323
9
Magneto- and Quantum-Confined Transport
The magnetic eld is indispensable in the characterization and performance
evaluation of semiconductor devices. Magnetoresistance (MR) is the relative
change Δρ/ρ
o
in resistivity ρ from zero-eld resistivity ρ
o
when an external
magnetic eld is applied to them. In ordinary magnetoresistivity (OMR),
the effect is rather small and can be classically described as a result of the
force acting on conducting electrons in the magnetic and electric eld. A 3D
sample can be converted into a 1D conguration when the magnetic eld is
in the quantum domain. Unfortunately, for transverse conguration with
the electric eld perpendicular to the applied magnetic eld, the traditional
framework based on FD statistics and Boltzmann transport equation (BTE)
is of limited use and density matrix takes on an increasing importance [1].
A quantum-mechanical theory using a density matrix for electrical con-
ductivity in the presence of a magnetic eld is described. The conventional
methods using the BTE are not satisfactory because the magnetic eld intro-
duces a curvature in the free path of an electron. The expectation value of the
current density and the components of the conductivity tensor in a magnetic
eld are obtained. The advantages of the density matrix in electric transport
problems with a magnetic eld are discussed.
9.1 Classical Theory of MR
As discussed earlier, the conductivity of a 3D material is the reciprocal of
resistivity:
σ
ρ
τ
3
3
3
2
3
2
3
1
== =
nq
m
nq
mv
i
**
(9.1)
In the following equations, σ
0
is used to mean the 3D zero-magnetic-eld
conductivity in place of σ
3
, and ρ
o
is used for ρ
3
.
In the presence of the electric eld E and magnetic eld B, the force on the
electron is
Fq
vB
=− +×
()
E
(9.2)
324 Nanoelectronics
The transient effect of the change in velocity in response to this force can
now be described by the differential equation
m
dv
dt
qvB
v
*( )
=− +× −E
τ
(9.3)
where τ is used to mean the collision time τ
c
. The steady-state solution
()
dv dt
/ = 0
of this equation is obtained as
v
B
B
q
m
o
=−
+
()
+×
(
)
=
µ
µ
µµ
τ
0
0
2
0
1
EE with
*
(9.4)
For a magnetic eld in the z-direction, the components of velocity vector
v
are obtained as
v
B
B
xx
y
=−
+
()
−
(
)
µ
µ
µ
0
0
2
0
1
EE
(9.5)
v
B
B
xy
x
=−
+
()
+
(
)
µ
µ
µ
0
0
2
0
1
EE
(9.6)
v
zo
z
=µ
E
(9.7)
The current density
jn
qv=−
3
is now obtained as
j
xxxx xy y
=+σσEE
(9.8)
j
yy
xyyy
=+σσEE
x
(9.9)
j
zzzz
oz
==σσEE
(9.10)
with
σσ
σ
µ
xx yy
B
==
+
()
0
0
2
1
(9.11)
σσ
zz o
=
(9.12)
σσ
σ
µ
µ
xy yx
B
B=− =−
+
()
0
0
2
0
1
(9.13)
325Magneto- and Quantum-Confined Transport
The z-component of the current is not affected by a magnetic eld in the
z-direction. The diagonal components σ
xx
and σ
yy
of the magnetoconductiv-
ity decrease monotonically as the magnetic eld is increased. The magnitude
of the off-diagonal components σ
xy
and σ
yx
rst increases and then decreases
as B is increased. The relative MR in the classical model vanishes as
∆ρ
ρ
σσ
σσ
xx xx
xx xy0
0
22
10
=
+
−=
(9.14)
when the conductivity tensor is inverted into the resistivity tensor. Since
conductivity is a tensor, the inversion of ρ
xx
= 1/σ
xx
is not appropriate, but is
used in the published literature to extract mobility from MR experiments.
This incorrect form of the relative incremental MR is given by
∆ρ
ρ
σ
σ
µ
xx
xx
B
0
0
0
2
1=−=
()
(9.15)
Comparison of Equation 9.15 with Equation 9.14 clearly indicates that
Equation 9.15 is not correct for a constant collision time. Arora et al. [2]
resolved the apparent contradiction between the rise of MR mobility and
fall of drift mobility with increasing channel concentration. They attribute
this to the scattering-dependent MR factor that is evaluated by μ
MR
= MRμ
o
.
Inversion of tensors cannot be term by term as is done in Equation 9.15.
However, as shown later, the expression can be reproduced when the energy
dependence of the relaxation time is taken into account with a proportional-
ity factor A
o
so that Δρ
xx
/ρ
0
= A
o
(μ
0
B)
2
with A
o
= 0 for the energy-independent
mean-free time τ and A
o
= MR
2
when the energy dependence of collision
time is taken into account.
9.2 Rationale for Density Matrix
There have been many studies of electronic transport in solids by using the
BTE [3]. BTE has been very successful for problems involving zero or magnetic
eld in the order of a fraction of a tesla (T) that is a few kilogauss (kG). For
high magnetic elds, a density-matrix approach is required. Unfortunately,
a density-matrix approach used in the literature is quite difcult to under-
stand. A simple quantum-mechanical derivation of the eld terms of the
BTE based on the gage-independent density-matrix formalism was given by
Weisenthal and de Graaf [4]. With the magnetic eld present, extreme care is
required in the correct interpretation of the physical quantities involved for
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