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Using the
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theory, the coupling effect between the Δ1 and Δ2’ bands on the energy band structure of different energy valleys is studied. The analytical model of the energy-dispersion relationship applicable to uniaxial stress for arbitrary crystal plane and orientation as well as different energy valleys is established. For typical crystal orientations, the main parameters of energy band structure such as band edge level, splitting energy, density-of-state (DOS) effective mass and conductivity effective mass are calculated. The calculated results are in good agreement with the data reported in related literature. Finally, the relationship between the DOS effective mass, conductivity effective mass and the change of stress and orientation of different crystal planes is given. The proposed model and calculation results can provide a theoretical reference for the design of nano-electronic devices and TCAD simulation.

As an important method of extending Moore’s Law, strained silicon technology can significantly improve the mobility of carriers in devices [

The key to obtaining the structure of the conduction band in uniaxial strained silicon is establishing the E(k)-k relation near the minimum value of the conduction band. Using the traditional k∙p perturbation method and deformation potential theory, the calculation results in [

The bottom of the conduction band of bulk silicon is located on the Δ axis of the Brillouin zone, and the energy band at the bottom of the conduction band and its neighbor band at the boundary of Brillouin zone are denoted as the Δ1 band and Δ2’ band. The Δ1 band and the Δ2’ band are non-degenerate at the Γ point of the centre of Brillouin zone, but they are degenerate at the position X. Under shear stress, the degeneracy of the Δ1 and Δ2’ bands at X is eliminated. The two bands couple with each other, changing the dispersion relationship of the conduction band and the valley minimum [_{v} . Here the subscripts v = 1, 2, 3 denote three equivalent energy valleys [00±1], [0±10] and [±100]. Consequently,

T 1 = ( 1 0 0 0 1 0 0 0 1 ) , T 2 = ( 0 1 0 0 0 1 1 0 0 ) , T 3 = ( 0 0 1 1 0 0 0 1 0 ) .

In the coordinate system of each valley, the z-axis coincides with the rotation axis of each ellipsoid. For any energy valley v, then the wave vector k_{v} and the strain tensor ε_{v} can be respectively written as

k v = T v k and ε v = T v T ε T v .

For each energy valley, the shear strain tensor can be expressed as

ε shear , v = ε v , 12 = ε v , 21 .

Not any uniaxial stress can lead to splitting at position X. If the shear strain tensor of a valley is ε_{shear} = 0, then Δ_{1} and Δ_{2’} at X are still degenerate. Therefore, under uniaxial stress without shear stress, the uniaxial stress only leads to movement of the energy level of conduction band, without changing its band structure.

If the z-axis of the lattice coordinate system (x, y, z) is rotated by φ and the x-axis is rotated by θ, the stress coordinate system (x’, y’, z’) is obtained. The wave vector

k ε = ( k x ′ k y ′ k z ′ ) T .

under stress coordinate system can be expressed as

k ε = U k ,

where

k = ( k x k y k z ) T

is the wave vector in the lattice coordinate system. The transformation matrix is

U = ( cos φ cos θ − sin θ sin φ cos θ cos φ sin θ cos θ sin φ sin θ − sin φ 0 cos φ )

For different valleys,

k ε , v = T v k ε = T v U k .

According to the two-band k∙p theory, the strain Hamiltonian of different valleys of the conduction band in bulk silicon at position X of the boundary of the Brillouin zone is a 2 × 2 matrix [_{1} and Δ_{2’} bands. For any energy valley, they can be written as

H i , v ( k ) = E C , 0 + Δ E C , v + ( ℏ 2 2 ) k T M v k + ( − 1 ) i − 1 ( 0 0 ℏ m 0 p ) T v k . (1)

with

Δ E C , v = D d ( ε 11 + ε 22 + ε 33 ) + D u ε v v ,

M v = T v T ( m t − 1 m t − 1 m l − 1 ) T v .

where E_{C}_{,0} = 1.119 eV is the energy of the conduction band in its unstressed state, ΔE_{C}_{,v} is the energy change of the conduction band edge, and i = 1, 2 denote the energy bands of Δ_{1} and Δ_{2’}. In addition, p denotes the electron momentum at point X when Δ_{1} and Δ_{2’} are not strained. m_{0} stands for the mass of free electrons, m_{t} and m_{l} represent the transverse and longitudinal effective mass of the electrons, respectively. If the point X is assumed to be the origin of k-space, the minimum value of the conduction band [00±1] can be −k_{0} = −0.15(2π/a_{0}), where a_{0} is the lattice constant. When no uniaxial stress is applied, then Δ_{1} is the lowest band, and for any energy valley, the following relationship holds:

( ∂ H 1 , v ( k ) ∂ k i ) k = T v ( 0 , 0 , − k 0 ) T = 0 ,

that yields

p = ( m 0 m l ) ℏ k 0 . (2)

The subscripts ( v , i ) = ( 1 , z ) , ( 2 , y ) , ( 3 , x ) denote the [00±1], [0±10] and [±100] energy valleys and their rotation axes. Substituting Equation (2) into Equation (1), and diagonalizing the strain Hamiltonian, the dispersion relation of the conduction band energy at X point can be obtained:

E v ( k ) = E C , 0 + Δ E C , v + ( ℏ 2 2 ) k T M v k − ( 0 0 ℏ 2 k 0 m l ) T v k + ( k T P v k − D ε shear , v ) 2 (3)

where

P v = T v T ( 0 P 0 0 0 0 0 0 0 ) T v ,

here P = ℏ 2 / M , the parameter M can be obtained by the empirical pseudo-potential method [_{1} and Δ_{2’} at the X point under the action of shear stress leads to a change in the valley minimum. Using the dispersion relation of Equation (3), the minimum of each valley can be obtained from

( ∂ E v ( k ) ∂ k i ) k = T v ( 0 , 0 , − k 0 , ε ) T = 0 . (4)

By expanding Equation (3) at the valley minimum and neglecting the higher order terms, one obtains the conduction band dispersion relation near the minimum:

E v ( k ) = E C , 0 + Δ E C , v + ( H 1 , v ) k z = k 0 , ε + ( ℏ 2 ( k z − k 0 , ε ) 2 2 m l − ℏ 2 k 0 , ε ( k z − k 0 , ε ) m l ) . (5)

The calculation of the effective mass uses the stress coordinate system in the [00±1] valley. Starting from the dispersion relation of the conduction band from Equation (3) or Equation (5), the following inverse transform

k ↦ U − 1 T v − 1 k ε , v

is carried out in Equation (3) to obtain E_{v}(k_{ε}_{,v}). The effective mass of valley [00±1] can then be obtained by the following expression:

m i , v = ( 1 ℏ 2 ∂ 2 E v ( k ε , v ) ∂ k i 2 ) k ε , v = T v U ( 0 0 − k 0 , ε ) T − 1 , (6)

where, i = x ′ , y ′ , z ′ . The 6-degree degenerate conduction band can be split by the uniaxial stress into valleys in different degenerate states, leading to the change of the distribution of electron concentration in the valley. Under the action of stress, the quantum state density of each energy valley is

g v ( E ) = M v ( 4 π ( 2 m n , v ) 3 / 2 h 3 ) E − E C , v , (7)

where M_{v} is the degeneracy of each valley, and E_{C}_{,v} is its minimum energy. The effective mass of each valley can be expressed as

m n , v = m v , x ′ m v , y ′ m v , z ′ 3 . (8)

Assume that electrons are subject to the Boltzmann distribution

f B ( E ) = exp ( E f − E k B T ) ,

and ΔE_{split1} and ΔE_{split2} represent the split energy of the valley of the conduction band. For the general application, the electron concentration of the conduction band at equilibrium state can be represented as

∑ v = 1 3 ( ∫ E C , v ∞ g v ( E ) f B ( E ) d E ) = 2 ( 2 π m DOS k B T h 2 ) 3 / 2 exp ( E f − E C k B T ) . (9)

E_{C}_{,v} also represents the minimum energy of each valley. Taking E_{C}_{,1} = 0 as the reference point of energy, then E_{C}_{,2} = ΔE_{split1}, E_{C}_{,3} = ΔE_{split2} . The expression of the effective mass of the electron density at the bottom of the silicon conduction band under arbitrary uniaxial stress is obtained by the following formula:

m DOS = ( M 1 m n , 1 2 / 3 + M 2 m n , 2 2 / 3 exp ( − Δ E split 1 k B T ) + M 3 m n , 3 2 / 3 exp ( − Δ E split 2 k B T ) ) 2 / 3 . (10)

