1. Basic Equations for Device Operation
Transport equations: The total e- and hole current densities are the sum of the drift due to electric field, e and diffusion due to concentration gradient components:
where
Jn and Jp are e- and hole current densities, respectively
n and p are e- and hole concentrations, respectively
mn and mp are e- and hole mobilities, respectively.
The total current density is given by: J = Jn + Jp
(1)
(2)

2. Continuity Equations
Continuity equations: based on the conservation of mobile charge, e- and hole continuity equations are given by:
where
Gn = e- generation rate
Rn = e- recombination rate
Gp = hole generation rate
Rp = hole recombination rate
(3)
(4)

3. Poisson’s Equation Poisson’s equation: where e = electric field (5)
r(x) = space charge density (charge/cm3)
K = dielectric constant
eo = permittivity of free space = 8.854x10-14 F/cm
(5)
(6)

4. Poisson’s Equation Another form of Poisson’s Equation (Gauss’s Law):
where QS = integrated charge density
In silicon, r consists of:
mobile charges: e- and holes with concentrations n and p, respectively.
fixed charges: ionized acceptors and donors with concentrations Na- and Nd+, respectively.
(7)
(8)

5. Carrier Concentration - Boltzmann’s Relations
(9)
(10)

6. Carrier Concentration
(11)
(12)
(13)

7. MOS Capacitors - 1. Band Diagrams
MOS structure
p-Si
SiO2
Metal
(Al)
Oxide
(SiO2)
Semiconductor
(p-Si) E0
qFm= 4.1eV
EFM
qcox=0.95eV
Eg » 8eV Ec Ev EF
qFS=4.9eV
qcS= 4.05eV
Eg=1.12eV
Note:
three materials in contact, EF = constant at equilibrium.
currents through SiO2 are very small.
holes flow from semiconductor ® metal on contact.
e- flow from metal ® semiconductor on contact.
bands will bend downwards in silicon at the interface (Fm < Fs)
where Fm = metal work function
Fs = p-type semiconductor work function = cS + Eg/2 + fB.

8. MOS Capacitor at Equilibrium: Applied V = 0Ec Ev EF
Al SiO2 p-Si
3.8 eV
3.10 eV
3.15 eV
0.4 eV
Abrupt transition in Ec and Ev levels at the material interfaces.
For Fm < Fs, band bends downward.
A typical potential drop ~ 0.4 eV across SiO2. This depends on EF in Si. This potential can be supported because no current flows through SiO2.
Substantial barriers exist to current flow from:
S ® M (~ 3.10 eV)
M ® S (~ 3.15 eV).
Depletion region exists near the surface because EF near the surface is further from Ev than the bulk region.
An applied voltage, DVfb = Fm - Fs = Fms can make the band flat.

9. MOS Capacitor - Flat Band, V = Vfb
EF is constant in Si because SiO2 prevents any current flow (i.e., Si is at equilibrium).
Ec and Ev are flat.
hole concentration is uniform in Si and SiO2 (as shown by flat Ec and Ev).
Þ FLAT-BAND.
The voltage required to achieve flat band condition is Vfb = fms.

10. MOS Capacitor – Accumulation, V < VfBqV
EFM Ec EF Ev
Al SiO p-Si
- ve
EF is still constant in the Si since SiO2 prevents any current flow.
EF is closer to Ev at the surface.
more holes accumulated near the surface than the bulk.
Þ ACCUMULATION.

11. MOS Capacitor – Depletion, V > VFBqV
EFM Ec EF Ev
Al SiO p-Si
+ ve Ei fB Qg Qd
EF is still constant in Si (I = 0).
EF is farther from Ev at the surface.
\ The surface is depleted of holes.
Þ DEPLETION.

12. MOS Capacitor – Inversion, V >> VFBYs
EF is still constant in Si (I = 0).
EF is closer to Ec at the surface than it is to Ev.
\ more e- than holes at the surface.
If Ys = 0, the surface becomes intrinsic
If 0 < Ys < - fB, the surface will become slightly n-type:
Þ WEAK INVERSION.
If fs > - fB, the surface will have an e- concentration > hole concentration in the bulk Þ STRONG INVERSION.

