Chapter 4 - Excess Carriers and Relaxation

ECE 4339
Han Q. Le (copyrighted) U. of Houston

0. Physical constants or frequently used formulas

Execute the package below will give you the full list of physical constants in Mathematica.

<<PhysicalConstants`

Execute the package below helps do unit conversion.

<<Units`

0.1 Fundamental constants

ℏ=(PlanckConstant/(2 π ))
e=ElectronCharge
m=ElectronMass
c=SpeedOfLight
(Subscript[k, B]) = BoltzmannConstant

The above constants are given in MKS unit. Another very useful unit system is the CGS unit:

ℏ=(PlanckConstant/(2 π))  /. Joule->10^(7) Erg
e=ElectronCharge /. Coulomb->2.99792458*10^(9) Esu
m=ElectronMass /. Kilogram->10^3 Gram
c=SpeedOfLight  /. Meter->100 Cm
(Subscript[k, B]) = BoltzmannConstant /. Joule->10^(7) Erg

0.2 Other useful physical constants

0.2.1 Bohr radius

Bohr radius is the "effective" radius of the electron orbit in the lowest energy state of the ideal hydrogen atom. It is NOT the actual orbit, of course, but only a secondary physical constant useful in many calculations.
It is defined as (in CGS units)

(Subscript[a, B])=((ℏ^2)/(m (e^2)))

(Subscript[a, B])=(Subscript[a, B]) /. Esu-> (Sqrt[Erg Cm]) /.  Erg->Gram ((Cm^2)/(Second^(2 )))

0.2.2 Rydberg

Rydberg is the binding energy of the electron in the hydrogen atom (ideal atom).

(Subscript[R, ∞])=((e^2)/(2 (Subscript[a, B])))

(Subscript[R, ∞]) = (Subscript[R, ∞]) /. Esu-> (Sqrt[Erg Cm])

But nobody really use (Subscript[R, ∞]) in erg unit. The more convenient unit is eV: electron volt. In this unit:

(Subscript[R, ∞])=(e/(2 (Subscript[a, B])))c(Second/Esu) 10^(-8)  eV

Thus: (Subscript[R, ∞])=13.605691932782346` eV

What is the value of eV in other energy unit?

eV=ElectronCharge Volt

eV = eV /. Volt-> Joule/Coulomb

eV = eV /. Joule->10^7 Erg

0.3 Use of physical constants as "natural" units

As the natural unit of length, we can use Bohr radius. For energy, we can use Rydberg, or eV (semi natural). Boltzmann constant is often used with eV

(Subscript[k, B]) = (BoltzmannConstant/(ElectronCharge ))

(Subscript[k, B])  =  (Subscript[k, B]) /. Joule-> Coulomb  eV

Thus, sometimes, we use this value:
                                 (Subscript[k, B]) =8.61734×(10^(-5))  eV/K

Overview

1. Excess carriers and quasi-Fermi levels

Introductory concept. Behavior of gas molecules when additional population is introduced.

http://phet.colorado.edu/en/simulation/gas-properties

1.1 Fermi-Dirac statistics - review only

Electrons and all particles with odd spin obey Fermi-Dirac statistics

FermiDirac[En_,Ef_,kB_,T_]:=(1/(1+(E^((En-Ef)/(kB T))))) ;

We use (Subscript[k, B]) in unit of eV:   8.617342790900663*(10^(-5))

App Demo: Fermi-Dirac distribution

1.2 Density of states - review only

Density of state D[E] is a function that determines carrier concentration per unit volume per unit energy as a function of energy. It is a function of the band structure. For parabolic band, the DoS can be estimated as a function ~(Sqrt[|E -(Subscript[E, g])|]) where (Subscript[E, g]) is the band edge energy. Usually the band gap if we take the energy origin as the valence band. We'll learn more about this in Chapter 4.
Below is an example (for simplicity - everything here will be normalize to be unit less) - see Chapter 3 for real units.

