The optical gain variation of Ge nanowires induced by L-valley splitting under the [110] direction stress

The electronic structures of Ge nanowires under the [110] direction stress are calculated via effective-mass k·p theory, and the results manifest eight equivalent L-valleys will be split into fourfold degenerate L 1-valleys and L 2-valleys. With increasing stress, the electron levels at the L 1-valleys and L 2-valleys can be pushed close to and away from those at the Γ-valley, respectively, which causes the appearance of a rising inflection point in the Γ-valley filling ratio and gain peak intensity at around 2.5 GPa stress. Moreover, we prove the positive net peak gain with small diameters is apt to be obtained considering the free-carrier absorption loss.

The electronic structures of Ge nanowires under the [110] direction stress are calculated via effective-mass k•p theory, and the results manifest eight equivalent L-valleys will be split into fourfold degenerate L 1 -valleys and L 2 -valleys.With increasing stress, the electron levels at the L 1 -valleys and L 2 -valleys can be pushed close to and away from those at the Γ-valley, respectively, which causes the appearance of a rising inflection point in the Γ-valley filling ratio and gain peak intensity at around 2.5 GPa stress.Moreover, we prove the positive net peak gain with small diameters is apt to be obtained considering the free-carrier absorption loss.© 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd T he efficient lasers or light emitters compatible with Si-based CMOS are still elusive, which restricts the successful realization of optoelectronic devices such as modulators and photodetectors on the basis of Si.Although group IV Ge can be integrated into Si monolithically, it is an indirect-bandgap semiconductor, which leads to low luminous efficiency.Fortunately, the indirect-bandgap L-valley of Ge is only about 136 meV lower than its direct bandgap Γ-valley, enabling the adjustment of band structures by applying the stress. 1)][12] Among them, Ge nanowires have been deemed to be a potential candidate in the field of optoelectronics due to their easy integration into photonic circuits. 13)n the one hand, compared with the bulk structure, Ge nanowires can withstand large fracture strain, 14) which is almost two orders of magnitude higher than that of the bulk form; 15) on the other hand, Ge nanowires with a size below 10 nm have been grown experimentally, 16) which enables people to produce more compact light sources with high spontaneous rate.Therefore, these characteristics make it possible to impose large stresses to modulate the band structures of Ge nanowires to realize efficient luminescence.Moreover, recent theoretical calculations indicate that the optical gain of Ge nanowires can be continuously improved under the uniaxial stress along the [001] or [100] direction. 17,18)The reason is that the eight equivalent Lvalleys of Ge nanowires will not be split, and the energy difference between the Γ-valley and L-valley will reduce with the increase in stress, resulting in more electrons occupying the Γ-valley in these two cases.
In view of the fact that the effective-mass theory has been successfully used to describe the electronic structures of lowdimensional nanostructures, [19][20][21][22] especially nanowires or quantum wires, the band structures of Ge nanowires at the Γ-valley and L-valley under the [110] direction uniaxial stress are investigated firstly using the k•p theory in this paper, and the calculated results show that the eight degenerate L-valleys of Ge nanowires will be split into two group of L 1 -valleys and L 2 -valleys with fourfold degeneracy. Iterestingly, the electron energy levels at the L 1 -valleys and L 2 -valleys will approach and go away from those at the Γ-valley, respectively, with increasing stress.Therefore, we uncover that both the filling ratio at the Γ-valley and gain peak intensity will increase first and then decrease with the stress of 0 ∼ 4 GPa, and reach the maximum at around 2.5 GPa.Further, when the free-carrier absorption (FCA) loss of Ge nanowires is considered, it is found that the positive net peak gain with small diameters of nanowires under the [110] direction stress can be easily acquired.Our work deepens the applications of strained Ge nanowires as devices in optoelectronics.