13. 2. Electrostatic Potential and Charge Distribution
We now derive the relation among the surface potential, electric field, and charge (Qs) by solving Poisson’s eqn in the surface region of silicon.
For a metal/oxide/p-type silicon system, we define potential at any point x:
f(x) º fi(x) - fi(x ® µ) = amount of band bending at any point x.
where
f(x = 0) º fs = surface potential; fi(x ® µ) =ff = Fermi potential
Thus,
f(x) < 0 when bands bend upward in accumulation
f(x) > 0 when bands bend downward in depletion and inversion

14. Band Bending
(14)
(15)
(16)

15. Band Bending
Thus, the charge density at a depth x in the substrate is given by:
Substituting n(x), p(x), Na, and Nd in Poisson’s equation we can determine surface charge density in silicon substrate.
(17)
(18)

16. Band Bending \ Multiplying both sides of (19) by 2d(f(x))/dx, we get:
Poisson’s equation:
\ Multiplying both sides of (19) by 2d(f(x))/dx, we get:
(19)
(20)
(21)

17. Band Bending
Integrate from the bulk (f = 0, df/dx = 0) toward the surface at any point x to get:
(22)
(23)
At x = 0, f = fs and E = Es, then from Gauss’s law (7), the total charge per unit area induced in silicon (equal and opposite to the charge on the metal gate is: Qs = - Ke0Es

18. Band Bending
(24)
The charge expression above is valid in all regions of MOS capacitor operation - accumulation, depletion, and inversion. The Eq can, also, be expressed as:
(25)
(26)

19. Surface Charge Density vs. Surface Potential
Plot of (24) showing the variation of the induced surface charge density as a function of surface potential for p-type substrate.
Note from (24):
for fs < 0: Qs > 0
Þ ACCUMULATION
Qs ~ exp(-fs/2vt)
for fs = 0: Qs = 0
Þ FLAT-BAND
for fs > 0: Qs < 0
Þ DEPLETION and WEAK-INVERSION
Qs ~ SQRT(fs)
for fs > 2ff
Þ INVERSION
Qs ~ exp(fs/2vt)

20. Depletion Approximation
For the simplicity of analytical integration, we can assume that in the depletion region fB < fs < 2fB, we can approximate (23) as:
We know: at x = 0, f(x) = fs; Therefore,
(27)
(28)
(29)

21. Depletion Approximation
(30)
(29) is a parabolic Eq with the vertex at f(x) = 0, x = Wd. Thus, Wd = distance to which band-bending extends = width of the depletion region.
Then the depletion layer charge density is given by:
In an MOS system, Wd reaches a maximum, Wd,max at the onset of inversion defined by fs º 2fB = (2kT/q)ln(Na/ni).
(31)
(32)

22. Strong Inversion
Beyond strong inversion, the concentration of inversion charge n(x) becomes significant. Therefore, from (23) we get:
N(sub) = 1016 cm-3
fs = 0.88 V
fs = 0.85 V
This equation must be solved numerically with boundary condition, f = fs at x = 0. From the solution of f(x), the inversion carriers, n(x) can be calculated from (16).
The numerically calculated n(x) Vs. depth plot is shown in Fig. It is seen that inversion charge distribution is extremely close to the surface with an inversion layer width < 50 A.

23. Strong Inversion
Generally, inversion-carriers must be treated quantum-mechanically (QM) as a 2-D gas. According to QM model:
inversion layer carriers occupy discrete energy bands
peak distribution is A away from the surface.
When the inversion charge density Qi > Qd, we get from (24):
(33)
(34)

24. 3. Low Frequency C – V Characteristics
Assuming silicon at equilibrium. The total capacitance of an MOS structure is: C = d(-Qs)/dVg
where the applied gate bias, Vg of an MOS system is:
Vg = Vfb + Vox + fs
or, dVg = dVox + dfs
where Vox = voltage drop in the oxide
Thus, the total capacitance equals the oxide capacitance, Cox and the silicon capacitance, Cs connected in series.
(35)

25. Low Frequency C – V Characteristics
From (24), we get:
From (36), we can calculate C - V characteristics in different regions of MOS operation.
(36)

26. Low Frequency C – V: Accumulation, 0 << fs
In strong accumulation, 0 << fs . Then from (36), we can show:
Since 0 << fs, for large fs, Cs becomes very large.
(37)
(38)

