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next generation 3D nano device simulator

1D Tutorial

Self-consistent 6-band k.p calculations of holes in strained Si/SiGe MOSFETs

Authors: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
-> 1DstrainedSi_SiGe_OberhuberPRB1998_step1_selfconsistent.in       - self-consistent single-band Schrödinger equation
   1DstrainedSi_SiGe_OberhuberPRB1998_step2_6x6kp_selfconsistent.in - self-consistent 6-band k.p Schrödinger equation which reads in potential of single-band results as initial guess
   1DstrainedSi_SiGe_OberhuberPRB1998_6x6kp_selfconsistent.in       - self-consistent 6-band k.p Schrödinger equation
 

Self-consistent 6-band k.p calculations of holes in strained Si/SiGe MOSFETs

This tutorial is based on the following paper:

[Oberhuber]
Subband structure and mobility of two-dimensional holes in strained Si/SiGe MOSFET's
R. Oberhuber, G. Zandler, P. Vogl
Physical Review B 58, 9941 (1998)

Structure

We calculate the hole energy levels, wave function and density of the following p-channel MOS structure:

     Gate - SiO₂ - strained Si - Si_0.8Ge_0.2

• Gate (applied voltage: V_G = 1.6 V and 1.1 V)
• 5 nm SiO₂
• 15 nm strained Si(001) with respect to Si_1-xGe_x, x=0.2, homogenously n-type doped 5 * 10¹⁶ cm^-3
• 500 nm unstrained Si_1-xGe_x buffer, x=0.2,                    homogenously n-type doped 5 * 10¹⁶ cm^-3

Both, Si and SiGe are n-type doped.

The method is 6-band k.p where we have to solve the Schödinger equation (6-band k.p Hamiltonian) not only for k_|| = 0
but also for a lot of k_|| vectors, i.e. for k_|| = (k_x,k_y) /= 0.

Our algorithm is the following:

1. 1DstrainedSi_SiGe_OberhuberPRB1998_step1_selfconsistent.in       - self-consistent single-band Schrödinger equation
   
   First, we solve the single-band Schrödinger equation and the Poisson equation self-consistently.
   This is useful because we want to obtain a reasonable start value for the electrostatic potential.
   
  $simulation-flow-control
   ...
   raw-potential-in = no             ! STEP 1 only

  $quantum-model-holes
   ...
   model-name       = effective-mass ! STEP 1 only (single-band)
 
2. 1DstrainedSi_SiGe_OberhuberPRB1998_step2_6x6kp_selfconsistent.in - self-consistent       6-band k.p Schrödinger equation which reads in potential of single-band results as initial guess
   
   Then we read in the electrostatic potential calculated in 1) (which serves as a start value for the electrostatic potential) and
   solve the 6-band k.p Schrödinger equation and the Poisson equation self-consistently (for both k_|| = 0 and k_|| /= 0).
   
  $simulation-flow-control
   ...
   raw-potential-in = yes            ! STEP 2      (read in electrostatic potential)

  $quantum-model-holes
   ...
   model-name       = 6x6kp          ! STEP 2      (6-band k.p)
 
3. 1DstrainedSi_SiGe_OberhuberPRB1998_6x6kp_selfconsistent.in       - self-consistent       6-band k.p Schrödinger equation
   In principle, step 1 can be omitted but it helps to save some CPU time for step 2.
   Note: If you want to avoid step 1, raw-potential-in must be set to no. This is then equivalent to this input file.
   Note: "3" leads to the same results as "1" + "2"-
    

In agreement with [Oberhuber] (fur the purpose of this tutorial), we do not allow the wave functions to penetrate into the SiO₂ layer (i.e. we assume infinite barriers at the SiO₂/Si interface) although nextnano³ is general enough to also include the oxide into the quantum region if desired.

Material parameters

We use the same material parameters as quoted in the [Oberhuber] paper. The only modification is the band offset.
We use a Ge-Si valence band offset, i.e. average of the three valence band edges (E_v,av), of 0.58 eV (unstrained) and
for Si we take the following value for the absolute valence band deformation potential:

 absolute-deformation-potential-vb = 2.05d0 ! [eV]

This results in a band offset with respect to the topmost valence band edges:
       heavy/light hole of Si_0.8Ge_0.2 (unstrained) - light hole strained Si = 0.046027 eV

Results

The following figure shows the valence band edge profile and the hole density for two different gate voltages (V_G = 1.6 V, V_G = 1.1 V).

  $poisson-boundary-conditions
   ...
   applied-voltage = -1.6d0 ! V_G = 1.6 V   Fig. 2(a) of [Oberhuber]
 ! applied-voltage = -1.1d0 ! V_G = 1.1 V   Fig. 2(b) of [Oberhuber]
 

Due to tensile strain, the topmost valence band in the strained Si layer is the light hole band.
The heavy/light hole splitting for strained Si on Si_0.8Ge_0.2 substrate: 79.7 meV ([Oberhuber] 80 meV)
In light gray, the squares of the topmost wave functions are indicated.






One can clearly see, that the inversion channel has a width of less than 10 nm.
At room temperature (300 K), in the Si layer only the 3-4 highest hole subbands are significantly occupied (not shown).
The small density ("parasitic channel") in the SiGe layer is not due to these 3-4 states but due to other states.
In our calculations, the influence of the parasitic channel in the Si_0.8Ge_0.2 layer is negligible whereas [Oberhuber] found that only for high voltages (e.g. V_G = 1.6 V) it is negligible and for V_G = 1.1 V almost 30 % of this density is contained in the parasitic channel.

The topmost subband (-31.3 meV for V_G = 1.6 V, -89.4 meV for V_G = 1.1 V) has predominantly light-hole character whereas the second, heavy-hole related subband lies
- 65.4 meV below the topmost subband (for V_G = 1.6 V).
- 54.6 meV below the topmost subband (for V_G = 1.1 V).
([Oberhuber] 55 meV)

The inversion layer sheet density (integrated over whole device) is found to be:
- for V_G = 1.6 V: n_s = 1.85 * 10¹² cm^-2    ([Oberhuber], Fig. 2(a): n_s = 1.2 * 10¹² cm^-2)
- for V_G = 1.1 V: n_s = 2.08 * 10¹¹ cm^-2    ([Oberhuber], Fig. 2(b): n_s = 4    * 10¹¹ cm^-2)
 

Our results are in reasonable agreement with Fig. 2(a) and 2(b) of [Oberhuber] considering the uncertainty in some assumptions (e.g. k_|| space resolution, valence band offset, grid resolution).

Density of states (DOS)

The following figure shows the density of states (i.e. all eigenstates are considered) for the two different gate voltages.

Note:

• At V_G = 1.6 V, the valence band edge maximum is approximately at E_v =   0.106 eV.
• At V_G = 1.1 V, the valence band edge maximum is approximately at E_v = -0.010 eV.

The total DOS output is contained in the following file: Schroedinger_kp/DOS_hl_sum_norm.dat

The following figure shows the density of states for the ground state (light hole state) for the two different gate voltages.
(The DOS for the spin up state is very similar to the DOS of the spin down state.)

The DOS output for each eigenstate is contained in the following file: Schroedinger_kp/DOS_hl_norm.dat

The DOS, as calculated within this 6-band k.p approach is used for obtaining the self-consistent quantum mechanical hole density, thus taking into account nonparabolicity rather than employing a parabolic energy dispersion E(k_x,k_y).

For more details on the density of states, see this tutorial:
Electron density of states (DOS) of a GaAs quantum well with infinite barriers and
hole density of states (DOS) of a Si hole channel (triangular potential)

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