In Equation (10), the subscripts 1, 2, and 3 of the effective mass m and the degeneracy M in the above equation represent the lowest valleys, the intermediate valleys and the highest valleys, respectively. If the conduction band is split into a lower 2-degree degenerate valley and a higher 4-degree degenerate valley, then Δ E split1 = Δ E split2 = Δ E , and m n , 1 = m n , low , m n , 2 = m n , 3 = m n , high and the DOS effective mass becomes

m DOS = ( 2 m n , low 3 / 2 + 4 m n , high 3 / 2 exp ( − Δ E k B T ) ) 2 / 3 . (11)

If the valley is divided into a lower 4-degree degenerate valley and a higher 2-degree degenerate valley, the density calculation method is the same as the former. In particular, if there is no split of valley under a single uniaxial stress, then the shear strain component of any energy valley becomes due to symmetry, and Δ E split1 = Δ E split2 = 0 . Then the effective mass of the density is

m DOS = ( 6 2 / 3 ) m n , 1 = ( 6 2 / 3 ) m n , 2 = ( 6 2 / 3 ) m n , 3 . (12)

The electron effective mass of the valley m_{n}_{,v} can be obtained by Equation (6) and Equation (8).

For conduction bands of strained silicon, the conductivity effective mass depends on the electron occupancy in each valley and the effective mass along the stress direction. In general, R_{v} is used to represent the electronic possession of each valley, where v = 1, 2, 3 is the conduction band with the lowest, intermediate and the highest valley, respectively. According to Equation (7) we can see:

R v = ( m n , v ) 3 / 2 exp ( − E c , v k B T ) ∑ v = 1 3 ( m n , v ) 3 / 2 exp ( − E c , v k B T ) . (13)

Following the same method as the conductivity effective mass of the unstrained silicon, the expression of the conductivity effective mass m_{c} of the electrons in conduction band under uniaxial stress on arbitrary crystalline plane can be written as

m c = ( ∑ v = 1 3 ( R v m v , x ′ ) ) − 1 . (14)

It should be noticed that if there is no stress applied, then R 1 = R 2 = R 2 = 1 / 3 , and Δ E split1 = Δ E split2 = 0 , hence Equation (14) can be simplified to the unstrained conductivity effective mass

m c = 3 / ( 2 m t − 1 + m l − 1 ) .

In order to make this model suitable for arbitrary crystal plane and orientation, let the angle θ between x’ axis and x axis reflect the crystal plane in any direction of uniaxial stress. Using Hooke’s law, we can derive the strain tensor ε of the three typical high-symmetry planes (001), (101), and (111). Here c_{ij} is the elastic stiffness coefficient, and σ is the corresponding tensile stress. A negative value corresponds to compressive stress. From the following strain tensor model, the conduction band structure can be calculated and analyzed.

ε ( 001 ) = σ × ( ( c 11 cos 2 θ + c 12 cos 2 θ c 11 2 + c 11 c 12 − 2 c 12 2 ) ( sin 2 θ 2 c 44 ) 0 ( sin 2 θ 2 c 44 ) ( c 11 sin 2 θ − c 12 cos 2 θ c 11 2 + c 11 c 12 − 2 c 12 2 ) 0 0 0 − ( c 12 c 11 2 + c 11 c 12 − 2 c 12 2 ) ) ,

ε ( 101 ) = σ × ( 1 2 ( c 11 cos 2 θ − 2 c 12 sin 2 θ c 11 2 + c 11 c 12 − 2 c 12 2 ) ( sin 2 θ 4 c 44 ) − ( cos θ 2 c 44 ) ( sin 2 θ 4 c 44 ) 1 2 ( c 11 sin 2 θ − 2 c 12 cos 2 θ c 11 2 + c 11 c 12 − 2 c 12 2 ) − ( sin θ 2 c 44 ) − ( cos θ 2 c 44 ) − ( sin θ 2 c 44 ) − 1 2 ( c 11 c 11 2 + c 11 c 12 − 2 c 12 2 ) ) ,

ε ( 111 ) = σ × ( 1 3 ( 2 c 11 cos 2 θ + c 12 ( 1 − 4 sin 2 θ ) c 11 2 + c 11 c 12 − 2 c 12 2 ) 1 3 ( sin 2 θ c 44 ) − 1 3 ( cos θ 2 c 44 ) 1 3 ( sin 2 θ c 44 ) 1 3 ( 2 c 11 sin 2 θ − c 12 ( 3 − 4 sin 2 θ ) c 11 2 + c 11 c 12 − 2 c 12 2 ) − 1 3 ( sin θ 2 c 44 ) − 1 3 ( cos θ 2 c 44 ) − 1 3 ( sin θ 2 c 44 ) 1 3 ( 1 c 11 + 2 c 12 ) ) .