Based on the same eight basis functions in Ref. 18, the only difference between the Hamiltonian of Ge nanowires grown along the [001] direction at the direct Γ-valley under the stress along [100] and [110]  Obviously, the form of the strain tensors under U 110 is unlike those under U 100 in Ref. 18.We can see the symmetry broken of nanowires under U 100 arises from the relations ε xx ≠ ε yy and ε xy = 0.In regard to the [110] direction stress, the cylindrical symmetry of nanowires will also be destroyed, but the reason is that ε xx = ε yy and ε xy ≠ 0 in Eq. ( 5), which can be found in Eqs.(7) and (8).Owing to the broken cylindrical symmetry, the envelope wavefunction of the carriers in Ge nanowires under U 110 can take the same Bessel function expansion form as Ref. 18.For the indirect [111] L-valley, we can write the effectivemass Hamiltonian of Ge nanowires grown along the [001] direction under the [110] direction stress U 110 as follows 25)  k z , and V L (r, θ) are the same as those in Ref. 25.The last term H L 110,e in Eq. ( 9) is the strained Hamiltonian introduced by the [110] direction stress, which can be expressed as 26) H I a a : 1 0 where a L i is defined as the unit vector parallel to the wave vector of ith equivalent L-valley, and ε 110 is the strain tensor mentioned above.{} and : are the dyadic product and double dot product, respectively.In Ref. 18, it has been clarified that the degenerate L-valleys of Ge nanowires will not be split when U 100 is imposed, but the situation is different for the uniaxial stress along the [110] direction.For example, with regard to the L-valley along the [111] direction, the strained Hamiltonian H L 110,e can be written as 110,e can be derived as e is destroyed when U 110 is exerted.According to the calculation, it is found that the Hamiltonian of eight degenerate L-valleys will be split into two groups under U 110 .The strained Hamiltonian of the first group is as follows and this set of four degenerate L-valleys is labeled as L 1 .The other group of different strained Hamiltonian is and this set of four degenerate L-valleys is labeled as L 2 .We adopt the same method as in Ref. 25 to calculate the electron energy levels at each L-valley under U 110 , and the convergence of the electron levels at the L-valley is guaranteed, which is also consistent with the results calculated by the sp 3 d 5 s * tight-binding model in Ref. 27.Then, we can acquire the quasi-Fermi level E fc of the conduction band in strained Ge nanowires via the equation 15 where N d and Δn are the donor doping concentration and injected electron concentration, respectively.Owing to the heavy n-type doping in strained Ge nanowires, the acceptor doping concentration N a is neglected.Thus, the quasi-Fermi level E fv of the VB can be determined Finally, we will compute the optical gain of Ge nanowires under U 110 by the formula in Ref. 17 after the band structures and quasi-Fermi-levels are obtained.Figure 1(a) shows the band structures of the six lowest electron states of unstrained Ge nanowires at the Γ-valley when the diameter D = 12 nm and the temperature T = 300 K.In the calculation, we take the same effective-mass parameters of Ge material as Ref. 18, and the effect of bandgap E g variation with the temperature is also considered by the empirical Varshni formula.Meanwhile, it is well known that, when computing the electronic structures of lowdimensional nanostructures grown along the [001] direction, the axis approximation is exact, 21) which is the case in this paper.Due to the unstrained situation, the cylindrical symmetry of Ge nanowires is kept, thus the notation in Ref. 17 can be used to label each electron state.In the figure, the physical meaning of the superscript J can be found in Ref. 17, and every electron state with the same |J| is double degenerate because of the preserved symmetry.As seen in Fig. 1(a), each electron state presents a slight non-parabolic behavior, and the energies of e 1 1 2 and e 0 3 2 are so close that it is difficult to distinguish them in the figure.For example, the energy difference between these two electron states at the Γ point is only about 3.0 meV.Figures 1(b) and 1(c) show the dispersion relation of the eight lowest electron states of the L 1 group and L 2 group valleys in Ge nanowires with U 110 = 0 GPa when the diameter D = 12 nm, respectively.Obviously, the shape of each electron state at the L 1 -valleys and L 2 -valleys in two figures is almost parabolic, and the reason has been analyzed in Ref. 25.According to Eq. ( 8), the strain tensor ε xy is equal to 0 if U 110 = 0 GPa, which will lead to eight equivalent L-valleys not split into L 1 -valleys and L 2 -valleys.In this case, the energy of the lowest electron state at the minimum L-valley is about 166.40 meV lower than that at the minimum Γ-valley.Therefore, the electrons will preferentially fill the L-valley, resulting in the low luminous efficiency of Ge nanowires.When U 110 = 2 GPa is exerted on Ge nanowires, several lowest electron states are displayed at the Γ-valley, L 1 -valley and L 2 -valley in Figs.1(d)-1(f), respectively.In this case, another deformation potential parameter d v = −5.3eV should be employed, 26) and the cylindrical symmetry of Ge nanowires is broken by the applied U 110 , thus the double degeneracy of every electron state in Fig. 1(d) disappears, which makes J no longer a good quantum number. 18)Because the splitting energy of each degenerate electron state under U 110 is small, the electron states such as e 1 and e 3 are not shown in Fig. 1(d).Comparing Fig. 1(a) with 1(d), we find the energies of the electron states decrease under U 110 .At the Γ point, the energy of e 0 1 2 in Fig. 1(a) is about 70.76 meV higher than that of e 0 in Fig. 1(d).Importantly, unlike the case of U 110 = 0 GPa, the eight equivalent L-valleys of Ge nanowires will be split into fourfold degenerate L 1 group and L 2 group valleys due to ε xy ≠ 0. Compared with the stress-free case, it can be clearly found that the energy of every electron state in Figs.1(e) and 1(f) moves upward and downward, respectively.Moreover, the energies of the lowest electron state at the minimum L 1 -valleys and L 2 -valleys are about 27.986 meV and 190.661 meV lower than that at the minimum Γ-valley, respectively, which will lead to the interesting electron occupation at the Γvalley, L 1 -valley and L 2 -valley.Figures 1(g)-1(i) are the same as the corresponding Figs.1(d)-1(f) but for U 110 = 4 GPa.Compared with the case of U 110 = 2 GPa, each electron level in Figs.1(h) and 1(i) will further move upward and downward, respectively.Therefore, the energy of the lowest electron state at the minimum L 1 -valleys increases to nearly 110.27 meV greater than that at the Γ-valley, while the energy of the lowest electron state at the minimum L 2 -valleys decreases to nearly 215.077 meV less than that at the Γ-valley.
As discussed above, the eight degenerate L-valleys of Ge nanowires will be spilt into L 1 -valleys and L 2 -valleys under U 110 , and the splitting energy is u L xy , which is proportional to U 110 .Therefore, with the increase of U 110 , the rule of electron filling at the Γ-valley will be different from that when the [001] or [100] direction stress is imposed. 17,18)igure 2  concentration Δn are chosen as 4 N 0 and 2 N 0 , respectively to determine the quasi-Fermi-levels of Ge nanowires via Eqs.( 15) and ( 16) in which N 0 is equivalent to 1 × 10 19 cm −3 .We can see that the occupation ratio N e G /N e at the Γ-valley is about 1.425% without U 110 when D = 6 nm, which will increase to 4.659% with U 110 = 2 GPa.Therefore, more electrons can be filled into the Γ-valley by applying U 110 with the magnitude of 0 ∼ 2 GPa.However, if U 110 is further increased, the ratio N e G /N e is reduced, and the reason can be analyzed from Fig. /N e will decrease and increase with rising U 110 , respectively.Because the electron energy levels at the L 1 -valleys and L 2 -valleys will be pushed to move close to and away from those at the Γvalley, respectively by the increased U 110 , which has been discussed in Fig. 1.The numerical results indicate that, when U 110 = 0 GPa, N L e 1 /N e and N L e 2 /N e are both 49.927%, which will become about 8.705% and 90.823% with U 110 of 2 GPa, and 0.041% and 99.677% with U 110 of 4 GPa.Therefore, the splitting of the L-valley under U 110 results in the unique variation rule of N e G /N e , which is different from the case of exerting [001] or [100] direction stress. 