Fig.4.22 Energy levels (bottoms of subbands) and density of states for a triangular quantum well structure (j = 1, 2, …, correspond to the different subbands).

• After evaluating this integral and adding the contribution from all subbands, one obtains

• The quantized energy levels for the subbands can be found using a numerical self-consistent solution of the dinger and Poisson's equations.
  
• However, an excellent approximation for the exact solution can be found by assuming a linear potential profile (i.e., constant effective field Feff) in the semiconductor and close to the semiconductor-insulator interface.
  
• In this case, the energy levels are given by

where is the effective mass for electron motion perpendicular to the (100) surface, and Ec(0) is the minimum conduction band energy at the Si-SiO2 interface.

• The effective field Feff is expressed through the surface field FS and the bulk field FB.
  
• For electrons, the relationship linking Feff, FB, and FS, giving the best fit to the self-consistent solution of dinger and Poisson's equation is given by and Feff = (FS + FB)/2, where ns is the interface electron sheet density, and qnB (= qNAddep(av)) is the sheet density of depletion charge.
  
• Similarly, for holes, FS = q(ps + where ps is the interface hole sheet density, and qpB (= qNDddep(av)) is the sheet density of depletion charge.
  
• In reality, it has been found that a slightly different form of the effective field Feff1 = (FS + 2FB)/3 gives a better fit to the measured data.
  
• Solving these equations iteratively, one can obtain the relation between ns and the Fermi level [EF Ec(0)].

Fig.4.23 Comparison of the interface carrier density versus EF Ec(0) characteristics for different substrate doping densities in (a) semilog plot and (b) linear plot. Symbols: calculations based on a 2DEG formulation, solid lines: charge sheet model, straight line in b): linear approximation to 2DEG formulation, the slope gives

• In the calculation, it can be assumed that the maximum value of nB is given by

• In the subthreshold region, the calculation agrees reasonably well with the classical charge sheet model (CCSM) given by Brews:

especially at low levels of substrate doping.

• The difference between the curves at high substrate doping levels is caused by the fact that the large bulk field quantizes the energy levels even in the subthreshold region.
  
• However, at strong inversion, the difference between the charge sheet model and the 2DEG formulation is large.
  
• As can be seen from Fig.4.23, the dependence of ns on EF in the above threshold regime can be approximated by a straight line: where EF0 is the intercept of this linear approximation with ns = 0.
  
• This approximation means that a fraction of the applied voltage, equal to is accommodated by a shift in the Fermi level with respect to the bottom of the conduction band.
  
• The shift in the Fermi level with respect to the bottom of the conduction band changes the above-threshold capacitance from to where the parameter can be interpreted as a correction to the insulator thickness.
  
• From the straight-line approximation in Fig.4.23b), is obtained, which is much smaller than that of
  
• This difference is caused by
  
  • a much larger effective mass in the conduction band in Si, which makes quantum effects much less pronounced, and
    
  • the large difference in the dielectric constants between the insulator and the semiconductor for the MOS system.

Practice Problems

4.1 Clearly draw the band diagrams for an ideal MOS structure and no oxide charge) on n-type Si for i) accumulation, ii) depletion, and iii) inversion. If the oxide thickness tox = 40 nm and VG = 1 V, determine the magnitude and sign of the charge density in the semiconductor. What is the status of the surface?

4.2 Show that for an MOS structure on p-type Si, the electron and hole concentrations as functions of position are given by where n0 and p0 are the equilibrium electron and hole concentrations respectively, and is defined by = [Ei(bulk) Ei(x)]/q.

4.3 Continuing with the derivation given in Section 4.2, show that the electric field E in the semiconductor in an MIS capacitor can be given by where all the notations carry their usual meanings.

4.4 Sketch the electric field and voltage distribution in an MOS structure at the threshold gate voltage. Data: substrate voltage = 0, and VFB = 0. Compute the threshold voltage VTH from the voltage distribution.

4.5 Calculate and plot the semiconductor surface charge per unit area for an MIS structure as a function of the surface potential

4.6 Starting from Eqn.(4.16), show that at flatband (i.e., when Vs = 0), the flatband capacitance per unit area Hence, compute its magnitudes for substrate dopings of

4.7 Consider the energy band diagram of a metal-SiO2-Si-SiO2-metal structure as shown in Fig.P7. Assume symmetric bands with
(a) What is the flatband voltage for this structure?
(b) Sketch the band diagram of the structure when the left metal plat is at 2 V and the right metal plate is grounded. Assume What is the strength of the electric field in Si? What are the positions of the Imrefs in Si? In the band diagram, all the appropriate voltage levels must be specified. Neglect induced charges in Si.