From the dispersion relation model, the dispersion relation curve of any valleys along an arbitrary crystal plane can be obtained. Crystal plane and orientation have an infinite variety of options. These cannot be calculated one by one, so the [_{shear,v}. Let

k = k min = ( 0 0 − k 0 , ε ) T

in Equation (3), then

Δ E shear , v = E v ( k min , ε shear , v ) − E v ( k min , 0 ) .

compressive stress. The position of the wave vector k corresponding to the minimum energy level of the conduction band does not change, since the shear strain tensor of the [_{1} band in the [00±1] valley is −k_{0}, and the minimum value of band Δ_{2’} is k_{0}. There is no coupling between the two at point X, the non-diagonal element in Equation (3)

k T P 1 k = P k x k y = 0 ,

and the Δ_{1} and Δ_{2’} bands are coincident and the minimum of the energy band is still at point X. Similarly, the change of energy band in [0±10] valley has the same trend as the [00±1] valley.

In order to reflect the effect of shear stress on the band structure, the [00±1] energy valley is used to calculate under the application of stress in the [_{1} and Δ_{2’} bands in [±100] and [0±10] are still degenerate at the point X. The band curvature does not show apparent change.

The band edge level of the silicon conduction band is a necessary parameter for calculating the density-of-state (DOS) effective mass and conductivity effective mass of electrons. Under the action of uniaxial stress, the degenerate energy level in the conduction band is split, and the movement ΔE_{C}_{,v} of each energy level can be described by deformation potential theory. For each valley, the wave vector k_{min} corresponding to the valley minimum is substituted into Equation (1), obtaining the band edge level corresponding to each energy valley. For bands with an energy minimum at point X, the band edge level under uniaxial stress can be reduced to

E C , v = E C , 0 + Δ E C , v + Δ E shear , v .

In order to illustrate the relationship between stress and the splitting energy of the conduction band,

valley in the silicon conduction band along with the [

In the design of scaled CMOS devices, stresses in the [

The change of conductivity effective mass of electrons under stress in the [

For general application,

Based on the two-band k∙p theory, the analytical model of energy-dispersion relationship of different energy valleys under arbitrary uniaxial stress on arbitrary crystal plane was established. In this process, the coupling effect between the Δ_{1} band and the Δ_{2’} band on the dispersion of different valleys and the variation of valley minimum by shear stress under different uniaxial stresses were considered through the transformation of stress and lattice coordinate systems. Based on the established dispersion relationship on the conduction band, three typical high-symmetry crystal planes (001), (101) and (111) were taken as examples, and major energy structure parameters such as the band edge level, splitting energy, density-of-state (DOS) effective mass, conductivity effective mass, and mobility in the [

In addition to the above mentioned, in fact, strain technology can be involved not only in MOS devices, but also in bipolar and optoelectronic devices. Recent studies of TCAD simulation pointed out that the strain effect has an influence on the electrical parameters of bipolar transistors, it is possible to introduce uniaxial stress induced by the functional oxides into silicon photonics applications. The proposed band-parameter model and calculation results can provide a theoretical reference for the design and optimization for silicon-based nano-electronic devices, bipolar devices or optoeletronic devices, and TCAD simulation.

This work is financially supported by National Natural Science Foundation of China (Grant Nos.: 61404019, 61704147), the Science Fund from the Education Department of Hebei Province, China (Grant No.: QN2017150).

Wang, G.Y., Yu, M.D. and Zhou, C.Y. (2018) Modelling and Calculation of Silicon Conduction Band Structure and Parameters with Arbitrary Uniaxial Stress. Journal of Applied Mathematics and Physics, 6, 183-197. https://doi.org/10.4236/jamp.2018.61018