17,18)For D = 6 nm and 18 nm, N e G /N e will also change similarly as increasing U 110 , and it is concluded that N e G /N e can reach the maximum when U 110 is 2 GPa or 2.5 GPa whether the diameter of Ge nanowires is small or large.Even though, we notice that the ratio N e G /N e for D = 6 nm is around 10 times higher than that for D = 12 nm or 18 nm from Fig. 2(a).The main reason is that the intensity of quantum confinement effect on the electron at the Γ-valley and L-valley is different, and the energy reduction of the electron at the Γ-valley is smaller than that at the L-valley due to the change of quantum confinement effect caused by the increase of the diameter of Ge nanowires.The filling behavior of electrons at the Γ-valley will lead to the interesting variation of optical gain versus the magnitude of U 110 , which is demonstrated in Fig. 3.During the calculation of the optical gain, the transitions from 64 lowest electron states to 96 highest hole states are considered.Sufficient electron states and hole states should be taken into account to ensure the accuracy of the calculation.Moreover, for the same diameter of Ge nanowires, the energies of the hole states are more dense than those of the electron states, so more hole states need to be considered in the calculation of the optical gain.Meanwhile, N d and Δn are set as 4 N 0 and 2 N 0 , respectively.There are two main reasons.Firstly, we investigate the variation law of optical gain of Ge nanowires with U 110 in this paper, so N d and Δn are chosen to be fixed; secondly, as illustrated by the results in Ref. 25, there is almost no optical gain owing to the indirect-bandgap nature of Ge nanowires even if Δn = 4 N 0 when N d is zero for the case of U 110 = 0 GPa, and a remarkable optical gain can be obtained when Δn is close to 10 N 0 .When U 110 is imposed, more electrons will fill at the Γ-valley compared with the case of U 110 = 0 GPa, thus it is possible to obtain obvious optical gain and net optical gain by setting the values of N d and Δn as the order of 10 19 cm −3 .In Fig. 3(a) with D = 6 nm, it is found that the gain peak intensity along the z direction enhances greatly as increasing U 110 from 0 GPa to 1 GPa, and this peak originates from the optical transition e h 0 1 2 0 1 2


with U 110 of 0 GPa and e 0 ⇒h 0 with U 110 of 1 GPa.There are two main reasons for the enhancement.First, the proportion of electrons at the Γvalley has increased from about 1.425% under U 110 = 0 GPa to 3.226% under U 110 = 1 GPa; secondly, the quasi-Fermilevels E fc and E fv will change when U 110 is imposed, which results in the remarkable increment of the expression in the optical gain formula.We notice that, when U 110 rises from 2 GPa to 3 GPa, the gain peak intensity along the z direction only increases slightly.In fact, the ratio N e G /N e at the Γ-valley will reduce from 4.659% with 2 GPa stress to 4.085% with 3 GPa stress in Fig. 2(a), and f f 1 is almost unchanged.Whereas, the redshift of the gain peak position occurs as the increase of U 110 , which causes the gain peak intensity to increase a little instead.The gain peak intensity will shrink as U 110 is further increased from 3 GPa to 4 GPa, In this case, the ratio N e G /N e will drop to 3.095% with 4 GPa stress, meanwhile, it is also accompanied by the decrease of f f 1 to some extent.Figures 3(b) and 3(c) show the same variation rule of optical gain with U 110 as Fig. 3(a), but the diameter is 12 nm and 18 nm, respectively.We can clearly find that the gain peak intensity along the z direction will increase firstly and then decrease, reaching the maximum at nearly U 110 = 3 GPa in these two figures, and a similar analysis can be performed as in Fig. 3(a).Therefore, through the calculation, we can conclude that although the where the physical meaning of the parameters in Eq. ( 17) can be found in Ref. 17.In Fig. 4(a), the peak gain along the z direction, FCA loss and net peak gain of Ge nanowires with D of 6 nm as a function of U 110 is presented.It can be found that the FCA loss experiences a linear increase approximately as increasing U 110 and reaches about 1000 cm −1 when U 110 = 4 GPa, which is far less than the peak gain along the z direction.In this case, a considerable positive net peak gain is gained by exerting U 110 .If D is increased to 12 nm and 18 nm, the calculated results are displayed in Figs.4(b) and 4(c), respectively.Comparing Figs.4(a) with 4(b), we can see that the FCA loss becomes larger under the same U 110 , mainly because the electron and hole energy levels are redshifted when D is increased, which can result in the increase of λ in Eq. (17).For example, λ will go up from about 1292.73 nm in Fig. 4(a) to 1569.55 nm in Fig. 4(b) when U 110 = 2 GPa