4.8 (a) Find the voltage VFB required to reduce to zero the negative charge induced at the semiconductor surface by a sheet of positive charge located below the metal.
(b) In the case of an arbitrary distribution of charge in the oxide, show that

where = oxide capacitance per unit area = where d = oxide thickness.

4.9 Charge density of is distributed in the oxide (d = 40 nm) in a Si MOS capacitor. Assume Find the flatband voltage required to be applied at the gate to compensate these charges if: i) the charges are uniformly distributed in the oxide, ii) the charge distribution is linear with the peak at the metal-SiO2 interface and zero at the Si-SiO2 interface, and iii) same as ii) but now with the peak at the Si-SiO2 interface and zero at the metal-SiO2 interface. Physically justify the answers.

4.10 An Al-gate where m is the Al work function to vacuum) MOS structure is made on p-type % where is electron affinity for Si) substrate. The SiO2 thickness d = 50 nm, and the effective oxide interface charge Find Wmax, VFB, and VTH. Sketch the C-V curve for this device giving all relevant details.

4.11 Find VTH for an MOS structure in Si with p-type substrate and d = 80 nm. Repeat for n-substrate with the same parameters (note: the new can be calculated from the change in EF).

4.12. Calculate and plot the maximum width of the depletion region for an ideal (i.e., VFB = 0) MIS capacitor on p-type Si with as a function of the substrate bias Vsub for -2 V < Vsub < 0.1 V. Assume that the voltage difference between the inversion layer at the interface and the gate contact is maintained constant when the substrate potential is changed (charge screening), so that the substrate voltage reverse biases the inversion layer/p-type substrate junction. Also, calculate the threshold voltage VT, and the capacitance of the structure at low and high frequencies for V >> VT for Vsub = 0. Data: ni =

4.13 Calculate and plot the surface potential as a function of the gate voltage VG in depletion and inversion for a two-terminal MIS structure. Identify the weak inversion, moderate inversion, and the strong inversion regions in the plot (as per Tsividis). Can the plot be really linearized in subthreshold? Determine an effective value of in subthreshold from the plot.

4.14 Calculate and plot the gate-to-substrate capacitance Cmis as a function of the gate voltage VG for a two-terminal MIS structure with area = The plot should show all the regions of operation (i.e., accumulation, depletion, weak inversion, and strong inversion). Mark Cso in the plot, with the magnitude shown. (Note: the externally measured capacitance includes the oxide capacitance).

4.15 Calculate and plot the temperature dependence of the surface charge per unit area for the surface potential i) in the temperature range between 150 K and 450 K. Data: effective densities of states in conduction and valence bands and respectively at 300 K (with both of them having a dependence), and the energy gap Eg = 1.12 eV (the variation of the energy gap with temperature may be neglected).

4.16 From the equivalent circuit for an MIS structure, determine the expression for the impedance across its two terminals as a function of frequency. Hence, calculate and plot the effective capacitance of the structure as a function of the gate voltage VG (varying from -5 V to +5 V) for frequencies of

4.17 As a practice problem, draw any arbitrary C-V curve of your choice, and following the parameter extraction algorithm discussed in Section 4.4.1, obtain the i) oxide thickness, ii) threshold voltage, iii) substrate doping, iv) flatband capacitance, v) flatband voltage, and vi) fixed oxide charges.

4.18 The C-V curve of a two-terminal MIS structure shows a shift of 10 mV in the flatband voltage after a bias-temperature stress test. If the flatband voltage before the stress test is -1 V, and the surface state density is determine the oxide fixed charges.

4.19 Derive Eqn.(4.40).

4.20 Derive Eqn.(4.43).

4.21 (a) Compute and plot the surface electron concentration ns as a function of [EF - EC(0)] under the 2DEG approximation for
(b) Repeat part (a) under the 3D approximation (i.e., the 3D charge sheet model as given by Brews).
Data: (Note: the constant energy surface for Si consists of six ellipsoids of revolution, and ml ( ) and mt ( ) represents the lateral and transverse effective mass respectively. For {100} direction four of these ellipsoids will lye on the surface and two ellipsoids will be perpendicular. Refer to Problem 22 also.)

4.22 In the classical limit, the separation of the energy subbands in a 2D electron gas is small compared to the thermal energy kT. In this case, the sheet density of the 2D electron gas is given by the classical charge sheet model, given by Eqn.(4.46), which is derived using a conventional 3D electron gas approach. Show that in this limit (i.e., Ej Ej 1 << kT), the equation

reduces to Eqn.(4.46). In the above equation, mpi is the parallel effective mass for the valley i, and Eji is the energy level of the jth subband in valley i. Note: the effective mass mpi is mt for two valleys, and for four valleys, where mt and ml is the transverse and lateral effective mass respectively.