E-k Relation of Valence Band in Arbitrary Orientation / Typical Plane Uniaxially Strained

Uniaxial strain technology is an effective way to improve the performance of the small size CMOS devices, by which carriermobility can be enhanced. The E-k relation of the valence band in uniaxially strained Si is the theoretical basis for understanding and enhancing hole mobility. The solving procedure of the relation and its analytic expression were still lacking, and the compressive results of the valence band parameters in uniaxially strained Si were not found in the references. So, the E-k relation has been derived by taking strainedHamiltonian perturbation into account. And then the valence band parameters were obtained, including the energy levels at Γ point, the splitting energy, and hole effective masses. Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design.


Introduction
Uniaxially strained Si technique is the preferred option to the design and application for the high-speed and highperformance nanoscale CMOS devices and circuits, which can enhance significantly the carrier mobility [1][2][3][4][5][6][7].It is theoretically essential to study the E-k relation of valence band in uniaxially strained Si for the understanding and improvement of hole mobility enhancement.
Although wide reports [8][9][10][11][12][13] about the valence band of uniaxially strained Si can be found, there still have been some questions to discuss carefully.(1) The E-k relation is key to obtain the physical parameters of valence band.However, reports about how to derive the relation and what its analytical expression is are not found.(2) The cell structures and physical properties are different when the uniaxial stress is applied on Si materials along different orientation/different planes.So, it is very important to give the valence band parameters under different strained situation for the device design.Only some valence band structures in uniaxially strained Si are reported and overall analysis is still needed.
In view of those, we aim to discuss how to solve the eigenvalue matrix of the valence band structure in uniaxially strained Si and establish the analytical E-k model for the valence band.Moreover, the valence band parameters inarbitrary orientation/typical plane uniaxially strained Si will be studied, including the energy levels at Γ point, the splitting energy, and the hole effective mass.Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design.

The E-k Relations
2.1.Strain Tensor.Strain tensor model is the basis of the calculation of the deformation potential energy in uniaxially strained Si materials [14,15].The strain tensors for arbitrary orientation/(001), (101), and (111) planes uniaxially strained Si are, respectively, shown as follows: where  11 ,  12 , and  44 are the elastic stiffness coefficients of Si materials;  is the applied uniaxial stress; the azimuth angle  is shown in Figure 1.

Strained kp Hamilton.
The degeneracy case of valence band in unstrained Si is Γ  25 threefold degenerate states without the coupling of spin-orbit taken into consideration.And the corresponding kp Hamilton can be written as follows: The applied uniaxial stress will lead to the displacement of wave vector  and the corresponding variation can be described by the strained tensors in Section 2.1.And thus the matrix for the strained Hamilton is similar to expression (2): where the values for , , , , , and  can be found in [17,18].In order to obtain the finerband structures, we must take the coupling of spin-orbit into consideration.And we need to change (2) and (3) to the expression for the double group firstly: Then we also take the following spin-orbit Hamilton (H SO ) into consideration [17].Solving the eigenvalues of the matrix ( KP +  Str +  SO ) and the results we desired will be obtained.The final E-k relations of the valence band in uniaxially strained Si are shown as follows: where   (,  = 1, 2, 3) is the element of the matrix   +

Results and Discussion
3.1.Energy Levels at Γ Point.For unstrained Si, the maximum of valence band energy level is at the Γ point; that is,  k = (0, 0, 0).The relations between the energy levels of the first valence band ("Heavy Hole Band, " "HH"), the second valence band ("Light Hole Band, " "LH"), and the third valence band ("Spin-Orbit Coupling Band, " "SO") and the stress can be given by (5).
The energy levels at the Γ point of the "HH, " "LH, " and "SO" in uniaxially strained Si (001), (101), and (111) are shown in Figures 2, 3, and 4, respectively.As indicated in the figures, splitting between the first valence band and the second valence band in all the strained Si occurs and the degeneracy at the top of the valence band lifts, which are obviously different from one of the unstrained Si.Under the tensile strain, the energy levels of the first and the second bands increase with the increasing stress.The energy level of the first band also increases with the increasing compressive stress while the level of the second band decreases with the increasing compressive stress.Moreover, the change in the energy levels with the angle of applied stress can also be seen in Figures 2, 3, and 4.

