1. EE 5340 Semiconductor Device Theory Lecture 6 - Fall 2003
Professor Ronald L. Carter
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2. Approximate m func- tion for extrinsic, compensated n-Si1
Nd > Na  n-type
no = Nd - Na
s = no q mn
NI = Nd + Na
NAs > NP  As par
NP > NAs  As par
po = ni2/no
Param. As P
mmin
mmax
Nref 9.68e e16 a
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3. Approximate m func-tion for extrinsic, compensated p-Si1
Na > Nd  p-type
po = Na - Nd
s = po q mp
NI = Nd + Na
Na = NB  B par
no = ni2/po
Parameter B
mmin 44.9
mmax 470.5
Nref 2.23e17 a
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4. Summary
The concept of mobility introduced as a response function to the electric field in establishing a drift current
Resistivity and conductivity defined
m(Nd,Na,T) model equation developed
Resistivity models developed for extrinsic and compensated materials
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5. Equipartition theorem
The thermodynamic energy per degree of freedom is kT/2
Consequently,
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6. 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) E9 T E8 T-0.52
Ec (V/cm) T T1.68
b E-2 T T0.17
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7. Carrier velocity2 carrier velocity vs E for Si, Ge, and GaAs (after
Sze2)
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8. Carrier velocity saturation (cont.)
At 300K, for electrons, mo = v1/Ec = 1.53E9(300)-0.87/1.01(300) = 1504 cm2/V-s, the low-field mobility
The maximum velocity (300K) is vsat = moEc = v1 = 1.53E9 (300) = 1.07E7 cm/s
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9. Diffusion of Carriers (cont.)
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10. 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)
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11. 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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12. Current density components
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13. Total current density
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14. 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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15. 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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16. 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
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17. Silicon Planar Process1
M&K1 Fig. 2.1 Basic fabrication steps in the silicon planar process:
(a) oxide formation,
(b) oxide removal,
(c) deposition of dopant atoms,
(d) diffusion of dopant atoms into exposed regions of silicon.
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18. LOCOS Process1
1Fig 2.26 LOCal Oxidation of Silicon (LOCOS). (a) Defined pattern consisting of stress-relief oxide and Si3N4 where further oxidation is not desired, (b) thick oxide layer grown over the bare silicon region, (c) stress-relief oxide and Si3N4 removed by etching, (d) scanning electron micrograph (5000 X) showing LOCOS-processed wafer at (b).
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19. Al Interconnects1
1Figure (p. 104) A thin layer of aluminum can be used to connect various doped regions of a semiconductor device. 1
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20. Ion Implantation1
1Figure (p. 80) In ion implantation, a beam of high-energy ions strikes selected regions of the semiconductor surface, penetrating into these exposed regions.
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21. Phosphorous implant Range (M&K1 Figure 2
Phosphorous implant Range (M&K1 Figure 2.17) Projected range Rp and its standard devia-tion DRp for implantation of phosphorus into Si, SiO2, Si3N4, and Al [M&K ref 11].
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22. Implant and Diffusion Profiles
Figure Complementary-error-function and Gaussian distribu-tions; the vertical axis is normalized to the peak con-centration Cs, while the horizon-tal axis is normal-ized to the char-acteristic length
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23. Diffused or Implanted IC Resistor (Fig 2.451)
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24. An IC Resistor with L = 8W (M&K)1
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25. Typical IC doping profile (M&K Fig. 2.441)
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26. Mobilities**
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27. IC Resistor Conductance
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28. An IC Resistor with Ns = 8, R = 8Rs (M&K)1
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29. The effect of lateral diffusion (M&K1)
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30. A serpentine pattern IC Resistor (M&K1)
R = NSRS NCRS
note: RC = 0.65RS
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31. Band model review (approx. to scale)
metal
n-type s/c
p-type s/c Eo Eo Eo
qcs ~ 4+V
qcs ~ 4+V
qfm
~ 4+V
qfs,n
qfs,p Ec Ec
EFm
EFn
EFi
EFi
EFp Ev Ev
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32. Making contact be- tween metal & s/c
Equate the EF in the metal and s/c materials far from the junction
Eo(the free level), must be continuous across the jctn.
N.B.: qc = 4.05 eV (Si),
and qf = qc + Ec - EF Eo
qc (electron affinity) qf
(work function) Ec EF
EFi
qfF Ev
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33. Ideal metal to p-type barrier diode (fm<fs)
p-type s/c
metal
No disc in Eo
Ex=0 in metal ==> Eoflat
fBn=fm- cs = elec mtl to s/c barr
Vbi= fBp-fs,p = hole s/c to mtl barr Eo
qfm
qcs
qVbi
qfs,p
qfBn Ec
EFi
EFm
EFp Ev
qfp<0
Depl reg
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34. Ideal metal to n-type barrier diode (fm>fs,Va=0)
n-type s/c
No disc in Eo
Ex=0 in metal ==> Eoflat
fBn=fm- cs = elec mtl to s/c barr
Vbi=fBn-fn= fm-fs elect s/c to mtl barr Eo
qfm
qcs
qVbi
qfBn
qfs,n Ec
EFm
EFn
EFi
Depl reg Ev
qf’n
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35. Ideal metal to n-type Schottky (Va>0)
qVa = Efn - Efm
Barrier for electrons from sc to m reduced to q(Vbi-Va)
qfBn the same
DR decr Eo
qcs
qfm
q(Vbi-Va)
qfs,n
qfBn Ec
EFm
EFn
EFi Ev
Depl reg
qf’n
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36. References
1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 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.
3Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997.
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