Transcript EE 5340 Lecture 13 - Semiconductor Device Theory Fall 2003

EE 5340
Semiconductor Device Theory
Lecture 13 - Fall 2003
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc
L 13 Oct 7
1
Direct carrier
gen/recomb
(Excitation can be by light)
-
gen
+
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rec
+
E
Ec
Ef
Efi
Ec
Ev
Ev
k
2
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
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3
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
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4
Shockley-ReadHall Recomb
Indirect, like Si, so
intermediate state
ET
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E
Ec
Ef
Efi
Ec
Ev
Ev
k
5
S-R-H trap
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”
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6
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
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7
S-R-H
recombination
• Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
sn (capture cross sect for electrons),
sp (capture cross sect for holes), with
tno = (Ntvthsn)-1, and
tpo = (Ntvthsp)-1, where sn,p~p(rBohr,n.p)2
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8
S-R-H net recombination rate, U
• In the special case where tno = tpo = to
= (Ntvthso)-1 the net rec. rate, U is
d dp
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)
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9
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
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10
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
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11
Minority
electron
lifetimes,
taken from
Shur**
p. 101.
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12
Minority
hole
lifetimes,
taken from
Shur**
p. 101.
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13
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)
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14
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
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15
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
L 13 Oct 7
      
n  
i
j
k n,
y
z 
 x
16
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
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17
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
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18
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".
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19
The Continuity
Equation (cont.)
dn dp
 The LHS,
or
 -U, 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).
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20
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).
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21
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
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22
References
* Device Electronics for Integrated Circuits, 2nd ed., by Muller
and Kamins, Wiley, New York, 1986.
** Physics of Semiconductor Devices, M. Shur, Wiley.
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