434
15 External Fields
(a)
(b)
(c)
(d)
(e)
Fig. 15.7 a–d Geometry for van-der-Pauw Hall measurements. a Best geometry (cloverleaf), b acceptable square
geometry with small contacts on the corners, c,d not recommended geometries with contacts on the edge centers or
inside the square, respectively. e Current distribution, as visualized by lock-in thermography [1432], in epitaxial ZnO
layer on sapphire with Hall geometry as in part b. Grey dashed line indicates the outline of the 10 × 10 mm 2 substrate,
grey areas indicate gold ohmic contacts
σ zz = σ 0 =
q
2 n τ
m ∗ = q n μ .
(15.9d)
Perpendicular to the magnetic field, the conductivity (σ zz ) is given by (8.5). If only one type of carrier
(charge q, density n) is considered, the condition j y = 0 leads to E y = μB E x and j x = σ 0 E x . The
Hall coefficient is defined as R H = E y /( j x B) or more precisely as
R H =
ρ xy
B
,
(15.10)
where the resistivity tensor ρ is the inverse of the conductivity tensor σ ,
ρ = σ
−1
=
⎛
⎝
ρ xx ρ xy 0
ρ yx ρ yy 0
0 0 ρ zz
⎞
⎠
(15.11a)
ρ xx = ρ yy =
σ xx
σ 2
xx + σ 2
xy
(15.11b)
ρ xy = −ρ yx =
σ xy
σ 2
xx + σ 2
xy
(15.11c)
ρ zz =
1
σ zz
=
1
σ 0
.
(15.11d)
For a single type of carriers, the Hall coefficient is therefore given by
R H =
μ
σ 0
=
1
q n
.
(15.12)
It is negative (positive) for electron (hole) conduction. We note that electrons and holes are deflected
in the same y-direction by the magnetic field and collect at the same electrode. Thus the Hall effect
15 External Fields
(a)
(b)
(c)
(d)
(e)
Fig. 15.7 a–d Geometry for van-der-Pauw Hall measurements. a Best geometry (cloverleaf), b acceptable square
geometry with small contacts on the corners, c,d not recommended geometries with contacts on the edge centers or
inside the square, respectively. e Current distribution, as visualized by lock-in thermography [1432], in epitaxial ZnO
layer on sapphire with Hall geometry as in part b. Grey dashed line indicates the outline of the 10 × 10 mm 2 substrate,
grey areas indicate gold ohmic contacts
σ zz = σ 0 =
q
2 n τ
m ∗ = q n μ .
(15.9d)
Perpendicular to the magnetic field, the conductivity (σ zz ) is given by (8.5). If only one type of carrier
(charge q, density n) is considered, the condition j y = 0 leads to E y = μB E x and j x = σ 0 E x . The
Hall coefficient is defined as R H = E y /( j x B) or more precisely as
R H =
ρ xy
B
,
(15.10)
where the resistivity tensor ρ is the inverse of the conductivity tensor σ ,
ρ = σ
−1
=
⎛
⎝
ρ xx ρ xy 0
ρ yx ρ yy 0
0 0 ρ zz
⎞
⎠
(15.11a)
ρ xx = ρ yy =
σ xx
σ 2
xx + σ 2
xy
(15.11b)
ρ xy = −ρ yx =
σ xy
σ 2
xx + σ 2
xy
(15.11c)
ρ zz =
1
σ zz
=
1
σ 0
.
(15.11d)
For a single type of carriers, the Hall coefficient is therefore given by
R H =
μ
σ 0
=
1
q n
.
(15.12)
It is negative (positive) for electron (hole) conduction. We note that electrons and holes are deflected
in the same y-direction by the magnetic field and collect at the same electrode. Thus the Hall effect