Splitting Energy.
The splitting energy is closely related to the hole distribution in strained Si, which is the basis to establish the model of the density of states in valence band.
The relationship between the splitting energy and the magnitude of the stress  and the direction of the applied stress  in uniaxially strained Si (001), (101), and (111) are shown in Figure 5.As can be seen from the figure, the function relations between the splitting energy and the compressive tress are nearly linear.And the slope of the relation curve is greatest when  = /4.The splitting energies increase with the increasing tensile stress and they tend to be gradual when the magnitude of the stress is beyond 1 GPa.Moreover, the effect of the applied stress  on the splitting energy can be neglected under tensile stress, which can be indicated from the nearly coincide in the curves for  = 0, /2, and  = /4.In summary, the splitting energy under compressive strain is much greater than the one under tensile strain.That is, the change in splitting energy for the compressive case is more sensitive.This is one of the most important reasons why to choose compressive film to improve the pMOS performance.Our results shown in the right part of Figure 6 are consistent with the results from literature [16] is shown in the left part of Figure 6.It is with agreement in our result shown in the right part of Figure 6.  7, 8, and 9.The values of the 2D equienergy are 10, 25, 50, 75, and 100 meV, respectively, and the value of the 3D equienergy is 40 meV.
For instance, [110]/(001) uniaxially compressive strained Si (1 GPa) is shown in Figure 7.As can be seen, the protruding parts in equienergy surface of unstrained Si significantly shrink when [110]/(001) uniaxially compressive stress is applied on the unstrained Si.And the shape of the equienergy surface of [110]/(001) uniaxially compressive strained Si turns to be ellipsoid spherical.
It is worthy to note that the anisotropy of the valence band in strained Si is more significant compared with the one in unstrained Si.For example, [−110] direction and [110] direction are equivalent directions in unstrained Si and their corresponding effective masses are also equal.However, their values are not coincident in [110]/(001) uniaxially compressive strained Si, which can be seen from the 2D  In order to verify the accuracy of our results, the equienergy surface in [110]/(001) uniaxially compressive strained Si reported from [16] is shown in Figure 10.It is the same as our results in Figure 7, which proves the validity of our model.

Conclusions
In this paper, we establish the models of strain tensor and strain Hamilton firstly and derive the E-k relation of the valence band in uniaxially strained Si by kp method.Finally the valence band parameters were obtained, including the energy levels at Γ point, the splitting energy, and hole effective masses.
The results show that (1) splitting between the first valence band and the second valence band in all the strained Si occurs and the degeneracy at the top of the valence band lifts, which are obviously different from one of the unstrained Si. (2) The splitting energy under compressive strain is much greater than the one under tensile strain and the change in splitting energy for the compressive case is more sensitive.
(3) The anisotropy of the valence band in strained Si is more significant compared with the one in unstrained Si.Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design.

Figure 3 :
Figure 3: The levels at Γ point versus stress and angle in uniaxially strained Si (101).

Figure 4 :
Figure 4: The levels at Γ point versus stress and angle in uniaxially strained Si (111).

Figure 5 :
Figure 5: The splitting energy versus stress and angle in uniaxially strained Si.

3. 4 .
Physical Interpretation.Now we turn to discuss the reason why the valence band structures of Si materials will change under strain from two aspects.Strained Si has different valence band structure and hole mass from that of relaxed Si.These may be understood by their thin film cell structures.The cubic cell structure of relaxed Si goes triangular or tetragonal or monoclinic due Advances in Condensed Matter Physics 7