Matrix Techniques
2.3 Physical Models
2.3.1 Kubo-Greenwood Formula
The outcomes of the solver p-NEP contain the hole subband energy level, the Fermi level, the wave function, the DOS function, and the strained E-k structure. Then, the hole inversion-layer mobility can be calculated using the Kubo-Greenwood formula [6]-[8]:
∑ 1
(2.3.1.1) where e is the free electron charge, is the total hole density per unit area which sums along the z-direction, is the group velocity of subband μ along x direction, and f0 is the Fermi-Dirac distribution function in equilibrium. Under the momentum relaxation time approximation, the total scattering time of subband μin (1) can relate to acoustic phonon scattering, optical phonon scattering, and surface roughness scattering through the expression:
. In this dissertation, eighteen lowest subbands were used in the mobility calculation.
2.3.2 Acoustic Phonon Scattering
We followed the isotropic treatment by Fischetti et al. [9] concerning acoustic phonon scattering, but did not take into account the inelastic and dielectric screening effects of acoustic phonons in this work. The critical parameter, namely, the acoustic deformation potential Dac, is strongly connected to Bir-Pikus deformation potentials [4],[10],[11]. According to Lawaetz [11], Dac can be formulated as
⁄ ;
;
, (2.3.2.1) where and are the average longitudinal and transverse elastic coefficients, respectively. , , and are the elastic coefficient elements whose values are listed in Table II. The elastic acoustic phonon scattering rate model used in this work is
∑ , (2.3.2.2)
where is equal to ⋅ ∗ , the wavefunction overlap integral between the initial subband μ and the final subband ν. and denote the crystal density and the longitudinal sound velocity, respectively. Both intra- and inter-subband acoustic phonon scattering were considered in this work.
2.3.3 Optical Phonon Scattering
Optical phonon scattering involves the absorption and emission of optical phonons with the exchange of energy (61.2 meV in this dissertation). According to Wiley [12] and Costato and Reggiani [13], the optical deformation potential Dop can
have the following formalism:
, (2.3.3.1) where a0 is the lattice constant, d0 is the deformation potential of optical phonons, and
̅ is the average sound velocity consisting of the longitudinal and transverse sound velocity, and , respectively, with the formulation of ̅ 2 /3. Then, the isotropic absorbing and emitting optical phonon scattering rate can be written as
∑ ∓
∓ , (2.3.3.2) : optical phonon frequency,
: Bose occupation factor of optical phonons
Eq. (2.3.3.2) features both intra- and inter-subband optical phonon scattering but with no screening effect in this work.
2.3.4 Surface Roughness Scattering
To deal with surface roughness scattering, we first followed Pham, et al. [14] and De Michielis, et al., [15] to take only the intra-subband scattering. Then, the screening effect in the dielectric function as formulated by Yamakawa et al.[16] and Gámiz, et al. [17] was incorporated into the surface roughness scattering rate expression in the context of the exponential autocovariance function:
∑
,
′ ′, (2.3.4.1)
∶ rms height of the amplitude of the surface roughness,
∶ correlation length of the surface roughness,
∶ ′ ,
∶ angle between ′ and ,
∶ adjustable factor (= 1/2 in this work),
, ∶ , ∗ ⋅ ⋅ ⋅ ∗ ⋅ ⋅ ∗ ⋅ ,
: energy minimum of subband ν
: static wave-vector-dependent dielectric function,
1 ′ ′;
∑ ′ | | | | | ′|.
Importantly, Eq. (2.3.4.1) has the ability to adequately handle the angular dependence of surface roughness scattering, as shown in Fig. 2.1 in terms of the calculated unscreened scattering rate versus hole energy for both (001) and (110) substrates. It can also be seen from the figure that the scattering angular is more pronounced for GPa-level stress, especially on the (001) substrate.
2.3.5 WKB Based Hole Gate Direct Tunneling
According to the literatures [18]-[20],the isotropic hole direct tunneling current density contributed by the μth subband with WKB approximation can be written as
∙ ∙ ∙ , (2.3.5.1)
∑ , (2.3.5.2) where e denotes the elemental charge, is impact frequency of hole' wave packet on interface, is the inversion hole density of subband μ per energy per area, is the transmission probability through insulator of WKB part, is the transmission probability through insulator of reflection part. There are eighteen subbands taken into account in our calculation. The impact frequency is described as
eF 2⁄ 2 , (2.3.5.3) Worth noting that the constant quantized effect mass in Eq. (2.3.5.3) approximately obtained by the triangular potential method will be discussed in Chapter 3. For the inversion hole density, we calculate it through
. (2.3.5.4) Two terms to be considered in the transmission probability through oxidelayer, the first one is WKB part, which can be modeled as
where is the barrier height of the tunneling hole with total energy E at cathode side or gate/oxideinterface, and is that at anode side or oxide/n-well interface with is the barrier height of oxide/Si interface. is an important parameter featuring the quantum transport in oxide. The WKB approximation is only valid when the wavefunction phase change is much smaller than the amplitude change, In other words, the barrier potential of oxide must be very sharp and high enough to ensure the validity of WKB method. Another transmission probability is , which is given as incident and leaving oxide. The semi-classical forms can be simply depicted as
, , 0 ;
, | | | | . (2.3.5.7) Moreover, and are the magnitudes of the purely imaginary group velocities of holes at the cathode and anode side within the oxide. The semi-classical form can be expressed as
, (2.3.5.8) where is the virtual energy in the classical forbidden region of oxide.
2.3.6 Stress-to-Strain Tensor
The stress effect on the warping valance band fundamentally is complex. Instead, we can use approximate simplification to deal with the mechanics of materials such as a general form originating from Hooke’s law as below [21]
where σ refers to the normal stress component acting on the planes perpendicular to i-direction, while τ indicates the shear stress components oriented in the j-direction acting on the planes perpendicular to i-direction, denotes shear strain, is average shear strain and is defined as one half the . , and represent the Young’s modules, Poisson’s ratio and shear modulus of elasticity, respectively. We therefore establish the elastic strain-stress matrix as follows [22]11 12 12
where , and are the elastic stiffness constants. Then, the inverse matrix in Eq. (2.3.6.4) refers to the strain-to-stress tensor written by
0 0 0
The corresponding relationships between strain and stress under longitudinal, transverse, and out-of-plane stresses in (001) and (110) p-MOSFETs are shown in Fig.
2.2 and 2.3.