27. Low Frequency C – V: Flat Band, fs = 0
For fs = 0: the inversion charge term in (36) can be neglected to get:
Expanding the exponential term into power series and retaining the first five terms, we get:
(39)
(40)

28. Low Frequency C – V: Flat Band
(41)
(42)
Thus, Eq. (42) shows that Cfb is somewhat less than Cox.

29. Low Frequency C – V: Depletion, fB < fs < - fB
In depletion, the general expression for Cd is given by (36). However, we can derive an approximate expression from depletion approximation using Eq. (30) and (31).
We know that:
Thus, C decreases with the increase in Vg
(43)

30. Low Frequency C – V: Strong Inversion
In strong Inversion, fs >> 0. Then from (36), we can show that:
Negative sign indicates that the charge has changed sign.
Since fs >> 0, for large fs, Cs becomes very large.
(44)

31. 4. Intermediate and High Frequency C - V
There are plenty of majority carriers which can respond to the A.C. signal.
The minority carriers, on the other hand, are scarce and have to originate from:
diffusion from bulk substrate
generation in the depletion region
external sources (e.g. n+ diffusion region etc.)
Thus, the inversion charge can not respond to the applied A.C. signal higher than 100 Hz.
Therefore, at any high frequency applied signal to Vg:
Qi is assumed to be constant
Qd = Cs

32. Intermediate and High Frequency C - V
At any high frequency applied signal to Vg:
Qd will vary with the signal around its maximum value Qd,max
Wd will vary with the signal around its maximum value Wd,max.
Thus, the high frequency capacitance is given by:
(45)

33. MOS C - V Characteristics

34. 5. Oxide and Interface Charges

35. 5. Oxide and Interface Charges
Types of origins:
fixed interface charge, Qf: believed to be uncompensated Si-Si bonds, on the order of cm-2.
interface trapped-charge, Qit: due to un-terminated Si bonds, on the order of 1010 cm-2.
Oxide trapped-charge, Qot: due to defects in the SiO2 network, usually negligible in advanced MOSFET devices. Can be guaranteed after a large amount of charges have passed through the SiO2 and broken up the network.
mobile charge, Qm: sue to alkaline ions (e.g., Na, K), usually very low concentration (~ 109 cm-2) in advanced technology.

36. Effects of Qf and Qot on C - V
The flat-band voltage change due to Qf and Qot is:
If Qot is small, Qf can be estimated by DVfb.
(46)

37. Effects of Qf and Qot on C - V
Figure.

38. Effects of Qm on C - V
By bias-temperature stress, Qm can be moved from Si-SiO2 interface to the metal-SiO2 interface. The shift in VFB can be used to estimate Qm.
Figure.

39. Effects of Qit on C - V Interface trapped-charge, Qit:
the traps have energy levels distributed in the band gap
the charge state of the traps depends on Fermi level.
thus, as the surface potential is swept from flat band towards depletion, the traps will try to pin the surface potential, i.e. it will take more gate voltage to move the surface potential through the traps.

40. Effects of Qit on C - V Assume acceptor type traps:
Thus, Cs is in parallel with Cit. Hence, the capacitor model is:
(47)
(48)

41. 6. Polysilicon Gates At equilibrium (Vg = 0).
At Inversion (Vg >> 0). Ec Ev EF
n+ poly SiO p-Si Ei fB Ec Ev EF
n+ poly SiO2 p-Si Ei fS VG fp
Vox
Work-function difference:
n+poly with p-type Si: fMS = - Eg/2 - (kT/q)ln(NA/ni) (49)
p+poly with n-type Si: fMS = Eg/2 + (kT/q)ln(ND/ni) (50)

42. Polysilicon Depletion Effect
Note:
Capacitance at inversion, Cinv does not return to Cox.
Cinv shows a maximum value, Cmax < Cox.
Cmax­ as the polysilicon doping concentration, Np­.
As Np­, depletion width¯
Cmax ® Cox for higher Np.
Total capacitance at strong inversion is given by:
1/C = 1/Cox + 1/Csilicon + 1/Cpoly
As VG­, Csilicon­ but xd(poly)­ Þ Cpoly¯.
\Low frequency C - V shows a local maximum at a certain VG.

