+1/zph
super conductor= mercury Tc=4.2K, kammerlingh Ounes
T<Tc>Tc normal conductor</Tc>
Type 1 SC
TC=1-10K
Type 2
TC=10-30K
Type 3
Tc>30K
258K record
HOLY GRAIL Tc=300K
Superconductors exhibit unique features other
than their ability to perfectly conduct current
Super conductors expel magnetic fields during
the transition to the superconducting state- like
a perfect diamagnetic material
This property is called Meissner effect
Sommerfield’s quantum Free e- QFET
Valence e- occupy discrete energy states following Pauli’s exclusion principle. Only e- close to Fermi level participate in conduction process; Valence e- = n≅ 𝟏𝟎^𝟐𝟖𝒎^-3
energy states split into discrete and closely spaced to accommodate all the valence e-; At ‘0K’ arrangement leads to a sea
with small separation
* Occupation probability = Fermi Dirac distribution
function applicable to fermions
- The electrostatic interactions, the electron – electron and the
electron – ion are negligible
Sommerfield’s quantum free e- QFET
- moving electrons are waves, when impurity is present the e- waves collide, thus resistance is there, but normally it through orderly obstacles, it moves without collision
- e- in a metal are treated as if they are in an infinity 3D potential well
En = (nx^2+ny^2+nz^2)h^2/8ma^2
- Merit of QFET
1) Correct evaluation of electronic specific heat – from valence e- close to the Fermi level - Heat absorption happens due to this small fraction of e-
kBT/Ef is a fraction less than 1% and temperature dependent
CuTF approx = 8.110^-4K
it could explain the result CV = 10^-4
dU/dT=(π^2/2)neff*Kb^2T/Ef
2)scattering mechanism with thermally vibrating ionic array 𝝀 ∞ 𝟏/T
σ not only depends on
the number of e- per unit volume but also depends
on the λ ∝ 1/T
σ =ne^2τ/m=ne^2λ/mvf
*Conduction e- move in array of ‘+’ ions, colliding with ionic centers and other e- resulting in R
Amplitude of vibrations increase with T followed by
increase in e- scattering
*cross sectional area of scattering=πr^2
𝝆 ∞ T (mean free path)
3) It doesnt open with the statement σ ∝ 1/T
σ not only depends on
the number of e- per unit volume but also depends
on the λ/vF
σ =neff^2τ/m=neff^2λ/mvf
4) wiedman - Frans law
K(Thermal conductivity)/σ(electrical conductivity)
∝ T
K/σ = LT L=> Lorenz const
L=π^2/3e^2Kb^2T
(2.4510^-8) WΩK-2
Thermal Conductivity
K=1/3Cel/VvL
5) for a mole of electron gas, the fraction of e- gain energy KbT = neff=Na/Ef*KbT
6) As per QFET, the actual number of valence electrons depends on
concept of density of states; (g(E).dE) gives number of available electron states per unit volume per unit energy range at a certain energy
level, E
(g(E).dE) gives number of available electron
states per unit volume per unit energy range at a certain energy
level, E
DEMARKS QFET
metals are supposed to have HALL coefficients which are negatice
Rh = 1/n^e <0
Zn, Be etc = its positive R
These demerits reflects that real potentials of ionic centers in metal was ignored in the development of quantum free electron gas model
could not explain the behavior of se
Failed to explain
- differences in conduction in metal, semiconductor and insulator
comparison of fermi velocity and drift velocity
Fermi velocity is greater than the thermal velocity and drift velocity (from CFET) of electrons
Thermal velocity, typical order is 10^5 m/s
Typical order of Vd
for an electric field of 1 V/m is 10-4 m/sFg
classifications of solid
Case of conductors
Thus conductors (metals) characterized by a partially filled conduction
band (no band gap)
* Case of semi-conductors
Completely filled valence band and completely empty conduction band
Energy gap of 3-5 eV
At normal temperatures, possible for electrons in the valence band to
move into the conduction in the case of metals and semiconductors
* Case of insulators
Materials with energy band gap greater than 5eV
Electron conduction is impossible and attempts to excite lead to a
dielectric breakdown
