Chapter 3 - Carriers in Semiconductor

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

Part 3

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

0.4 Modified natural units for semiconductors

0.4.1 Donor Bohr radius

The system of donor-electron or acceptor-hole behaves like an atom. The difference is that the electron/hole does not behave like a free electron, but really a set of Bloch wavefunctions, which behaves like a particle with an effective mass. Also, the electrostatic force between the ionized donor or acceptor and the electron/hole is screened by the polarization of other electronic bonds, which is described in terms of a dielectric constant.

The donor Bohr radius is:
(SuperStar[(Subscript[a, B])])=(((ℏ^2) ϵ)/((SuperStar[m]) (e^2))) where ϵ  is the semiconductor dielectric constant, (SuperStar[m]) is the electron effective mass. 
We notice that: 
(SuperStar[(Subscript[a, B])])=(((ℏ^2) ϵ)/((SuperStar[m]) (e^2))) =((ℏ^2)/(m (e^2))) (m/(SuperStar[m]))ϵ =(Subscript[a, B]) (m/(SuperStar[m]))ϵ 
This makes it easy to estimate the effective Bohr radius in a semiconductor. Usually, effective mass is not given in terms of gram but as the ratio: ((SuperStar[m])/m)
Example, for GaAs:
m=0.0665 ; ϵ ~13.2 . Thus, the effective Bohr radius in GaAs is:

(Subscript[a, B]) * 13.6/0.0665

Similarly, one can estimate the donor binding energy, which is called the effective Rydberg energy:
(SuperStar[(Subscript[R, ∞])])=((e^2)/(2 ϵ  (SuperStar[(Subscript[a, B])])))

8. App: Carrier concentration and distribution at thermal equilibrium

8.1 Calculations for some semiconductors

App Demo:  Carrier Concentration and Distribution at Thermal Equilibrium

8.2 Example 1: GaAs

Refer to this website (or you can find other similar websites) to look up for semiconductor properties:
http://www.ioffe.ru/SVA/NSM/Semicond/

Find the density of state Nc, Nv, and find the intrinsic carrier concentration (Subscript[n, i]) for GaAs at 300 K. Plot (Subscript[n, i]) from 100 K to 300 K. Plot and compare (Subscript[n, i]) for GaAs with Si

8.1.1 Find Nc:

(Tutorial in Mathematica)

Quiet[ClearAll[(Subscript[R, ∞]) , (Subscript[a, B]), m, e, h,(Subscript[k, B]),(Subscript[En, gap])] ]

(Subscript[N, C])= (2/((Subscript[a, B])^3)) ((((Subscript[m, eff])))^(3/2))((((((Subscript[k, B])  T)/(4 π (Subscript[R, ∞])))))^(3/2)) /. (Subscript[a, B])->  5.29177*10^(-9)  Cm /. (Subscript[m, eff]) ->  0.0665 /. (Subscript[k, B])->  8.61739*10^(-5)   eV/K  /. (Subscript[R, ∞])->  13.6057 eV

% /. T-> 300 K

Hence: (Subscript[N, C])=4.3033726616693754`*^17 (Cm^(-3))

8.1.2 Find Nv (see exercise/HW)

(Subscript[N, V])= (2/((Subscript[a, B])^3)) ((((Subscript[m, eff])))^(3/2))((((((Subscript[k, B])  T)/(4 π (Subscript[R, ∞])))))^(3/2)) /. (Subscript[a, B])->  5.29177*10^(-9)  Cm /. (Subscript[m, eff]) ->  0.53 /. (Subscript[k, B])->  8.61739*10^(-5)   eV/K   /. (Subscript[R, ∞])->  13.6057 eV

% /. T-> 300 K

8.1.3 Find ni (see exercise/HW)

8.1.4 Plot intrinsic carrier concentration (see exercise/HW)

Neglecting temperature dependence of the band gap (see app for T-dependence). Below is the answer.

(Sqrt[(Subscript[N, C]) (Subscript[N, V])])

IntrinsicCarrier=(Sqrt[(Subscript[N, C]) (Subscript[N, V]) (Cm^6) (K^3) ]) (E^(-((Subscript[En, gap])/(2 (Subscript[k, B]) T))))  ;
Plot[ IntrinsicCarrier  /. (Subscript[En, gap]) -> 1.43  /. (Subscript[k, B]) -> 8.61739*10^(-5) 
 ,{T,100 ,350 }, PlotRange->All
,PlotStyle->{{ Thick,Blue}}
, Filling->{1-> { Bottom, Hue[0.14,1,1,0.3]}}
, Frame-> True, GridLines-> Automatic
,FrameLabel->{"Temperature (K)","Carrier density ((cm^(-3)))"}
,LabelStyle->Directive[14,FontFamily-> "Arial"]]

GaAsplot=LogPlot[  IntrinsicCarrier  /. (Subscript[En, gap]) -> 1.43  /. (Subscript[k, B]) -> 8.61739*10^(-5) 
	, {T,100 ,350 }, PlotRange->All
,PlotStyle->{{ Thick,Blue}}
, Filling->{1-> { Bottom, Hue[0.14,1,1,0.3]}}
, Frame-> True
,GridLines-> Automatic,FrameLabel->{"Temperature (K)","Carrier density ((cm^(-3)))"}
,LabelStyle->Directive[14,FontFamily-> "Arial"]]

