Transcript Document

CHAPTER 8
MOBILITY
1
8.1 INTRODUCTION
2
8.1 INTRODUCTION
• High mobility material has higher frequency response
and higher current.
• Electron-electron or hole-hole scattering has no firstorder effect on the mobility. Electron-hole scattering
reduces the mobility.
• Minority carriers has ionized impurity scattering and
electron-hole scattering, majority carriers has ionized
impurity scattering.
3
8.2 CONDUCTIVITY
MOBILITY
4
CONDUCTIVITY MOBILITY
For p-type material.
Measure the majority carrier concentration and the
conductivity/resistivity is sufficient to calculate the
conductivity mobility.
5
8.3 HALL EFFECT
AND MOBILITY
6
Basic Equations for Uniform Layers or Wafers
Schematic illustrating the Hall
effect in a p-type sample.
7
Hall angle is defined as the angle between Ex and Ey.
8
For p-type sample.
For n-type sample.
When both carriers
present.
9
Temperature and magnetic
field dependent Hall
coefficient for HgCdTe
showing typical mixed
conduction behavior.
T=220~300K, n=ni2/p, RH is independent of B.
T=100~200K, mixed conduction causes RH to
Decrease and depends on B.
T<100K, p dominates, RH is independent of B.
10
Hall coefficient and
electron density for GaAs.
No mixed conduction, and it is independent of B.
11
The above results are based on the assumption of
energy-independent scattering mechanisms.If it is
relaxed, the Hall scattering factor r must be Included:
12
(a) Bridge-type Hall sample, (b) lamella-type van der Pauw Hall sample.
R12,34=V34/I12, V34=V4-V3
F is a function of Rr=R12,34/R23,41
For symmetric samples F=1.
ΔR24,13 is the difference with and
without the magnetic field.
13
The van der Pauw F factor plotted against Rr.
For uniformly doped samples with thickness d, the
sheet Hall coefficient is
RHS=RH/d and μH=︳RHS︳/ρs, ρs=ρ/d.
14
Nonuniform Layers
Hall measurements give average values,
assuming r=1:
15
The depth profile can be measured by etch and
measure method or by a pn junction controlled
depletion width.
Schottky-gated thin film van der Pauw sample, (a) top view, (b) cross section
along the A-A showing the gate two contacts and the space-charge region of
width W. The Hall measurement is performed in the region d-W.
16
The spatial varying Hall parameters is
determined by:
This is the so called differential Hall Effect, DHE.
17
Dopant density profiles determined by DHE, spreading resistance profiling,
and secondary ion mass spectrometry.
18
DHE encounters difficulty with large parameter
variation in multi layer system.
Assume a upper layer has carrier density p1 and
mobility μ1, and a lower layer has carrier density p2
and mobility μ2. The Hall effect measured weighted
values are:
19
Multi layers
A two layer structure has a upper layer thickness of d1
a conductivity of σ1, and a lower layer thickness of d2
a conductivity of σ2, the Hall constant is given as:
At low magnetic field it becomes:
20
Multi layers
At high magnetic field the Hall constant becomes:
where RH1 is the layer 1 Hall constant, RH2 is the
layer 2 Hall constant, d=d1+d2 and σ is:
21
Hall coefficient of a p-type substrate with an n-type layer as a function
of n1t1 for two magnetic fields. For low and high n1d1 the Hall coefficient
is independent of the magnetic field.
If the upper layer is more heavily doped or type
inverted by surface charges, then the surface Hall
parameters are measured
  (d1 / d) 1 ; R  RH1 (d / d1)
22
Sample Shapes and
Measurement Circuits
(a) Bridge-type Hall configuration, (b)-(d) lamella-type Hall configuration.
23
Van der Pauw Hall sample shapes.
24
Effect of non-ideal contact length or contact placement on the resistivity
and mobility for van der Pauw samples.
25
Hall sample with electrically shorted regions at the end; (a) top view with
the gate not shown, (b) cross section along cut A-A.
26
(a) Hall sample with electrically shorted end regions,
(b) ratio of measured voltage VHm to Hall voltages VH.
G=VHm/VH. VHm: VH for W/L<3. VH: VH for W/L<3.
27
8.4 MAGNETORESISTANCE
MOBILITY
28
MAGNETORESISTANCE MOBILITY
(a) Hall sample, (b) short, wide sample, Hall voltage is nearly shorted;
(c) Corbino disk, Hall voltage is shorted. They can be used to measure
the magnetoresistance effect.
29
Physical magnetoresistance effect (PMR):
The sample resistance increases when it is placed in
a magnetic field. Because the conduction is
anisotropic, involves more than one type of carrier,
and carrier scattering is energy dependent.
Geometrical magnetoresistance effect (GMR):
The charge carrier path deviates from a straight line.
GMR =  H
ξ=(〈τ3〉〈τ〉/〈τ2〉2)2 is the magnetoresistance
scattering factor.
30
Geometric magnetoresistance ratio of rectangular samples versus μGMRB
as a function of the length-width ratio.
31
For Corbino disc.
For rectangular samples with low L/W ratio and
μGMRB<1
For determining the error in μGMR to be <10%,
then L/W must be <0.4.
32
8.5 TIME-OF-FIGHT
DRIFT MOBILITY
33
(a) Drift mobility measurement arrangement and normalized output voltage
pulse (μp=180cm2/V．s,τn=0.67 μs,
T=423K, =60V/cm)
34
(b) output voltage pulses (μn=1000 cm2/V．s,τn=1μs, T=300K, =100V/cm,
N=1011 cm-2), (c) output voltage pulses (μn=1000cm2/ V．s, d=0.075cm,
T=300K, =100V/cm, N=1011cm-2). td=0.75us.
35
TIME-OF-FLIGHT DRIFT MOBILITY
This method was first demonstrated in HaynesShockley experiment. The pulse shape is :
Where N is the electron density in the packet at t=0. The
first term in the exponent is drift and diffusion part and
the second term is the recombination part.
36
TIME-OF-FLIGHT DRIFT MOBILITY
The minority carrier mobility
is determined as
The diffusion constant is
where the pulse width Δt
is measured at half the
maximum amplitude.
37
The lifetime is determined by measuring the electron
pulse at td1 and td2, corresponding to two drift
voltages Vdr1 and Vdr2. If there is no minority carrier
trapping, the output pulse is V01and V02, then
If there is minority carrier trapping, the pulse area is Ap, then
plot log(Ap) vs. delay time td, the slope should be -1/τn.
 = I/ qtn
38
(a) Time-of-flight measurement schematic, (b) output voltage for tt«RC,
The dashed lines indicate the effect of carrier trapping. (c) output voltage
for tt»RC, (d) implementation with a p+πn+ diode, both carriers can be
measured.
39
From the above system, QN=qN=QA+QC
At t=0 QA=0, at t=tt QA=QN, where tt is the
transit time
When QA changes from 0 to QN the external
current is:
40
The output voltage I×R is
For tt<<RC
For tt>>RC
tt can be determined from either case.
41
Two drift mobility
measurement implementations as discussed in the
text.
42
In Fig. a, V1 is applied to the gate and the diode, so there
won’t have inversion layer under the gate. V2’s period is
100us, its pulse width is 200ns. The poly-Si resistivity is
10KΩ/ □. Holes drift into the substrate, electrons drift
along the surface. The time difference between two
peaks gives the drift velocity. The field dependence of the
mobility is obtained by varying V2. The gate voltage
dependence of the mobility is obtained by varying V1.
43
For short channel devices with Rs
Solve for ID,sat and drop high order terms
Plot 1/ ID,sat vs. Lm, gives (1/ ID,sat)int and Lm,int
44
Substituting (1/ ID,sat)int into Lm,int gives
The above equation contains no μeff.