App Demo: 3D Density-of-state of isotropic parabolic bands

1.3 Carrier concentration and Fermi level at equilibrium - review only

1.3.1 Carrier distribution

We obtain the carrier distribution vs. energy by taking the product of DoS function and FD statistics:
                                         n[En]=D[En] (1/(1+(e^((En-Ef)/(kB T)))))

App Demo: Electron Distribution in Conduction Band

App Demo:  Electron-Void/Hole Distribution in Valence Band

App Demo:  Electron and Hole Distribution at Thermal Equilibrium

1.4 Carrier concentration and quasi-Fermi levels (non-thermal equilibrium)

App Demo:  Carrier Concentration and Distribution - Non-thermal equilibrium with Quasi-Fermi Levels

1.5 Advanced topics: 2D systems, quantum well and lasers.

A quantum has a 2D density of state, which is actually a lot simpler than 3D. It is constant for each energy level and step up like a staircase:
                                       D[ E]:=((SuperStar[m])/(π (ℏ^2))) U[E-(Subscript[E, 0])] 
where U[E-(Subscript[E, 0])] is a uit-step function with a broadening edge.

DOSQWC[me_,en_,qw_,Γ_]:=
me*Sum[  (1/(1+(E^((qw[[n]]-en)/Γ))))  ,{n,1,Length[qw]}]  ;
DOSQWV[mh_,en_,qw_,Γ_]:=
mh*Sum[  (1/(1+(E^((en-qw[[n]])/Γ))))  ,{n,1,Length[qw]}]  ;

Example, a QW has 2 energy levels

App Demo: Density of State of a 2D System (Quantum Well)

The App Demo below shows how you can move the conduction band Fermi level and valence band Fermi level to affect the population in each band. 
Carrier density relation:            n=∫D[E] f[E]dE     
                                              n=∫((SuperStar[m])/(π (ℏ^2))) U[E-(Subscript[E, 0])](1/(1+(e^((En-Ef)/(kB T)))))dE

In a laser or LED, the active region where electrons and holes recombine, their population are equal per the principle of charge neutrality, there is a fixed relationship between the 2 Fermi levels.

App Demo: Carrier Distribution of a 2D System (Quantum Well) with Quasi-Fermi Level in Each Band

2. Carrier relaxation-microscopic mechanism

2.1 Absorption

Absorption in materials obeys Beer's Law:
                                               P[x]=P[0] (e^(-α x)) 
where P is the transmitted light intensity (or power) and α is the absorption coefficient.

App Demo: Light Absorption - Beer’s Law

For virtually any material, absorption coefficient is a function of wavelength λ, expressed as α[λ],Hence:
                                                            P[x]=P[0] (e^(-α[λ] x)) 
For semiconductors, α[λ] is very small below the band gap (no electron transition from Val to Cond band), and grows very large very quicky above the bandgap (high density electron transitions from Val to Cond band).

(source: http://pveducation.org/pvcdrom/pn-junction/absorption-coefficient)

Plot the transmitted power of a beam of light as a function of semiconductor thickness from 0.1 um to 5 um.

App Demo: Light Absorption in Semiconductor - λ vs. Band Gap

Review of III-V (not including Nitride) and II-VI semiconductors

2.1.1 Direct bandgap

When the band extremas are vertically aligned, the bandgap is called direct. Optical transition is strong thanks to momentum conservation

source: http://www.ioffe.ru/SVA/NSM/Semicond/

2.1.2 indirect bandgap

When the band extremas are NOT vertically aligned, the bandgap is called indirect. Optical transition is weak because a phonon is required to achieve momentum conservation. 
Observation of the absorption edge for different SC. Note Si, Ge vs. the rest. Which ones are indirect and which ones are direct?
Indirect bandgap materials are not good light emitting. They can still be used as detectors, but not as light emitters.

source: http://www.ioffe.ru/SVA/NSM/Semicond/

2.2 Radiation (emission) - luminescence

2.2.1 Luminescence spectrum

A mechanism for an electron and a hole to recombine is to emit a photon, called radiative recombination. The phenomenon of emitting light is called luminescence.
If the luminescence comes from electrical injection, (Ohmic injection) we call electroluminescence (LED, laser)  
If from external light excitation (like from a laser, or a lamp), we call photoluminescence (optically pumped devices)
If from electron beam excitation (like in a cathode ray tube), we call cathodoluminescence.
It doesn't matter how we excite, the luminescence spectrum of different excitation methods are similar to each other, because it mostly depends on the behavior of electrons and holes that are relaxed at the band extrema.