GaAsplot2=Plot[ Log[10, IntrinsicCarrier]  /. (Subscript[En, gap]) -> 1.43  /. (Subscript[k, B]) -> 8.61739*10^(-5) 
	, {T,100 ,350 }, PlotRange-> All,PlotStyle->{{ Thick,Blue}}
, Filling->{1-> { Bottom, Hue[0.14,1,1,0.3]}}
, Frame-> True
,GridLines-> Automatic,FrameLabel->{"Temperature (K)","Carrier density (Bel (cm^(-3)))"}
,LabelStyle->Directive[14,FontFamily-> "Arial"]]

8.3 Comparing intrinsic GaAs and Si

8.3.1 Calculation for silicon:

For Si, we can use the same formulas, but with a modification for the Si conduction band peculiarity:

From: http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm

It has 6 valleys: we call it 6-fold degeneracy. Also, the energy surface is not spherical: it has different effective mass in different direction. A good approximation of the spatially averaged mass is:  
(SuperStar[(Subscript[m, net])]) = ((( (Subscript[m, long]) ((Subscript[m, trans])^2)))^(1/3))

Quiet[ClearAll[(Subscript[R, ∞]) , (Subscript[a, B]), m, e, h,(Subscript[k, B]),(Subscript[En, gap])]]

(Subscript[N, C])=(Subscript[g, C]) (2/((Subscript[a, B])^3)) ((((Subscript[m, eff])))^(3/2))((((((Subscript[k, B])  T)/(4 π (Subscript[R, ∞])))))^(3/2)) /. (Subscript[a, B])-> 5.29177*10^(-9)   /. (Subscript[m, eff])->  (((0.98*(0.19^2)))^(1/3)) /.   (Subscript[g, C])-> 6  /. (Subscript[k, B])-> 8.61739*10^(-5)     /. (Subscript[R, ∞])->  13.6057

Alternatively, we can define an DOS effective mass that takes into account anisotropy and degeneracy:

(Subscript[N, C2])= (2/((Subscript[a, B])^3)) ((((Subscript[m, eff])))^(3/2))((((((Subscript[k, B])  T)/(4 π (Subscript[R, ∞])))))^(3/2)) /. (Subscript[a, B])-> 5.29177*10^(-9)   /. (Subscript[m, eff])->  1.08   /. (Subscript[k, B])-> 8.61739*10^(-5)     /. (Subscript[R, ∞])->  13.6057

(Subscript[N, V])= (2/((Subscript[a, B])^3)) ((((Subscript[m, hh])))^(3/2))((((((Subscript[k, B])  T)/(4 π (Subscript[R, ∞])))))^(3/2))   /. (Subscript[a, B])-> 5.29177*10^(-9) /. (Subscript[m, hh]) -> 0.81 /. (Subscript[k, B])-> 8.61739*10^(-5)    /. (Subscript[R, ∞])-> 13.6057

(Subscript[n, i])=(Sqrt[(Subscript[N, C])  (Subscript[N, V])]) (E^(-(Subscript[En, gap])/(2(Subscript[k, B]) T) ))  /. (Subscript[En, gap])-> 1.1 /. (Subscript[k, B])-> 8.61739*10^(-5)   /.  (Subscript[R, ∞])->  13.6057

8.3.2 Comparison

Siplot=Plot[ Log[10., (Subscript[n, i]) ]
	, {T,100 ,350 }, PlotRange->All,  PlotStyle->{Thick,Red}
,Frame->True, GridLines->Automatic
,GridLines-> Automatic,FrameLabel->{"Temperature (K)","Carrier density (Bel (cm^(-3)))"}
,LabelStyle->Directive[14,FontFamily-> "Arial"]
]

Show[Siplot,GaAsplot2]

8.4 Example 2: p-dope and n-dope

A piece of Si is doped with boron up to (10^16)/ (cm^3). Assume that all boron atoms are ionized at 300 K.

Is it p-type or n-type?

http://demonstrations.wolfram.com/DopedSiliconSemiconductors/ 
Contributed by: S. M. Blinder

What is the free electron concentration? (find (Subscript[n, o]) for boron-doped Si )

Since (Subscript[p, o]) >> (Subscript[n, i]),  approximate (Subscript[p, o]) = (Subscript[N, A])

(Subscript[n, o])=(((Subscript[n, i])^2)/(Subscript[p, o ]))

(Subscript[p, 0]) = 10^16 ;

(Subscript[n, 0])  /. T->300

Alternative answer: by looking up, the value of (Subscript[n, i]) for Si at 300 K is 1.5× (10^10) /(cm^3)

(Subscript[n, 0])=((((1.5× (10^10)))^2)/(10^16))

8.5 Example 3: Find po for Sb-doped Ge (see exercise/HW)

8.5.1 Solving for carrier concentrations

A piece of Ge is doped with Sb at 5×(10^13)/(cm^3). Is it n-type or p-type?
Assume that acceptor doping is insignificant, what is the electron concentration?

Since  (Subscript[N, A]) is unspecified, it is negligible: 0
Condition of charge neutrality:
(Subscript[n, o])+(SuperMinus[(Subscript[N, A])])=(Subscript[p, o])+(SuperPlus[(Subscript[N, D])])
Equilibrium:
(Subscript[n, o]) (Subscript[p, o]) =((Subscript[n, i])^2) 
How many Eqs. do we have? How many unknown?

8.5.2 Plot electron concentration as a function of doping (see exercise/HW)

Plot the electron concentration calculated above as a function of doping from 10^12 to 5*10^14 at 300 K

9. Fermi level across different materials

9.1 Overview of concept

9.2 Example (see exercise/HW)

A Te-doped GaAs with 2*(10^17) density layer is epitaxially grown on a Mg-doped GaAs substrate with (10^18) concentration. Calculate and draw the relative band diagram of the 2 layers.