1. EE 5340 Semiconductor Device Theory Lecture 7 - Fall 2003
Professor Ronald L. Carter
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2. Fermi Energy
The equilibrium carrier concentration ahd the Fermi energy are related as
The potential f = (Ef-Efi)/q
If not in equilibrium, a quasi-Fermi level (imref) is used
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3. Electron quasi-Fermi Energy (n = no + n)
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4. Hole quasi-Fermi Energy (p = po + p)
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5. Ex-field when Ef - Efi not constant
Since f = (Ef - Efi)/q = Vt ln(no/ni)
When Ef - Efi = is position dependent,
Ex = -df/dx = -[d(Ef-Efi)/dx] = - Vt d[ln(no/ni)]/dx
If non-equilibrium fn = (Efn-Efi)/q = Vt ln(n/ni), etc
Exn = -[dfn/dx] = -Vt d[ln(n/ni)]/dx
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6. Si and Al and model (approx. to scale)
metal
n-type s/c
p-type s/c Eo Eo Eo
qcsi~ 4.05 eV
qcsi~ 4.05 eV
qfm,Al ~ 4.1 eV
qfs,n
qfs,p Ec Ec
EFm
EFn
EFi
EFi
EFp Ev Ev
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7. Equilibrium Boundary Conditions w/ contact
No discontinuity in the free level, Eo at the metal/semiconductor interface.
EF,metal = EF,semiconductor to bring the electron populations in the metal and semiconductor to thermal equilibrium.
Eo - EC = qcsemiconductor in all of the s/c.
Eo - EF,metal = qfmetal throughout metal.
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8. 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
fi=fBn-fn= fm-fs elect s/c to mtl barr Eo
qfm
qcs
qfi
qfBn
qfs,n Ec
EFm
EFn
EFi
Depl reg Ev
qf’n
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9. Metal to n-type non-rect cont (fm<fs)
n-type s/c
No disc in Eo
Ex=0 in metal ==> Eo flat
fB,n=fm - cs = elec mtl to s/c barr
fi= fBn-fn< 0
Accumulation region Eo
qcs
qfm
qfs,n
qfi
qfB,n Ec
EFm
EFn
EFi Ev
qfn
Acc reg
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10. 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
fBp= fm- cs + Eg = hole m to s
fi = fBp-fs,p = hole s/c to mtl barr Eo
qfm
qcs
qfi
qfs,p
qfBn Ec
EFi
EFm
EFp
qfBp Ev
qfi
qfp<0
Depl reg
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11. Metal to p-type non-rect cont (fm>fs)
n-type s/c
No disc in Eo
Ex=0 in metal ==> Eo flat
fB,n=fm- fs,n = elec mtl to s/c barr
fBp= fm- cs + Eg = hole m to s
Accumulation region Eo
qcs
qfm
q(fi)
qfs,n
qfBn Ec
EFm
EFi
EfP
qfp
qfBp
qfi Ev
Accum reg
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12. Metal/semiconductor system types
n-type semiconductor
Schottky diode - blocking for fm > fs
contact - conducting for fm < fs
p-type semiconductor
contact - conducting for fm > fs
Schottky diode - blocking for fm < fs
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13. Real Schottky band structure1
Barrier transistion region, d
Interface states
above fo acc, p neutrl
below fo dnr, n neutrl
Ditd -> oo, qfBn = Eg- fo
Fermi level “pinned”
Ditd -> 0, qfBn = fm - c
Goes to “ideal” case
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14. Fig 8.41 (a) Image charge and electric field at a metal-dielectric interface (b) Distortion of potential barrier at E=0 and (c) E0
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15. Poisson’s Equation
The electric field at (x,y,z) is related to the charge density r=q(Nd-Na-p-n) by the Poisson Equation:
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16. Poisson’s Equation n = no + dn, and p = po + dp, in non-equil
For n-type material, N = (Nd - Na) > 0, no = N, and (Nd-Na+p-n)=-dn +dp +ni2/N
For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = dp-dn-ni2/N
So neglecting ni2/N
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17. 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
fbi=fBn-fn= fm-fs elect s/c to mtl barr
0 xn xnc Eo
qfm
qcs
qfbi
qfBn
qfs,n Ec
EFm
EFn
EFi
Depl reg Ev
qf’n
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18. Depletion Approximation
For 0 < x < xn, assume n << no = Nd, so r = q(Nd-Na+p-n) = qNd
For xn < x < xnc, assume n = no = Nd, so r = q(Nd-Na+p-n) = 0
For x = 0-, there is a pulse of charge balancing the qNdxn in 0 < x < xn
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19. Ideal n-type Schottky depletion width (Va=0)r Ex xn
qNd x
Q’d = qNdxn x xn
-Em d
(Sheet of negative charge on metal)= -Q’d
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20. n x xn Nd
Debye length
The DA assumes n changes from Nd to 0 discontinuously at xn.
In the region of xn, Poisson’s eq is E = r/e --> dEx/dx = q(Nd - n), and since Ex = -df/dx, we have -d2f/dx2 = q(Nd - n)/e to be solved
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21. Debye length (cont)
Since the level EFi is a reference for equil, we set f = Vt ln(n/ni)
In the region of xn, n = ni exp(f/Vt), so d2f/dx2 = -q(Nd - ni ef/Vt), let f = fo + f’, where fo = Vt ln(Nd/ni) so Nd - ni ef/Vt = Nd[1 - ef/Vt-fo/Vt], for f - fo = f’ << fo, the DE becomes d2f’/dx2 = (q2Nd/ekT)f’, f’ << fo
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22. Debye length (cont)
So f’ = f’(xn) exp[+(x-xn)/LD]+con. and n = Nd ef’/Vt, x ~ xn, where LD is the “Debye length”
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23. Debye length (cont)
LD estimates the transition length of a step-junction DR. Thus,
For Va = 0, i ~ 1V, Vt ~ 25 mV d < 11%  DA assumption OK
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24. Effect of V  0
Define an external voltage source, Va, with the +term at the metal contact and the -term at the n-type contact
For Va > 0, the Va induced field tends to oppose Ex caused by the DR
For Va < 0, the Va induced field tends to aid Ex due to DR
Will consider Va < 0 now
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25. Effect of V  0
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26. Ideal metal to n-type Schottky (Va > 0)
qVa = Efn - Efm
Barrier for electrons from sc to m reduced to q(fbi-Va)
qfBn the same
DR decr Eo
qcs
qfm
q(fi-Va)
qfs,n
qfBn Ec
EFm
EFn
EFi Ev
Depl reg
qf’n
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27. Test 1 - 25Sept03 8 AM Room 206 Activities Building
Open book - 1 legal text or ref. Only
Calculator allowed
A cover sheet will be included with full instructions. See for examples from Fall 2002.
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28. References
1 Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997.
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