Temperature Dependence of the Fermi-Energy in Semiconductors

Introduction

Silicon is beside germanium and some III-V semiconductors like GaAs the most important material in semiconductor industry. The fabrication of state of the art processors are impossible without silicon. The reason is that nowadays fabricated silicon single crystals are extremely clean and perfect which is an essential requirement to obtain adequate yield during silicon processing. Every chip possess a lot of active region predominantly pn-junctions. For engineering these pn-junctions the Fermi-Energy E_F plays an important role in technology. The properties of a fabricated pn-junction depends strongly on the energy level of the Fermi-Energy.

In this section I will treat the temperature dependence of the Fermi-Energy.

Basics to Fermi-Energy

The electron-density in the conduction-band can be calculated with the following equation:

In a Similar way the hole-density in the valence-band can be calclated:

where

If the band-gap E_G = E_C - E_V is not too narrow, then (E - E_F)/kT >> 1 and the Fermi-Distribution can be replaced by the Boltzmann-Distribution: f_n(E) ~ f_B(E) = exp[ - (E - E_C)/kT].

Let's have a look at a semiconductor containing acceptors with the concentration N_A and donators with the concentration N_D. At any time the condition of neutrality has to be satisfied:

N_D⁺ is the concentration of ionized donators and N_A^- is the concentration of ionized acceptors:

Insertion of the equations for N_D⁺, N_A^-, p and n (see above) into the condition of neutrality yields a transcendent equation for the Fermi-Energy E_F:

*

Numerical Method for Calculation the Fermi-Energy

In order to define an Error-Function Err(E) one have to bring the all terms in equation (*) to the right side and replaced zero at the left side by Err(E). Now you have to guess the right value for E. If the guessed value was wrong then Err(E) has a value unequal to zero. Only when Err(E) = 0 you have guessed the correct value and E = E_F, see next figure.

a) Error-Function in dependence of the energy. When the Error-Function cross the E-axis then E = EF.		b) Iteration-steps to approximate the value of EF.

In order to estimate E_F you have to find the start-values E_L and E_R. These value must satisfy the condition Err(E_L) · Err(E_R) < 0 or in this special case Err(E_L) < 0 and Err(E_R) > 0. Then take the midpoint of the interval [E_L,E_R]: E_M = (E_L + E_L)/2 and calculate Err(E_M). The sign of Err(E_M) decide the selection of the interval for the next iteration (see interval labeled "2" in the above figure):
If Err(E_L) · Err(E_M) < 0 then leave E_L unchanged and set E_R = E_M.
If Err(E_L) · Err(E_M) > 0 then set E_L = E_M and leave E_R unchanged.
Now repeat this procedure as long as the wideness of the current interval E_R - E_L becomes smaller than a chosen limit. If this condition is valid then terminate the iteration procedure and set: E_F = E_M. Repeat this procedure for every temperature T.

It follows the program-structure of the above described procedure using C:

double Calculate_Fermi_Energy(double NA, double ND, double Temp)
{
  double E0 E1 EF;

  void Calulate_Start_Values()
  {
     EL = ...;
     ER = ...;
  }

  void Interval_Nesting()
  {
     ...
     EF = ...;
  }

  Calculate_Start_Values;
  Interval_Nesting;
  return EF;
}

To see the complede program code click here .

Results:

The temperature dependence was computed for the following parameters:

	EV	=	7.00eV	(arbitrary value)
	EG = EC - EV	=	1.12eV	(silicon)
	EC - ED	=	0,03eV	(donator-level)
	NA	=	0	(n-type silicon)

The left part of the next picture shows temperarure dependance of the Fermi-Energy E_F in the temperature range of 0 < T < 1500K for different donator concentrations N_D. Because the melting point of silicon is T_M = 1688K( = 1415°C), the chosen upper limit for the temperature is reasonable.

The right part of the picture shows the temperature dependence of the electron concentration n(T). To compute the electron concentration n, the result for E_F must be inserted (for a given temperature T and donator concentration n_D) in the equation for n, see above. The same holds for the hole concentration p q :

Fermi-Energy EF and electron concentration n in dependence of temperature T for different donator concentrations ND.