Transcript Solid-State Electronics Chap. 5

Chap 5. Carrier Motion





Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
Hall Effect
Homework
Solid-State Electronics
Chap. 5
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Carrier Drift
 When an E-field (force) applied to a semiconductor, electrons and
holes will experience a net acceleration and net movement, if there are
available energy states in the conduction band and valence band. The
net movement of charge due to an electric field (force) is called “drift”.
 Mobility: the acceleration of a hole due to an E-field is related by
* dv
F  mp
 qE
dt
If we assume the effective mass and E-field are constants, the we can
obtain the drift velocity of the hole by
eEt
vd  *  vi  t , E
mp
where vi is the initial velocity (e.g. thermal velocity) of the hole and t is
the acceleration time.
Solid-State Electronics
Chap. 5
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
E=0
 In semiconductors, holes/electrons are involved in collisions with
ionized impurity atoms and with thermally vibration lattice atoms. As
the hole accelerates in a crystal due to the E-field, the velocity/kinetic
energy increases. When it collides with an atom in the crystal, it lose s
most of its energy. The hole will again accelerate/gain energy until is
again involved in a scattering process.
Solid-State Electronics
Chap. 5
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
 If the mean time between collisions is denoted by cp, then the average
drift velocity between collisions is
vdp e cp
 e cp 
m

 *
p
vd   *  E  m p E
 m 
E
mp
p


where mp (cm2/V-sec) is called the hole mobility which is an important
parameter of the semiconductor since it describes how well a particle
will move due to an E-field.
 Two collision mechanisms dominate in a semiconductor:
– Phonon or lattice scattering: related to the thermal motion of atoms; mL T-3/2
– Ionized impurity scattering: coulomb interaction between the electron/hole and the


ionized impurities; mI T3/2/NI., N I  N d  N a : total ionized impurity conc. , mI 
If T, the thermal velocity of hole/electron carrier spends less time in the
vicinity of the impurity.  less scattering effect  mI 
Solid-State Electronics
Chap. 5
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
Electron mobility
Solid-State Electronics
Chap. 5
Hole mobility
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Drift Current Density
 If the volume charge density of holes, qp, moves at an average drift
velocity vdp, the drift current density is given by
Jdrfp = (ep) vdp = emppE.
Similarly, the drift current density due to electrons is given by
Jdrfn = (-en) vdp = (-en)(-mnE)=emnnE
 The total drift current density is given by Jdrf = e(mnn+mpp) E
Solid-State Electronics
Chap. 5
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Conductivity
 The conductivity  of a semiconductor material is defined by Jdrf   E,
so = e(mnn+mpp) in units of (ohm-cm)-1
 The resistivity  of a semiconductor is defined by   1/ 
Solid-State Electronics
Chap. 5
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Resistivity Measurement
 Four-point probe measurement
  2s
Solid-State Electronics
Chap. 5
V
Fc ; Fc : correction factor
I
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation
 So far we assumed that mobility is indep. of E-field, that is the drift
velocity is in proportion with the E-field. This holds for low E-filed. In
reality, the drift velocity saturates at ~107 cm/sec at an E-field ~30
kV/cm. So the drift current density will also saturate and becomes
indep. of the applied E-field.
Solid-State Electronics
Chap. 5
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation of GaAs
 For GaAs, the electron drift velocity reaches a peak and then decreases
as the E-field increases. negative differential mobility/resistivity,
which could be used in the design of oscillators.
 This could be understood by considering the E-k diagram of GaAs.
Solid-State Electronics
Chap. 5
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation of GaAs
 In the lower valley, the density of state effective mass of the electron
mn* = 0.067mo. The small effective mass leads to a large mobility. As
the E-field increases, the energy of the electron increases and can be
scattered into the upper valley, where the density of states effective
mass is 0.55mo. The large effective mass yields a smaller mobility.
 The intervalley transfer mechanism results in a decreasing average
drift velocity of electrons with E-field, or the negative differential
mobility characteristic.
Solid-State Electronics
Chap. 5
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Carrier Diffusion
 Diffusion is the process whereby particles flow from a region of high
concentration toward a region of low concentration. The net flow of
charge would result in a diffusion current.
Solid-State Electronics
Chap. 5
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Diffusion Current Density
 The electron diffusion current density is given by Jndif = eDndn/dx,
where Dn is called the electron diffusion coefficient, has units of cm2/s.
 The hole diffusion current density is given by Jpdif = -eDpdp/dx,
where Dp is called the hole diffusion coefficient, has units of cm2/s.
 The total current density composed of the drift and the diffusion
current density.
1-D J  enm n E x  epm p E x  eDn dn  eD p dp
dx
or
3-D
Solid-State Electronics
Chap. 5
dx
J  enm n E x  epm p E x  eDnn  eD p p
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Graded Impurity Distribution
 In some cases, a semiconductors is not doped uniformly. If the
semiconductor reaches thermal equilibrium, the Fermi level is constant
through the crystal so the energy-band diagram may qualitatively look
like:

 Since the doping concentration decreases as x increases, there will be a
diffusion of majority carrier electrons in the +x direction.
 The flow of electrons leave behind positive donor ions. The separation
of positive ions and negative electrons induces an E-field in +x
direction to oppose the diffusion process.
Solid-State Electronics
Chap. 5
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Induced E-Field
d
d ( E /( e))
1 dE
Fi
Fi

 The induced E-field is defined as E x    
dx
dx
e dx
that is, if the intrinsic Fermi level changes as a function of distance
through a semiconductor in thermal equilibrium, an E-field exists.
 If we assume a quasi-neutrality condition in which the electron
concentration is almost equal to the donor impurity concentration, then
 N d ( x) 
 E  Ei 


no  ni exp  F

N
(
x
)

E

E

kT
ln
d
F
i

 kT 
 ni 
d ( E F  Ei ) d ( Ei )
kT dN d ( x)



dx
dx
N d ( x) dx
 kT  1 dN d ( x)
 E x   
 e  N d ( x) dx
 So an E-field is induced due to the nonuniform doping.
Solid-State Electronics
Chap. 5
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Einstein Relation
 Assuming there are no electrical connections between the
nonuniformly doped semiconducotr, so that the semiconductor is in
thermal equilibrium, then the individual electron and hole currents
must be zero.
dn
 J n  0  enm n E x  eDn
dx
 Assuming quasi-neutrality so that n  Nd(x) and
dN d ( x)
dx
dN d ( x)
 kT  1 dN d ( x)
 0  en m n N d ( x) 
 eDn
dx
 e  N d ( x) dx
J n  0  eN d ( x) m n E x  eDn
Dn
kT
- - - -Einstein relation
mn
e
D p kT


 Similarly, the hole current Jp = 0
mp
e

Solid-State Electronics
Chap. 5

16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Einstein Relation
 Einstein relation says that the diffusion coefficient and mobility are not
independent parameters.
Typical mobility and diffusion coefficient values at T=300K
(m = cm2/V-sec and D = cm2/sec)
Silicon
GaAs
Germaium
Solid-State Electronics
Chap. 5
mn
Dn
mp
Dp
1350
8500
3900
35
220
101
480
400
1900
12.4
10.4
49.2
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
 The hall effect is a consequence of the forces that are exerted on
moving charges by electric and magnetic fields.
 We can use Hall measurement to
– Distinguish whether a semiconductor is n or p type
– To measure the majority carrier concentration
– To measure the majority carrier mobility
Solid-State Electronics
Chap. 5
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
 A semiconductor is electrically connected to Vx and in turn a current Ix
flows through. If a magnetic field Bz is applied, the electrons/holes
flowing in the semiconductor will experience a force F = q vx x Bz in
the (-y) direction.
 If this semiconductor is p-type/n-type, there will be a buildup of
positive/negative charge on the y = 0 surface. The net charge will
induce an E-field EH in the +y-direction for p-type and -y-direction for
n-type. EH is called the Hall field.
 In steady state, the magnetic force will be exactly balanced by the
induced E-field force. F = q[E + v x B] = 0  EH = vx Bz and the Hall
voltage across the semiconductor is VH = EHW
 VH >0  p-type, VH < 0  n-type
Solid-State Electronics
Chap. 5
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
 VH = vx W Bz, for a p-type semiconductor, the drift velocity of hole is
vdx 

Jx
Ix
I B
I B

 VH  x z  p  x z
ep ep Wd 
epd
edVH
for a n-type,
n
I x Bz
edVH
 Once the majority carrier concentration has been determined, we can
calculate the low-field majority carrier mobility.
 For a p-semiconductor, Jx = epmpEx.  m p 
 For a n-semiconductor,
Solid-State Electronics
Chap. 5
 mn 
20
IxL
epVxWd
IxL
enVxWd
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
Hall-bar with “ear”
Solid-State Electronics
Chap. 5
van deer Parw configuration
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
 5.14
 5.20
Solid-State Electronics
Chap. 5
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Solid-State Electronics
Chap. 5
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU