Download Electron Velocity, v - NC State University

Electrical
Characterization
Techniques
Hall
Effect
C-V
DLTS
• Electrical Parameters :
• Carrier concentration (ionized donors,
acceptors)
• Carrier type (electrons or holes)
• Carrier mobility
Derive Ohm’s law: V = IR
• Consider a block of material
• What is the current, I, for an applied voltage, V ?
I
-
V
+
Total Current, I
n = # electrons per unit volume
v = velocity of electrons in the material
# electrons in volume AL,
time to travel distance L,
N = nAL
t=L/v
 I = Q/t = qN/t = q (nAL) / (L/v) = A nqv
volume AL
A
L
I
-
V
+
Electron Velocity, v
What is the electron velocity, v ?
E = V/L
F = qE
a = F/m
Electric field in material,
Force on an electron,
Electron acceleration,
L
a
E
I
-
V
+
Electron Velocity, v
Electrons accelerate until they collide with
atoms in the material
Assume electron loses all its energy (v=0)
after each collision
L
E
I
-
V
+
Mobility
Electron velocity, v = at
t ~ 10-12 s
v=Ft/m
= qE t / m
= (qt / m) E
= mE
m = electron mobility = qt / m
Conductivity
I = A nqv
= A nq m E
Current density,
J=I /A
= nqm E
=sE
s = conductivity = nqm = nq2t / m
Resistivity, r = 1 / s
Ohm’s Law
I = A nqm E = A nqm V/L
Rearranging gives V = I (L/Anqm)
Resistance, R = L / Anqm = L/sA = rL/A
Units: [R] = W
[r] = W cm
[s] = (W cm)-1
[m] = cm2 / Vs
Conductivity Measurement
V
Jx = Ix / A = Ix / tW
Ix
x
t
W
Ex = Vx / L
s = Jx / Ex = Ix L / Vx tW
L
A = tW
• Usually use symmetric samples (L = W):
• s = Ix / Vxt
• Measure Ix, Vx, t  Can determine s
• s = nqm  Can determine n if mobility is
known (or vice versa)
• Need another technique to determine n or m
Hall Effect : Simple Analysis
• Discovered by Hall in 1879 on Au foils
• Reference (review article):
D.A. Anderson and N. Apsley, “The Hall
Effect in III-V Semiconductor Assessment”,
Semicond. Sci. Technol. 1, 187 (1986)
Hall Effect : Simple Analysis
V
x
Vy
Bz
Bz
t
vx
Ey
W
L
Ix
A = tW
Ix Ex
• Carriers experience force from :
Applied electric field,
Applied magnetic field,
Fx = qEx
Fy = qvxBz
• Typical field ~ 0.05 - 1 Tesla = 500 - 104 Gauss
• Carriers are deflected
producing an electric field,
Ey = Vy / W
(Hall voltage)
• Electric field builds up that counteracts magnetic field
force
• Sign of Hall voltage gives dominant carrier type
Hall Effect
At equilibrium,
qEy + qvxBz = 0
Ey = - vxBz
vx = Jx / nq
Jx = Ix / tW
Ey = Vy / W
Vy t / BzIx = - (nq)-1
Define Hall coefficient,
RH = - (nq)-1 = Vy t / BzIx
Measure Hall coefficient
Then
n = - (RHq)-1
Measure conductivity (at B=0)
Then
m = s / nq
Or
m = - RHs
Van der Pauw Technique
D
C
B
A
• Gives RH, s for arbitrary sample shapes
• Assumptions :
• Contacts are at circumference of sample
• Contacts are much smaller than sample area
• Sample is uniformly thick
• Sample has no holes
• Sample thickness << contact spacing
References :
L.J. van der Pauw, Philips Research Reports 13, 1
(1958)
L.J. van der Pauw, Philips Research Reports 20, 220
(1961)
Van der Pauw Technique
D
C
B
A
Conductivity measurement (when B = 0) :
• Apply IBC, measure VDA, define RBC-DA = VDA/IBC
• Apply IAB, measure VCD, define RAB-CD = VCD/IAB
van der Pauw analysis :
r=(s)-1 = (p/ln2) t [(RBC-DA + RAB-CD)/2] F(RAB-CD / RBC-DA
)
4.53
correction
factor
Previous simple analysis gave :
r=(s)-1 = t Vx / Ix
Van der Pauw Technique
Correction factor, F
from
Schroder,
Fig. 1.7,
p. 15
RAB-CD / RBC-DA
• Usually use symmetric samples (F ~ 1):
D
A
C
B
Van der Pauw Technique
D
C
A
B
Hall effect measurement :
• Apply B perpendicular to surface
• Apply IBD
• Measure DRBD-AC = VAC(B) / IBD – VAC(0)/IBD
• Then RH = ( t / B) DRBD-AC
Previous simple analysis gave :
RH = ( t / B ) (Vy / Ix)
Application to Thin Films
thin film
substrate
• Want to measure n, m of thin film not
substrate
• Conductance of substrate must be very
low compared to film
• No current flow in substrate
• Use semi-insulating (S.I.) substrates
• S.I. substrate created by doping with an
impurity producing deep traps (acceptors)
• e.g., Cr in GaAs
Fe in InP
Film Thickness
• What is the film thickness, t ?
• Depletion layers form at surfaces and
interfaces due to defects
• Fermi level is pinned at EF
Chandra et al., Solid State Electronics 22, 645
(1979)
t = d – Ls - Li
Substrate
Film
d
Ls
EF
Li
Film Thickness
• For GaAs with n ~ 1015 cm-3
Ls ~ 1 mm, Li ~ 1 mm
• Need thick films, d > 2 – 3 mm
Compensation
• Conductivity and Hall effect measure net free
carrier concentration
Or
n = ND+ - NAp = NA- - ND+
• Mobility can determine the compensation ratio :
q = NA- / ND+
Walukiewicz et al., J. Appl. Phys. 51, 2659 (1980)
n
Compensation
compensation ratio, q = NA- / ND+
Walukiewicz et al., J. Appl. Phys. 51, 2659 (1980)
Compensation
• Mobility is affected (reduced) by scattering mechanisms
between the free carriers (electrons and holes) and the sample
• Scattering
mechanisms:
• Phonons (acoustic + optical)
• Impurity atoms (neutral + ionized)
• Alloy disorder
• Scattering from surfaces and interfaces
• Defect scattering
Temperature-Dependent Hall
Effect/Conductivity
• Can determine scattering mechanisms by using
temperature-dependent measurements
At low T, ionized
impurity scattering
dominates
At high T,
phonon scattering
dominates
From Ibach & Luth, Fig. 12.13, p. 291
Temperature-Dependent Hall
Effect/Conductivity
• Can determine scattering mechanisms by using
temperature-dependent measurements
At low T, ionized
impurity scattering
dominates
At high T,
phonon scattering
dominates
From Ibach & Luth, Fig. 12.12, p. 291
Temperature-Dependent Hall Effect
• Can determine donor or acceptor energy levels
n ~ exp [ – (EC – ED)/kT ]
• Donors become increasingly ionized as T
increases
• slope of Arrhenius plot (log n vs 1/T)  EC – ED
C-V
• Gives n = ND+ as a function of depth
• Requires a device: Schottky diode, p-n junction
• e.g., apply metal contacts to semiconductor
sample to form Schottky diode
• Apply reverse bias voltage, V
W
-eV
++
++
+
+ + + ++ + + + + + + + + + + +
EF
C-V
• Apply small ac signal (dV~ 10 mV @ 1 MHz) on top of dc
reverse bias
• Depletion width varies (dW) with ac signal (dV)
• Causes donor ionization over width dW
• Measure capacitance change
• Can determine n = ND+
W
-eV
-e(V+ dV)
+++
++++
++++++++++++++
EF
++
++++
+++
+ ++ + +++++ ++
dW
EF
C-V
C = eA/W
dQ = - e ND+ A dW
C = - dQ/dV = eA ND+ dW/dV
ND+ =
2
eeA2 [ d(1/C2)/dV ]
• Can determine ND+ from slope of 1/C2 versus V
1/C2
slope = 2/ [eeA2 ND+]
V
• Can convert voltage scale to depth scale by W = eA/C
C-V
from Schroder, Fig. 2.2, p. 67
C-V
Usually assume n = ND = ND+ :
• All donors become ionized
• Minority carriers are neglected
• All majority carriers in depletion region are
removed
C-V
Interface characterization (MOSFETs)
C-V
Disadvantage :
• Maximum depth is limited by electrical
breakdown at high reverse bias
C-V Profiling :
• Can perform C-V measurement while
performing a chemical etch
Reference :
T. Ambridge et al., J. Appl. Electrochem. 5, 319
(1975)
Electrochemical C-V Profiling
• Replace metal contact with electrolytic solution
• Destructive method
DLTS
• Deep level transient spectroscopy
Reference :
D.V. Lang, J. Appl. Phys. 45, 3023 (1974)
DLTS
• What are traps ?
• Unwanted impurities or crystal
defects
→ e.g., Fe, Au in InP, GaAs
→ Introduces discrete energy
levels in the bandgap, usually
near midgap
→ Trap electrons or holes
EF
+ + + + + + + +
ET
Electron
traps
DLTS
• Electron traps
• Negative when an e- is captured
• Neutral when empty
• Acceptor-like
• Hole traps
• Positive when a hole is captured
• Neutral when empty
• Donor-like
EF
+ + + + + + + +
ET
Electron
traps
DLTS
• Requires Schottky diode or p-n junction
• e.g., apply metal contacts to sample to form
Schottky diode
• Apply reverse bias pulse and measure capacitance
transient
• Gives :
• NT vs W
• NT energy levels
DLTS
V
W0
C
0
C0
EF
t
t
WV0
Transient
V
C
-eV
-V
EF
t CV0
t
Steady-state
V
WV
C
-eV
-V
t
CV
EF
t
DLTS
V
-V
C
CV
t CV0
t
DC = CVNT / 2ND
• Capacitance transient gives trap concentration, NT
DLTS
WV0
V
C
-eV
-V
CV
t CV0
EF
t
• Capacitance transient is characteristic of the
emission of electrons from the traps:
DC(t) = DC exp (-ent)
emission rate
en ~ exp [ - (Ec – ET)/ kT ]
DLTS
DC(t) = DC exp (-ent)
en ~ exp [ - (Ec – ET)/ kT ]
• Capacitance transient varies with temperature
From Schroder, Fig. 5.12, p. 291
DLTS
DC(t) = DC exp (-ent)
From Schroder, Fig. 5.12, p. 291
• Define a “rate window” using two times, t1
and t2
• C(t2) – C(t1) is maximum when
(t1-t2)/ln(t1/t2) = 1 / en(T)
e.g., C(t2) – C(t1) is maximum at 260 K in above
figure
DLTS
• Vary the temperature and measure DC with t1 & t2 fixed
• Produces peak when (t1-t2)/ln(t1/t2) = 1 / en(T)
• en ~ exp [ - (Ec – ET)/ kT ]
• Each kind of trap has different ET and therefore en
• Produces a distinct peak for each trap
• Called the DLTS spectrum
From
D.V. Lang,
JAP 45,
3023 (1974)
DLTS
• Vary the rate window
• Peak moves to new position
From D.V. Lang, JAP 45, 3023 (1974)
DLTS
• Slope of Arrhenius plot (log en vs 1/T) gives
trap energy level, ET
From D.V. Lang, JAP 45, 3023 (1974)