Multicarrier conduction in semiconductors

Multicarrier conduction in semiconductors
Murthy O.V.S.N.
June 5, 2003
1 Introduction
Take a semiconductor piece, an n-type Silicon for instance, and put it in a conventional Hall effect
measurement setup. What you observe is as follows. The Hall voltage would appear to be
increasing linearly with magnetic field and the change in resistivity of the sample with magnetic
field, magnetoresistance, negligible. The carrier parameters like number density of carriers and the
mobility are obtained from the following relations
n = 1
R H e
(1a)
µ = 1
R S ne
(1b)
where
R H = V H
IB
=
V (B) − V (0)
IB
= ∆R MO,P N
B
(2a)
R S =
π R MN,OP + R NO,P M
f( R MN,OP
)
ln2 2 R NO,P M
if you are performing a van der Pauw measurement [3] on a square sample with corners marked
M,N,O and P. Both the quantities n, R S are usually expressed as 2D values (in units of cm −2 and
Ω resp.).
1
(2b)

2 Single carrier case
1 In order to see the conduction process microscopically, we can write out the equations of motion
for the electron
m e ˙v xe = −eE x − ev y B z
(3a)
m e ˙v ye = −eE y + ev x B z
(3b)
and cancel the second term of eqn. 3a since the current is zero in the y-direction. Physically
what happens is that the carriers (electrons) keep getting deflected until steady state is reached
where there is a transverse voltage developed which balances the Lorentz force on the carriers. The
second equation eqn. 3b then gives us (with ˙v ye = 0) the ratio E y /E x . In the time averaged sense,
the carriers undergo collisions with an average relaxation time (τ) and are left with a mean ‘drift
velocity’ v xe given as follows:
v x = v x0 − eE ∫ ∞
tP (t) = v x0 − eE ∫ ∞
tex[−t/τ]dt = v x0 − eEτ/m e (4)
τm e τm e
0
and since collision processes have no memory, the first term drops out. Armed with an expression
for averaged velocity in the direction of applied field, we can now calculate the ‘Hall angle’ (θ)
between the current and the resultant electric field.
tanθ = E y /E x = −Bµ e (5)
0
with the current J x = neµ e E x . Still, a bit further from experimentally observed quantities, we
calculate the ‘Hall constant’ or ‘Hall coefficient’ by R H = E y /J x B and as can be seen, this gives
along with our previous result eqn. 5, the carrier concentration from eqn. 1a.
1 The present analysis closely follows Smith [2].
2

3 Two-carrier case
What changes when we have two carriers instead, can be found out from the equations of motion
eqns 1a and 1b of the carriers written for both carriers. Here, we cannot assume that J y = 0
implies that both electron and hole currents are zero. In fact, both are non-zero in the presence
of a magnetic field, but due to their opposite signs, cancel each other again giving J y = 0. Going
back to eqn. 3a, we again neglect the second term assuming the Hall angle is small (e.g. in case
of a long Hall bar) and get a mean drift velocity v xe = −µ e E x . We use this in eqn. 3b and since
the right hand side of the equation is constant, we can apply the same analysis as above to get a
time-averaged velocity
v y e = −µ e E y − µ 2 eBE x
(6a)
and similarly for holes
v y h = µ h E y − µ 2 hBE x
(6b)
The total current, however, remains zero. Thus,
doing so, we obtain a Hall angle and a Hall coefficient
J y = −nev ye + pev yh = 0 (7)
tanθ = E y /E x = pµ2 h − nµ2 e
nµ e + pµ h
(8)
R H = E y /J x B =
pµ2 h − nµ2 e
e(nµ e + pµ h ) 2 (9)
3.1 Magnetic field dependence
All along we have been neglecting the magnetic force compared to the force due to the electric field,
but this assumption breaks down when the Hall angle is large. This happens when the dimensionless
number represented by µB >> 1 for any of the carriers. To see what happens in this scenario, let
us keep the term and introduce the angular ‘cyclotron frequency’ (ω = eB/m e )
˙v xe = −(e/m e )E x − ωv y
(10a)
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˙v ye = −(e/m e )E y + ωv x
(10b)
To solve this, we can introduce complex variables z = x + iy and Z = v x + iv y , the differential
equations become decoupled and merge to form
Z − iωZ = −(e/m e )(E x + iE y ) (11)
The solution is as follows
Z = Z 0 e iωt + (e/m e )(E x + iE y ) 1 − eiωt
iω
Again carrying out the time integration, we obtain the mean velocities as
(12)
v x + iv y = −(e/m e )(E x + iE y )τ/(1 − iωt) (13)
The current densities can be obtained by taking real and imaginary parts of the above equation
and writing J = −nev as follows
J x = ne2 τE x
{
m e 1 + ω 2 τ − ωτ 2 E y
2 1 + ω 2 τ } (14a)
2
J y = ne2 τE y
{
m e 1 + ω 2 τ + ωτ 2 E x
2 1 + ω 2 τ } (14b)
2
To put the equations in a closer to experimental form, we note that µ = eτ/m e , and also ωτ = µB
J x = σ xx E x + σ xy E y = neµ e E x − neµ 2 eBE y
(15a)
J y = σ yx E x + σ yy E y = neµ 2 eBE x + neµ e E y
(15b)
One thing particularly notable and which is not clearly expressed by above equations is that similar
expressions for the holes also exist and that the conductivities of both carrier species are additive.
In fact the above equations hold true for any number of carrier species and the conductivities add
up provided the basic assumptions like single parabolic band for conduction for each carrier species
(m e constant) and energy independent relaxation time (τconstant). The reader must be warned
that these assumptions are far from reality in many cases. These expressions will be our starting
point for further analysis of multicarrier conduction [1].
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References
[1] W.A. Beck and J.R. Anderson. Determination of electrical transport properties using a novel
magnetic field-dependent hall technique. J. Appl. Phys., 62(2):541, 1987.
[2] R. A. Smith. Semiconductors. Cambridge University Press, 1978.
[3] L.J. van der Pauw. A method of measuring the resistivity and hall coefficient on lamellae of
arbitrary shape. Philips Res. Repts., 13:1–9, 1958.
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