Hall Effect Experiment Lab Report

Hall effect applied in two dimensional material
Introduction
Hall Effect have been extensively applied in the past decades to determine the electron transport characteristic, such as hall mobility, concentration and types of transport carriers. Hall effect is also a good choice that characterize from the electric point of view the 2 dimensional materials. The purpose of the work are:
Design an experimental methods to determine electric characteristic of membrane materials by Hall Effect.
Determine the surface states of membrane by Hall effect.

Theory
From classical semiconductor physics, the conductivity σ and the Hall constant R_H are given by the follow relations[1]: δ(B)=e(〖nμ〗_n+pμ_p ) [1+(npμ_n μ_p (μ_p-μ_n )^2 B^2)/((nμ_n+pμ_p )^2+(p-n)^2

(3)
Because R_H (n,p) is monotonic decreasing function, the equation (3) confirm that R_H cannot determine which type of carriers increase in materials after applying an illumination.
Then the changing of the carriers in the material can be got clearly. The type of carriers which increase or decrease can be determined.

Van der Pauw method
Van der Pauw method was published by Van der Pauw in 1958. It can avoid the errors from structure geometry, and only four contacts are needed. So the method is very popular in the determination of Hall Effect in 2 dimensional structure.
To use the method, the geometries should satisfy the follow conditions [2]: The contacts are at the circumference of the sample. The contacts are sufficiently small The sample is homogeneous in thickness The surface of the sample is singly connected.

Figure 1. General van der Pauw geometries. Contacts are black

N_dep is the surface density of depleted carriers. Q_t is the charge trapped by interface states.
The charge trapped density can be obtained by the partial differentiation of (10) with respect to surface potential ϕ_s for a given temperature.
N_ss=(1/e) ((∂Q_t)/(∂ϕ_s ))_T=((∂N_ind)/(∂ϕ_s ))_T×(α∙ω∓1)-(C_ox/e)∓((∂N_dep)/(∂ϕ_s ))_T (11)
Where
ω=((∂N_H)/(∂N_ind ))_T (12)
The mobility ratio can also be obtained as shown below. r=N_ind/(∫_0^(N_ind)▒〖ω dN_ind 〗) (13)
Partial differentiation of (10) with respect to temperature T for a constant value of N_ind produce
N_ss=(∂T/(∂ϕ_s ))_(N_ind ) {β∓((∂N_dep)/∂T)_(N_ind ) }-(C_ox/e) (14)
From eq.(8) to eq.(13), ω, r and N_ss can be solved. ((∂N_ind)/(∂ϕ_s ))_T, ((∂N_dep)/(∂ϕ_s ))_T, (∂T/(∂ϕ_s ))_(N_ind ) and ((∂N_dep)/∂T)_(N_ind ) can be known from the space-charge region