43. 7. Inversion Layer Quantization
Typically, near the silicon surface, the inversion layer charges are confined to a potential well formed by:
oxide barrier
bend Si-conduction band at the surface due to the applied gate potential, VG.
Due to the confinement of inversion layer e- (in p-Si):
e- energy levels are grouped in discrete sub-bands of energy, Ej
each Ej corresponds to a quantized level for e- motion in the normal direction. E2 E1 E0
Edge of EC
Distance from the surface
Bottom of
the well E

44. Inversion Layer Quantization
Due to Quantum Mechanical (QM) effect, the inversion layer concentration:
peaks below the SiO2/Si interface
» 0 at the interface determined by the boundary condition of the e- wave function.
Solve Schrodinger and Poisson Eq self-consistently with the boundary conditions for wave function equal to:
0 for x < 0 in oxide
0 at x = ¥. QM
Classical
Depth
n (cm-3)

45. QM Effect on MOS Capacitors
At high fields, Vth­ since more band bending is required to populate the lowest sub-band, which is some energy above the bottom of EC.
Once the inversion layer forms below the surface, a higher Vg over-drive is required to produce a given level of inversion charge density. That is, the effective gate oxide thickness, tOXeff­ by:
DtOX = (eox/esi)Dz (51)
Inversion layer quantization can be treated as bandgap widening due to an increase in the effective bandgap by DEg given by:
Here, DEg = EgQM - EgCL.
(52) Þ ni¯ and n¯ due to QM effect.
(52)

46. Home Work 1: Due 0ct 6, 2005
1) An MOS capacitor is built with the structure shown below. The capacitor areas over the N and P regions are the same. The threshold voltages for the N and P substrates are -1 and +1 volts respectively. Sketch the shape of the high frequency (1 MHz) C-V curve that you would expect to measure for this structure. Label as many points as possible and explain.
2) In practical MOS systems, measurements of C - V sometimes show hysteresis effects; that is, the C - V curves look like the sketch in Fig. The sketch refers to measurements made when VG is swept with a very low frequency triangular wave (~ 1 Hz) and the A.C. measurement frequency is of the order of 1 kHz or higher. The sense of the hysteresis on such a curve can be observed experimentally to be either counterclockwise, as shown, or else be clockwise. N P
SiO2 Al

47. Home Work 1: Due Oct 6, 2005
The hysteresis sense allows one to differentiate between the two most common cause of non-ideal behavior.
(a) Show this by considering the following non-ideal effects:
(i) field-aided movement of positive ions in the insulator and (ii) trapping of free carriers from the channel in traps at the oxide-Si interface Qit.
(b) Using qualitative reasoning, prepare a table with sketches of the expected C - V plots for n- and p-type substrates; on each sketch indicate the sense of the hysteresis (i.e., clockwise or counterclockwise).

48. Home Work 1: Due Oct 6, 2005
3) MOS capacitors are fabricated on uniformly doped P-substrates with NA = 1x1017 cm-3, physical gate oxide thickness TOX = 6.7 nm, and N+ poly-silicon gate doping concentrations Np = 5x1018, 1x1019, and 5x1019 cm-3 as shown in the following C – V plots. Here C and Cox are the gate and oxide capacitances, respectively.
(a) Explain the observed variation in C/COX vs. Vg plots for each Np in the strong inversion region.
(b) Explain the observed increase in C/COX for Np = 5x1018 cm-3 and Vg ³ 3.5 V.
(c) Describe the possible reasons for C/COX < 1 even for a heavily doped poly with Np = 5x1019 cm-3 in the accumulation as well as in the strong inversion regions?

49. Home Work 5: Due Oct 6, 2005
4) A p-type MOS-capacitor with Na = 1x1018 cm-3 and TOX = 3 nm was fabricated to characterize van Dort’s analytical bandgap widening quantum mechanical (QM) model we discussed in class. Due to inversion layer quantization, the increase in the effective bandgap DEg = EgQM mV. Here EgQM and EgCL represent the QM and classical (CL) values of bandgap (Eg), respectively. Assume Qf = 0, VSUB = 0, and N+ poly gate.
(a) Show that the intrinsic carrier concentration due to QM effect is given by:
(b) Calculate the value of niQM at temperature T = 300oK.
(c) Calculate the value of threshold voltage Vth(CL) using the classical approach.
(d) Calculate the value of threshold voltage Vth(QM) due to QM effects.
(e) Calculate the shift DVth due to QM effects.