Transcript Document 7420693

EE 5340
Semiconductor Device Theory
Lecture 5 - Fall 2003
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc
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Assignment 2: 930AM students only*
• IF you have a class OTHER than 5340
at 8 AM, e-mail to [email protected]
– Subject line: “5340 8 AM Class”
– In the body of message include
• email address: ______________________
• Your Enrollment Name*: ______________
• Class you are taking at 8:00 AM: ________
* If you don’t do this, and don’t take Test 1 at 8:00
AM, you will not get credit for Test 1
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Classes of
semiconductors
• Intrinsic: no = po = ni, since Na&Nd << ni,
ni2 = NcNve-Eg/kT, ~1E-13 dopant level !
• n-type: no > po, since Nd > Na
• p-type: no < po, since Nd < Na
• Compensated: no=po=ni, w/ Na- = Nd+ > 0
• Note: n-type and p-type are usually
partially compensated since there are
usually some opposite-type dopants
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n-type equilibrium
concentrations
• N  Nd - Na , n type  N > 0
• For all N,
no = N/2 + {[N/2]2+ni2}1/2
• In most cases, N >> ni, so
# no = N, and
# po = ni2/no = ni2/N,
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(Law of Mass Action is always true in equilibrium)
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p-type equilibrium
concentrations
• N  Nd - Na , p type  N < 0
• For all N,
po = |N|/2 + {[|N|/2]2+ni2}1/2
• In most cases, |N| >> ni, so
# po = |N|, and
# no = ni2/po = ni2/|N|,
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(Law of Mass Action is always true in equilibrium)
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Intrinsic carrier
conc. (MB limit)
•
•
•
•
ni2 = no po = Nc Nv e-Eg/kT
Nc = 2{2pm*nkT/h2}3/2
Nv = 2{2pm*pkT/h2}3/2
Eg = 1.17 eV - aT2/(T+b)
# a = 4.73E-4 eV/K
# b = 636K
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Figure 1.9
Electron concentration vs.
temperature
for two n-type
doped semiconductors:
(a) Silicon doped
with 1.15 X 1016 As
atoms cm-3[1],
(b) Germanium
doped with 7.5 X
1015 As atoms cm3[2]. (p.12 in M&K1)
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Drift Current
• The drift current density (amp/cm2)
is given by the point form of Ohm Law
J = (nqmn+pqmp)(Exi+ Eyj+ Ezk), so
J = (sn + sp)E = sE, where
s = nqmn+pqmp defines the conductivity
• The net current is


I   J  dS
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Drift current
resistance
• Given: a semiconductor resistor with
length, l, and cross-section, A. What
is the resistance?
• As stated previously, the
conductivity,
s = nqmn + pqmp
• So the resistivity,
r = 1/s = 1/(nqmn + pqmp)
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Drift current
resistance (cont.)
• Consequently, since
R = rl/A
R = (nqmn + pqmp)-1(l/A)
• For n >> p, (an n-type extrinsic s/c)
R = l/(nqmnA)
• For p >> n, (a p-type extrinsic s/c)
R = l/(pqmpA)
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Drift current
resistance (cont.)
• Note: for an extrinsic semiconductor
and multiple scattering mechanisms,
since
R = l/(nqmnA) or l/(pqmpA), and
(mn or p total)-1 = S mi-1, then
Rtotal = S Ri (series Rs)
• The individual scattering mechanisms
are: Lattice, ionized impurity, etc.
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Net intrinsic
mobility
• Considering only lattice scattering
the total mobility is
1
m total
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
1
mlattice
, only,
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Lattice mobility
• The mlattice is the lattice scattering
mobility due to thermal vibrations
• Simple theory gives mlattice ~ T-3/2
• Experimentally mn,lattice ~ T-n where n
= 2.42 for electrons and 2.2 for holes
• Consequently, the model equation is
mlattice(T) = mlattice(300)(T/300)-n
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Net extrinsic
mobility
• Considering only lattice and impurity
scattering
the total mobility is
1
m total
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
1
mlattice