is fixed.Whereas, the peak gain along the z direction goes down, which leads to a relatively small positive net peak gain or even negative net peak gain at U 110 = 0 GPa and 1 GPa.Moreover, there is a rising inflection point in the net peak gain, and its maximum value will appear at U 110 = 2.5 GPa.The reason is that the peak gain will increase first and then decrease as the increase of U 110 , which has been discussed in Fig. 3.In Fig. 4(c), it will be difficult to obtain positive net optical gain regardless of U 110 with the further improvement of FCA loss.Therefore, from Figs. 4(b) and 4(c), it is found that in order to obtain the positive net peak gain of Ge nanowires with D = 12 nm or 18 nm and realize their applications in the field of optoelectronics, N d and Δn must be enhanced.Specially, for D = 12 nm, we only need to improve a little on the basis of N d = 4 N 0 or Δn = 2 N 0 , and a good net peak gain will be gained.Whereas, for D = 18 nm, we should make a big improvement based on N d = 4 N 0 or Δn = 2 N 0 to acquire a good positive net optical gain.However, due to the limitation of doping technology, both N d and Δn should not exceed 1 × 10 20 cm −3 whether D is 12 nm or 18 nm.In summary, the electronic structures of Ge nanowires at the Γ-valley and L-valley under the uniaxial stress U 110 are calculated by the effective-mass theory, and it is found that the eight equivalent L-valleys will be split into L 1 -valleys and L 2 -valleys with fourfold degeneracy, which differs from the case of imposing [001] or [100] direction stress.Importantly, as the increase of U 110 , the energy of electrons at the L 1 -valleys and L 2 -valleys will be pushed close to and away from those at the Γ-valley, which can lead to the appearance of an inflection point of the filling ratio at the Γ-valley when U 110 is around 2.5 GPa.Similarly, regardless of the diameter of nanowires, the optical gain and net peak gain along the z direction will increase first and then decrease as increasing U 110 from 0 GPa to 4 GPa by considering the FCA loss.Moreover, it is also proved that the positive net peak gain with small diameters of nanowires is easy to acquire.Our research further enriches the application of strained Ge nanowires in the field of silicon photonics.015004-5 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd

X = 6 .
973 eV and u L X = 16.3 eV are the deformation potentials of Ge material at the L-valley. 26)However, for the L-valley along the [1 ̄11] direction, the strained Hamiltonian H L (a) shows the ratio N e G to N e of Ge nanowires with different diameters as a function of U 110 , where N e G and N e are the electron concentration filled at the Γ-valley and total electron concentration, respectively.During the calculation, the doping concentration N d and injected electron

Fig. 1 .Fig. 2 .
Fig. 1.(a) The band structures of six lowest electron states of Ge nanowires without the stress at the Γ-valley when the diameter D = 12 nm; (b) and (c) are the band structures of eight lowest electron states of L 1 group and L 2 group valleys in Ge nanowires without the stress when D = 12 nm, respectively; (d)-(f) are the same as (a)-(c) but for the U 110 = 2 GPa; (g)-(i) are the same as (a)-(c) but for U 110 = 4 GPa.
2(b).As seen in Fig. 2(b), taking D = 12 nm as an example, the proportion of the electrons at the L 1 -valleys N L e 1 /N e and L 2 -valleys N L e 2

Fig. 3 . 4 ©
Fig. 3.The optical gain of Ge nanowires with D = 6 nm, 12 nm, and 18 nm as a function of U 110 is demonstrated in (a), (b), and (c), respectively, where the solid and dash lines represent the optical gain along the z direction and x direction, respectively.

Fig. 4 .
Fig. 4. (a) The relationship between the peak gain along the z direction, FCA loss, net peak gain of Ge nanowires and U 110 when D is 6 nm; (b) and (c) are the same as (a) but for D = 12 nm and 18 nm, respectively.
24)ection is the term introduced by the stress, that is, from H U groups of parameters a c , a v , b v , d v , and ε xx , ε yy , ε zz , ε xy are the deformation potentials at the Γ-valley and the [110] direction stress-induced strain tensors.It should be noticed that the strain tensors ε 110 is different from ε 100 , For the applied [110] direction stress U 110 , the elastic moduli C 11 = 128.53GPa,C 12 = 48.26GPaand C 44 = 66.8 GPa of Ge material can be used to calculate ε xx , ε yy , ε zz , and ε xy .24) e in which two