Above: Near-IR (very short-wave IR) LED

http://mbe.mntl.illinois.edu/workdone.htm

The emission has a strong peak near the band edge, and an exponentially tapered tail toward higher energy that exhibits the temperature dependence distribution:  (e^(-E/(Subscript[k, B]) T)) (the tail of Fermi-Dirac distribution). This temperature is the electron-hole population temperature, which is sometime “hotter” than lattice temperature, depending on the amount of excess energy of the excitation.
By looking at the emission peak, you can roughly tell the band gap of a semiconductor. A more accurate spectral line fitting, as well as low temperature measurements can give very accurate bandgap value.

The above example shows emission spectra from different semiconductor alloy. Shown is the InGaAs luminescence. The others are GaInNAs.

Above: Laser spectra for long haul optical communications

2.2.2 Model for luminescence spectrum

A simple model for the luminescence spectrum is so-called k-conservation model, in which each electron recombines only with hole of the opposite momentum. Then, the emitted energy is:
                                                    h ν =(Subscript[E, g]) +((ℏ (k^2))/(2 (Subscript[m, e]^( * )))) +((ℏ (k^2))/(2 (Subscript[m, h]^( * )))) =(Subscript[E, g]) +((ℏ (k^2))/(2 (SuperStar[μ]))) 
 where (SuperStar[μ]) is defined as the effective reduced mass of the electron-hole pair:
                                                             (1/(SuperStar[μ]))=(1/(Subscript[m, e]^( * ))) +(1/(Subscript[m, h]^( * )))        
 Let                                                      u=((SuperStar[μ])/(Subscript[m, e]^( * ))) =((Subscript[m, h]^( * ))/((Subscript[m, e]^( * ))+(Subscript[m, h]^( * ))))  
 Then:                 (Subscript[E, e])≡((ℏ (k^2))/(2 (Subscript[m, e]^( * )))) =u ( h ν-E)     ; (Subscript[E, h])≡((ℏ (k^2))/(2 (Subscript[m, h]^( * )))) =(1-u) ( h ν-E)    
 The spectrum is simply the density of occupied electrons and holes at momentum k.
 Thus:                                       R[h ν]  ∝ (Subscript[ρ, C])[(Subscript[E, e])] (Subscript[F, C])[(Subscript[E, e])] (Subscript[ρ, V])[(Subscript[E, h])] (1-(Subscript[F, V])[(Subscript[E, h])]) 
Here is the case we often have different electron and hole fermi levels.

Here, the QW bandgap is the bulk bandgap + quantum well energy.
We will plot the density of state 
The density of state:              (Subscript[ρ, C])[ (Subscript[E, e])]=((Subscript[m, e]^( * ))/(π (ℏ^2))) U[(Subscript[E, e])]=((Subscript[m, e]^( * ))/(π (ℏ^2))) U[u ( h ν-E)] 
                                               (Subscript[ρ, V])[ (Subscript[E, h])]=((Subscript[m, h]^( * ))/(π (ℏ^2))) U[-(Subscript[E, h])]=((Subscript[m, h]^( * ))/(π (ℏ^2))) U[-(1-u) ( h ν-E)]

Example: emission from GaAs/(Subscript[Al, x])(Subscript[Ga, 1-x])As QW LED or laser under lasing threshold

Below is an illustration of emission from (Subscript[Al, x])(Subscript[Ga, 1-x])As  (using QW model for convenience, but the QW energy levels are set at ~ 0)

App Demo: Illustration of GaAs/AlGaAs QW LED or Subthreshold Laser emission

2.2.3 Illustration

Why do we care about luminescence from electron-hole recombinations?

Once upon a time, noble gases (Helium, Neon, Krypton, Argon, Xenon...) were most importantly used to light up activities that are of somewhat dubious nobility...

Today, LEDs are relieving these gases from their “nobless oblige”..

LED are used from the theme of worship...

such as LED cathedral

to Sin City...

Demo

Electron-hole radiative recombination is what gives us

2.3 Other mechanisms:

Hot electron relaxation: phonon emission.
Auger recombination (non-radiative)