Physical Model and Simulation Technique

Two important scattering mechanisms are considered in the simulation: acoustic phonon scattering and optical phonon scattering. The scattering-matrix elements are approximated, such that phonon scattering can be considered as velocity randomizing.

The square of the matrix elements between the m-th subband and the n-th subband

Where M^B3D^B is the corresponding matrix element for a bulk hole and q is a phonon wave vector. The overlap integral I^Bmn^B is defined as

m 2

I , ( , ', )_{n D} k k q_z F_nα( , )k z F_mα( , )exp(k z iq z dz_z )

α

=

∑∫

∗

The coupling coefficient for the two dimensional hole is derived:

2

Using the matrix elements, the two dimensional hole scattering rates are calculated according to [5.2]. The acoustic phonon scattering rate is give by

2

where Ξ is the effective acoustic deformation potential, ρ is the material density, u^B1 ^Bis the sound velocity, T is lattice temperature and D^Bn^B(E) is the two dimensional density of hole states in n-th subband.

The optical phonon scattering is

where D^Bt^BK is the average optical deformation potential n^Bop^B is the Bose-Einstein distribution. The + and – represents the absorption and emission rates. The scattering parameters are listed in Table 5.1 [5.3, 5.4].

In the numerical implementation of the valence subband structure in a Monte Carlo simulation, a tabular form of the E-k relationship for the lowest four valence subbbands is established in the simulation. According to the 8-fold symmetry of the Brillouin zone of the subband structure in a quantum well, it’s only necessary to tabulate one-eighth of the zone, which is defined as

0≤k_x≤k_y ≤0

As in [5.5], the eigenvalues for k^B// ^B<0.6π/a, which significantly contribute to the low-field channel mobility, are evaluated. The above k-space region is discretized by mesh points and energy and its gradient at each k point are evaluated. A part of the tabular subband structure form of the k-E relationship is shown in Table 5.2. Table 5.3 lists the E-k relationship including a subband index.

A flowchart of a simple Monte Carlo simulation can be referred to [5.6]. In the Monte Carlo simulation, a sample hole is simulated under an external electric field. It travels freely between two successive scatterings. The free-flight time is determined by using a fixed time technique. During the free flight, the hole is accelerated by the field and its momentum and energy are updated according to the tabular form of the E-k relationship. If a scattering happens, a random number is generated to decide the

responsible scattering mechanism and subband index. Then, the new hole state is chosen according to the tabular form of the E-k relationship. This procedure is continued until the fluctuation due to the statistical uncertainty is less than 1%.

5.3 Results and Discussions

Fig. 5.1 shows the configuration of the simulated devices. The Si/Ge/Si and Si/Si^B0.75^BGe^B0.25^B/Si system are compared. The well width of 40Å is used in this simulation. Fig. 5.2 shows the calculated two dimensional density of hole states obtained from the realistic subband structure. Compared to Si/Si^B0.75^BGe^B0.25^B/Si system, the Si/Ge/Si system has lower density of states, which suggest higher mobility. In Fig.

5.3, hole velocity and average energy versus electric field in Si/Si^B0.75^BGe^B0.25^B/Si system are evaluated. The hole velocity and average energy increase with increasing electric field. The calculated phonon-limited low-field mobility is 422 cm^P2P/Vs under the assumption of no alloy scattering, as shown in Fig. 5.4.

In Fig. 5.5, the Ge quantum well exhibits higher hole mobility than that in Si/Si^B0.75^BGe^B0.25^B/Si system and the calculated phonon-limited low-field mobility is about 890 cm^P2P/Vs. The temperature dependence of the phonon-limited mobility, μ^Bph^B, is also evaluated. The approximated power-law dependence is illustrated by the dashed lines, as shown in Fig. 5.6.

Table 5.1 Scattering parameters for Si and Ge

8

9.2 [5.3] 10.8 [5.4]

13 [5.3] 8.8 [5.4] 10 62 [5.3] 38 [5.4]

eVcm t

op

Parameter Si Ge unit

eV D K

ω

meV Ξ

=

Table 5.2 A tabular form of k-E relationship

Table 5.3 A tabular form of k-E relationship

E

_F

E

₁

E

₂

E

₃

E

₄

E

_lim

Si Ge/

Si

_0.75

Ge

_0.25

Si 40Å

Fig. 5.1 Schematic section of the simulated structures: Ge and SiGe

quantum well.

0 50 100 150 200 0

2 4 6 8 10 12

Si_0.75Ge_0.25 Ge

Density of States (1014 eV-1 cm-2 )

Hole Energy (meV)

Fig. 5.2 Two dimensional density of states in Ge and SiGe quantum well.

1 10 10⁵

10⁶ 10⁷

Electric Field (kV/cm)

Hole velocity (cm/s)

0.05 0.1 0.15 0.2

Average Energy (eV)

Fig. 5.3 Hole velocity and average energy versus electric field.

0 1 2 3 0.0

5.0x10⁵ 1.0x10⁶

Hole velocity (cm/s) U=0

Electric Field (kV/cm)

Fig. 5.4 The phonon-limited low-field mobility in SiGe quantum well

structure.

0.0 0.5 1.0 1.5 0.0

4.0x10⁵ 8.0x10⁵

Si_0.75Ge_0.25 (U=0) Ge

Hole velocity (cm/s)

Electric Field (kV/cm)

Fig. 5.5 Comparisons of phonon-limited low-field mobility in Ge and

SiGe quantum well.

200 300 400 1000

2000 3000 4000 5000

Mobility

(

cm2 /Vs

)

Temperature (K)

Fig. 5.6 Temperature dependence of the phonon-limited mobility in

Ge quantum well.

Chapter 6 Conclusion

In this thesis, a self-consistent solution of the coupled Schrödinger and Poisson equation with the six-band Luttinger-Kohn model is presented. The hole subbands of the inversion layers in p-type MOS, SG SOI and DG devices are demonstrated.

In chapter 2, the Luttinger-Kohn model is introduced. In chapter 3, we have shown the physical characteristics in a (001) and (110) p-type MOS device. The subband energies, wave functions, two dimensional density of states are demonstrated.

The applied bias and substrate orientation effects are also included in the simulation.

In terms of the inversion layer capacitance, the (110) pMOS device shows higher performance than (001) pMOS device due to smaller peak depth of the hole concentration in the (110) MOS device. In chapter 4, we also have shown that symmetrical DG devices exhibit attractive advantages in comparison to SG SOI devices as a result of the double conducting channels and the improved low-field mobility resulting from the lower effective electric field in the silicon layer. In chapter 5, a two dimensional hole Monte Carlo simulation is developed to study the hole transport in Ge and SiGe quantum wells. The simulation results show that the phonon-limited low-field mobility in Ge quantum well is larger that that in SiGe quantum well due to lower scattering rate and lower effective mass.

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