45
Plotting Lm,int vs. Lm,int /(1/ ID,sat)int has the slope A.
Plotting A vs. 1/ (VGS-VT) gives the line with slope S.
The saturation velocity is
sat=1/ (WeffCox(S-2Rs))
46
8.6 MOSFET MOBILITY
47
MOSFET MOBILITY
There are many kinds of mobility due to :
lattice (phonon) scattering, ionized impurity
scattering,neutral impurity scattering, piezoelectric
scattering, or surface scattering. According to
Mathiessen’s rule
48
Effective Mobility
A MOSFET drain current is given as below,
the first term is drift current, the second term
is diffusion current
49
Surface conditions for gate-to-channel capacitance measurements
for (a) VGS＜VT, 2Cov is measured (b) VGS＞VTb 2Cov+Cch is measured.
50
(a) CGC and Qn versus VGS; (b) ID versus VDS. W/L=10μm/10μm,
tox=10nm, NA=1.6×1017cm-3.
51
Once Qn and gd are obtained,μeff can be
calculated.
μeff versus VGS for the data
of Fig. VT=0.5V.
52
Split C-V measurement arrangement.
Qn can be obtained from I1.
Qb and substrate doping profile can be found.
53
Capacitance as a function of gate voltage.
54
The electric field is
vertical field.
η=1/2 for electron mobility,
1/3 for hole mobility.
55
(a) Electron and (b) hole effective mobility as a function of effective field.
Data taken from the references in the inserts.
56
(a) Effective mobility, (b) normalized mobility versus VGS-VT.
57
Effect of Gate Depletion and Channel Location
Simulated Gate-to-channel Capacitance Versus Gate Voltage as a function of
poly-Si gate doping density. Oxide leakage current not considered. tox=2nm,
NA=1.69×1017cm-3 μeff =300 cm2/V-s
58
MOSFET Cross Section Showing
Drain and Gate Current, gate current
adds to source current and
subtracts drain current
CGC=
Cox
1+Cox/CG +Cox/Cch
ID,eff = ID + IG/2
ID,eff = ID = ID(VDS2) - ID (VDS1)
59
Drain and Gate Currents Versus Gate Voltage
for n-channel MOSFET
Gate insulator: HfO2~2nm thick.
With permission of W. Zhu and T.P. Ma, Yale University
60
Effect of Inversion Charge Frequency
Response
Showing source and drain resistance (RS
and RD),inversion layer resistance Rch,
overlap, oxide, channel and bulk
capacitance (Cov,Cox,Cch, and Cb)
CGC=
Cox Cch
Cox+Cch+Cb
Re (
tanh()

)
 = (j 0.25C’RchL2)½
61
Simulated Gate-to-channel Capacitance Versus
Gate Voltage
(a) Frequency
(b) Channel Length
Gate depletion and oxide leakage current not considered. Oxide leakage
current not considered. tox=2nm, NA=1017cm-3 μn =300 cm2/V-s
62
it = exp(E/kT)/nthNc= 4x10-11exp(E/kT) [s]
Cit = q2Dit / (1+2it2)
CGC=
Cox (Cch+Cit)
Cox+Cch+Cb+Cit
VG=VFB+s+ Qs/Cox± Qit/Cox
63
Simulated Gate-to-channel Capacitance vs. VG
as a Function Interface Trap Density
(a) Gate depletion and oxide leakage
current not considered
(b) Oxide leakage current not considered
tox=2nm, NA=1017cm17, μn =300
cm2/V-s, Dit=1012cm-2eV-1, τit=5×10-8s
64
Field-Effect Mobility
The MOSFET gm is:
Define the field effect
mobility as:
65
Effective and field-effect mobilities.
66
If we take the μeff dependence on VG into
consideration, then
67
When MOSFET operates in saturation region the
drain current is:
B represents the body effect. If m is the slope of (ID,sat)1/2
vs. (VGS-VT), then
68