Transcript Document 7653350

Semiconductor Device
Modeling and Characterization
EE5342, Lecture 5-Spring 2005
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc/
L5 February 02
1
Equipartition
theorem
• The thermodynamic energy per
degree of freedom is kT/2
Consequently,
1
2
mv
2
vrms
L5 February 02
thermal
3
 kT, and
2
3kT
7

 10 cm / sec
m*
2
Carrier velocity
1
saturation
• The mobility relationship v = mE is
limited to “low” fields
• v < vth = (3kT/m*)1/2 defines “low”
• v = moE[1+(E/Ec)b]-1/b, mo = v1/Ec for Si
parameter electrons
holes
v1 (cm/s) 1.53E9 T-0.87 1.62E8 T-0.52
Ec (V/cm) 1.01 T1.55
1.24 T1.68
b
2.57E-2 T0.66 0.46 T0.17
L5 February 02
3
vdrift
L5 February 02
[cm/s]
vs. E
[V/cm]
(Sze2, fig. 29a)
4
Carrier velocity
saturation (cont.)
• At 300K, for electrons, mo = v1/Ec
= 1.53E9(300)-0.87/1.01(300)1.55
= 1504 cm2/V-s, the low-field
mobility
• The maximum velocity (300K) is
vsat = moEc
= v1 = 1.53E9 (300)-0.87
= 1.07E7 cm/s
L5 February 02
5
Diffusion of
carriers
• In a gradient of electrons or holes,
p and n are not zero
• Diffusion current,`J =`Jp +`Jn (note
Dp and Dn are diffusion coefficients)

 p p
p 
Jp  qDpp  qDp  i 
j  k 
z 
 x y

 n
n
n 
Jn   qDnn   qDn  i 
j  k 
x y
z 

L5 February 02
6
Diffusion of
carriers (cont.)
• Note (p)x has the magnitude of
dp/dx and points in the direction of
increasing p (uphill)
• The diffusion current points in the
direction of decreasing p or n
(downhill) and hence the - sign in the
definition of`Jp and the + sign in the
definition of`Jn
L5 February 02
7
Diffusion of
Carriers (cont.)
L5 February 02
8
Current density
components

Note, since E  V



Jp,drift   pE  pqm pE  pqm pV



Jn,drift  nE  nqmnE  nqmn V

Jp,diffusion   qDpp

Jn,diffusion   qDn n
L5 February 02
9
Total current
density
The total current density is driven by
the carrier gradients and the potential
gradient





Jtotal  Jp,drift  Jn,drift  Jp,diff.  Jn,diff.

Jtotal    p  n V  qDpp  qDnn

L5 February 02

10
Doping gradient
induced E-field
•
•
•
•
•
If N = Nd-Na = N(x), then so is Ef-Efi
Define f = (Ef-Efi)/q = (kT/q)ln(no/ni)
For equilibrium, Efi = constant, but
for dN/dx not equal to zero,
Ex = -df/dx =- [d(Ef-Efi)/dx](kT/q)
= -(kT/q) d[ln(no/ni)]/dx
= -(kT/q) (1/no)[dno/dx]
= -(kT/q) (1/N)[dN/dx], N > 0
L5 February 02
11
Induced E-field
(continued)
• Let Vt = kT/q, then since
• nopo = ni2 gives no/ni = ni/po
• Ex = - Vt d[ln(no/ni)]/dx
= - Vt d[ln(ni/po)]/dx
= - Vt d[ln(ni/|N|)]/dx, N = -Na < 0
• Ex = - Vt (-1/po)dpo/dx
= Vt(1/po)dpo/dx
= Vt(1/Na)dNa/dx
L5 February 02
12
The Einstein
relationship
• For Ex = - Vt (1/no)dno/dx, and
• Jn,x = nqmnEx + qDn(dn/dx) = 0
• This requires that
nqmn[Vt (1/n)dn/dx] = qDn(dn/dx)
• Which is satisfied if
Dp
Dn kT