1
mimpurity
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Ionized impurity
mobility function
• The mimpur is the scattering mobility
due to ionized impurities
• Simple theory gives mimpur ~
T3/2/Nimpur
• Consequently, the model equation is
mimpur(T) = mimpur(300)(T/300)3/2
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Figure 1.17 (p. 32 in M&K1)
Low-field mobility in silicon as a function of
temperature for electrons (a), and for holes
(b). The solid lines represent the theoretical
predictions for pure lattice scattering [5].
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Mobility (cm^2/V-sec)
Exp. m(T=300K) model
for P, As and B in Si1
1500
1000
P
As
500
B
0
1.E+13
1.E+15
1.E+17
1.E+19
Doping Concentration (cm^-3)
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Exp. mobility model
function for Si1
max
min
min mn, p  mn, p
mn, p  mn, p 
a
 Nd, a 


Parameter
mmin
mmax
Nref
a
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1

N
 ref 
As
P
B
52.2
68.5
44.9
1417
1414
470.5
9.68e16 9.20e16 2.23e17
0.680
0.711
0.719
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Carrier mobility
functions (cont.)
• The parameter mmax models 1/tlattice
the thermal collision rate
• The parameters mmin, Nref and a model
1/timpur the impurity collision rate
• The function is approximately of the
ideal theoretical form:
1/mtotal = 1/mthermal + 1/mimpurity
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Carrier mobility
functions (ex.)
• Let Nd = 1.78E17/cm3 of phosphorous,
so mmin = 68.5, mmax = 1414, Nref = 9.20e16
and a = 0.711.
– Thus mn = 586 cm2/V-s
• Let Na = 5.62E17/cm3 of boron, so mmin =
44.9, mmax = 470.5, Nref = 9.68e16 and a
= 0.680.
– Thus mp = 189 cm2/V-s
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Net silicon (extrinsic) resistivity
• Since
r = s-1 = (nqmn + pqmp)-1
• The net conductivity can be obtained
by using the model equation for the
mobilities as functions of doping
concentrations.
• The model function gives agreement
with the measured s(Nimpur)
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Resistivity (ohm-cm)
Net silicon extr
resistivity (cont.)
1.00E+03
1.00E+02
P
1.00E+01
As
1.00E+00
B
1.00E-01
1.00E-02
1.E+13
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1.E+15
1.E+17
1.E+19
Doping Concentration (cm^-3)
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Net silicon extr
resistivity (cont.)
• Since
r = (nqmn + pqmp)-1, and
mn > mp, (m = qt/m*) we have
rp > rn, for the same NI
• Note that since
1.6(high conc.) < rp/rn < 3(low conc.), so
1.6(high conc.) < mn/mp < 3(low conc.)
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Net silicon (compensated) res.
• For an n-type (n >> p) compensated
semiconductor, r = (nqmn)-1
• But now n = N  Nd - Na, and the
mobility must be considered to be
determined by the total ionized
impurity scattering Nd + Na  NI
• Consequently, a good estimate is
r = (nqmn)-1 = [Nqmn(NI)]-1
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Summary
• The concept of mobility introduced as
a response function to the electric
field in establishing a drift current
• Resistivity and conductivity defined
• Model equation def for m(Nd,Na,T)
• Resistivity models developed for
extrinsic and compensated materials
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Equipartition
theorem
• The thermodynamic energy per
degree of freedom is kT/2
Consequently,
1
2
mv
2
vrms
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thermal
3
 kT, and
2
3kT
7

 10 cm / sec
m*
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Carrier velocity
saturation1
• 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
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Carrier velocity2
carrier
velocity
vs E
for Si,
Ge, and
GaAs
(after
Sze2)
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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
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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 

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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
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Diffusion of
Carriers (cont.)
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Current density
components

Note, since E  V



Jp,drift  s pE  pqm pE  pqm pV



Jn,drift  snE  nqmnE  nqmnV

Jp,diffusion   qDpp

Jn,diffusion   qDnn
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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   s p  sn V  qDpp  qDnn

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
34
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
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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
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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
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References
1Device
Electronics for Integrated
Circuits, 2 ed., by Muller and Kamins,
Wiley, New York, 1986.
– See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996,
for another treatment of the m model.
2Physics
of Semiconductor Devices, by
S. M. Sze, Wiley, New York, 1981.
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