 Vt , likewise
 Vt
mn
q
mp
L5 February 02
13
Direct carrier
gen/recomb
(Excitation can be by light)
-
gen
+
L5 February 02
rec
+
E
Ec
Ef
Efi
Ec
Ev
Ev
k
14
Direct gen/rec
of excess carriers
• Generation rates, Gn0 = Gp0
• Recombination rates, Rn0 = Rp0
• In equilibrium: Gn0 = Gp0 = Rn0 = Rp0
• In non-equilibrium condition:
n = no + dn and p = po + dp, where nopo=ni2
and for dn and dp > 0, the recombination
rates increase to R’n and R’p
L5 February 02
15
Direct rec for
low-level injection
• Define low-level injection as
dn = dp < no, for n-type, and
dn = dp < po, for p-type
• The recombination rates then are
R’n = R’p = dn(t)/tn0, for p-type, and
R’n = R’p = dp(t)/tp0, for n-type
• Where tn0 and tp0 are the minoritycarrier lifetimes
L5 February 02
16
Shockley-ReadHall Recomb
Indirect, like Si, so
intermediate state
ET
L5 February 02
E
Ec
Ef
Efi
Ec
Ev
Ev
k
17
S-R-H trap
1
characteristics
• The Shockley-Read-Hall Theory
requires an intermediate “trap” site in
order to conserve both E and p
• If trap neutral when orbited (filled)
by an excess electron - “donor-like”
• Gives up electron with energy Ec - ET
• “Donor-like” trap which has given up
the extra electron is +q and “empty”
L5 February 02
18
S-R-H trap
char. (cont.)
• If trap neutral when orbited (filled)
by an excess hole - “acceptor-like”
• Gives up hole with energy ET - Ev
• “Acceptor-like” trap which has given
up the extra hole is -q and “empty”
• Balance of 4 processes of electron
capture/emission and hole capture/
emission gives the recomb rates
L5 February 02
19
S-R-H
recombination
• Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
n (capture cross sect for electrons),
p (capture cross sect for holes), with
tno = (Ntvthn)-1, and
tpo = (Ntvthn)-1, where n~p(rBohr)2
L5 February 02
20
S-R-H
recomb. (cont.)
• In the special case where tno = tpo = to
the net recombination rate, U is
ddp
ddn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  dn, and p  po  dp, (dn  dp)
L5 February 02
21
S-R-H “U” function
characteristics
• The numerator, (np-ni2) simplifies in
the case of extrinsic material at low
level injection (for equil., nopo = ni2)
• For n-type (no > dn = dp > po = ni2/no):
(np-ni2) = (no+dn)(po+dp)-ni2
= nopo - ni2 + nodp + dnpo + dndp
~ nodp (largest term)
• Similarly, for p-type, (np-ni2) ~ podn
L5 February 02
22
S-R-H “U” function
characteristics (cont)
• For n-type, as above, the denominator
= to{no+dn+po+dp+2nicosh[(Et-Ei)kT]},
simplifies to the smallest value for
Et~Ei, where the denom is tono, giving
U = dp/to as the largest (fastest)
• For p-type, the same argument gives
U = dn/to
• Rec rate, U, fixed by minority carrier
L5 February 02
23
S-R-H net recombination rate, U
• In the special case where tno = tpo = to
= (Ntvtho)-1 the net rec. rate, U is
ddp
ddn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  dn, and p  po  dp, (dn  dp)
L5 February 02
24
S-R-H rec for
excess min carr
• For n-type low-level injection and net
excess minority carriers, (i.e., no > dn
= dp > po = ni2/no),
U = dp/to, (prop to exc min carr)
• For p-type low-level injection and net
excess minority carriers, (i.e., po > dn
= dp > no = ni2/po),
U = dn/to, (prop to exc min carr)
L5 February 02
25
Minority
hole
lifetimes.
Taken
from
Shur3,
(p.101).
L5 February 02
26
Minority
electron
lifetimes.
Taken
from
Shur3,
(p.101).
L5 February 02
27
Parameter example
•
tmin =
(45 msec)
1+(7.7E-18cm3Ni+(4.5E-36cm6Ni2
• For Nd = 1E17cm3, tp = 25 msec
– Why Nd and tp ?
L5 February 02
28
S-R-H rec for
deficient min carr
• If n < ni and p < pi, then the S-R-H net
recomb rate becomes (p < po, n < no):
U = R - G = - ni/(2t0cosh[(ET-Efi)/kT])
• And with the substitution that the
gen lifetime, tg = 2t0cosh[(ET-Efi)/kT],
and net gen rate U = R - G = - ni/tg
• The intrinsic concentration drives the
return to equilibrium
L5 February 02
29
The Continuity
Equation
• The chain rule for the total time
derivative dn/dt (the net generation
rate of electrons) gives
dn n n dx n dy n dz




.
dt t x dt y dt z dt
The definition of the gradient is
      
n  
i
j
k n,
x
y
z 

L5 February 02
30
The Continuity
Equation (cont.)
The definition of the vector velocity is
dx  dy  dz 
v 
i
j
k.
dt
dt
dt

 
Since A B  AxBx  AyBy  AzBz ,

dn n
then

 n  v
dt t
L5 February 02
31
The Continuity
Equation (cont.)
The gradient operator can be distributed



as   n v  n  v  n  v .
Considering the second term on the RHS,
 dx  dy  dz
 v 


 0, since
x dt y dt z dt

 dx d x

 0, etc.
x dt dt x
L5 February 02
32
The Continuity
Equation (cont.)
Consequently, since

Jn

 qn v , we have


n 1
dn n
   J n . So
  n v 

t q
dt t


dp p 1
dn n 1
   Jp

   J n , and

dt t q
dt t q
are the " Continuity Equations".
L5 February 02
33
The Continuity
Equation (cont.)
dp
dn
 The LHS,
or
 -V, of the Continuity Eq.
dt
dt
represents the Net Generation Rate of n
or p at a particular point in space (x, y, z).
n p
 The first term on the RHS,
or , is
t t
the " explicit" Local Rate of Change of n or
p at (x, y, z).
L5 February 02
34
The Continuity
Equation (cont.)

1
The second term on the RHS,    J n
q

1
or    J p is the local rate of n or p
q
concentrations flowing " out of" the
point (x, y, z). Note the difference in
signs for electrons (-q) and holes (  q).
L5 February 02
35
The Continuity
Equation (cont.)
So, we can re - write the continuity
equation for the holes as :

p dp 1

   Jp
t dt q
Which can be interpreted as :
Local rate of change 
net generation rate  rate of inflow
L5 February 02
36
References
• 1Device Electronics for Integrated
Circuits, 2 ed., by Muller and Kamins,
Wiley, New York, 1986.
• 2Physics of Semiconductor Devices,
by S. M. Sze, Wiley, New York, 1981.
• 3 Physics of Semiconductor Devices,
Shur, Prentice-Hall, 1990.
L5 February 02
37