Theory of effect of near-infrared laser polarization ... - OSA Publishing

Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors Houquan Liu and Weilong She * State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China * [email protected]

Abstract: In a semiconductor illuminated by a strong terahertz (THz) field, the electron–hole pairs excited by linear polarized near-infrared (NIR) laser can recombine to emit high-order THz sideband. Previous experimental results have shown, under the same condition of excitation intensity, the polarization direction of the NIR laser could affect the sideband intensity. In this letter, we theoretically investigate the effect of the NIR laser polarization direction on high-order terahertz sideband generation in bulk GaAs. ©2015 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.5970) Semiconductor nonlinear optics including MQW

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002). J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009). Y. Jie-Yun, “Dynamics of a pair of electron and hole in semiconductor superlattice under an intense electric field,” Chin. Phys. B 17(12), 4640–4644 (2008). F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013). F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014). B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012). R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455– 1456 (2007). J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78(7), 075204 (2008). J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014). X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013). J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014). B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013). H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).

1. Introduction Recently, the issue of semiconductor systems in the presence of intense terahertz (THz) fields is of great interest [1–6]. When linear polarized near-infrared (NIR) laser irradiates the semiconductors under an intense THz field being applied, high-order terahertz sideband

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Received 15 Jan 2015; revised 10 Mar 2015; accepted 23 Mar 2015; published 16 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.010680 | OPTICS EXPRESS 10680

generation (HSG) will occur [7–11], which presents a flat wide-band plateau similar to the high-order harmonic generation by atoms under super-intense optical lasers. The HSG in semiconductors has attracted much attention from both theoretical and experimental researchers. It has been ascertained that the physical mechanism of the HSG is electron–hole recollisions, which can be described by a three-step model [6,8]. In the first step, electron– hole pairs are created by NIR laser. In the second step, driven by the THz field, electron–hole pairs accelerate along different quantum trajectories. Finally, the electron-hole pairs annihilate and generate sidebands. It was first predicted by Liu et al. [7], and then a detailed theory study was given by Yan [8]. J.A. Crosse et al. gave theoretical prediction of lowpower ultra-broadband terahertz sideband generation in bi-layer grapheme [9] and Xie et al. explored the effects of excitation frequency on HSG [10]. The HSG has been verified by a series of experiments in InGaAs/AlGaAs quantum well [6], bulk GaAs [12] and GaAs/AlGaAs quantum well [13]. Zaks and associates [6] found in their experiment that, under the same condition of excitation intensity, the sideband intensity when the polarization direction of the NIR laser is parallel to the THz field is not same as that when the polarization direction of the NIR laser is perpendicular to the THz field, which means the polarization direction of the NIR laser could affect the sideband intensity. Therefore, in this letter, our purpose is to give a theoretical discussion about the effect of the polarization direction of the NIR laser on the HSG, which is performed in bulk GaAs. 2. Model and numerical result In the semiconductor, under an intense THz field, the kinetic energy acquired by the electronhole pair can be much greater than the exciton binding energy and the amplitude of the relative motion can be much greater than the exciton radius, so the essential physics of the HSG can be captured by the motion of free electrons and holes without Coulomb interaction [10]. The optical response of semiconductors under THz field is determined by i

∂ψ ( p, t ) ∂t

2  1  =  p − eA ( t )  + Eg − iγ 2 ψ ( p, t ) + d cv ⋅ E ( t ) , μ 2  

(1)

where, μ is the reduced mass of the electron-hole pair, Eg is band gap, γ 2 is interband dephasing rate due to photon scattering, p is momentum vector, A(t) is the vector potential of THz field A ( t ) = − F sin (ω0 t ) eˆz ω0 , E(t) is the NIR laser E ( t ) = E0 exp ( −iΩt ) and d cv is the interband transition dipole. The interband polarization can be obtained from the expectation values of the dipole moment operator P ( t ) = dˆ ( t ) = −

1

d cv ⋅ψ ( p, t ) dp, 3 ( 2π  )  ∗

(2)

from which the HSG spectrum can be calculated via the Fourier transform. By formally integrating Eq. (1), the expression for the electron-hole pair wave function can be found to be

ψ ( p, t ) = −

i ∞ − i ( Eg e  0

) t −τ dt ′′  p − eA( t ′′) 2

 − iγ 2 τ − i

t

2μ

d cv ⋅ E ( t − τ ) dτ .

(3)

Here τ denotes the delay between the recombination and the creation of the electron-hole pair. Substituting Eq. (3) back into Eq. (2), the polarization can be rewritten as

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P (t ) =

i

d cv* d cv ⋅ E0 eiSθ (τ ) dτ dp, 3 ( 2π  )  

(4)

where S = −  dt ′′  p − eA ( t ′′ )  2μ  − ( Eg  − Ω − iγ 2 )τ − Ωt. (5) t −τ From Eq. (4) we can see, to obtain the Fourier formula of P(t) one should first study d cv∗ d cv . In the previous theoretical works of Yan [8] and Xie et al. [10], they consider only the contribution of electron transition from the heavy hole band to conduction band, and they assume the transition dipole d cv as a constant, basing on which the NIR laser polarization direction dependent sideband intensity is not given in their results. Therefore, in our derivation the transition dipole is carefully treated and the influence of the optical transition from light hole band is taken into consideration. We begin our discussion from the wave functions of electrons of conduction band, light hole band and heavy hole band in bulk GaAs, which are respectively t

2

+ − = S ⊗ ↑↓ , 3 1 ,± = 2 2

X p ± i Yp 2 Z p ⊗ ↑↓  ⊗ ↓↑ , 3 6

(6)

X p ± i Yp 3 3 ,± = ⊗ ↑↓ . 2 2 2

Where ↑ ↓ presents spin up/down, S is the wave function of atomic s orbital, and X p , Xp

Yp , Z p

denote wave functions of atomic p orbitals. The transformations between

Yp , Z p

and the wave functions of the eˆx , eˆy , eˆz atomic p orbitals X , Y , Z are

,

given by  Xp   Yp    Zp

   cos θ p cos φ p  =  − sin φ p     sin θ p cos φ p 

cos θ p sin φ p cos φ p sin θ p sin φ p

− sin θ p   X    0   Y . cos θ p   Z 

(7)

Where, ( eˆx , eˆy , eˆz ) is global coordinate, and θ p and φ p are azimuth angle and polar angle of momentum p, respectively. According to Eqs. (6) and (7), d cv∗ d cv can be evaluated easily. Here we discuss the transition dipole of heavy hole band to conduction band and light hole band to conduction band separately. For the heavy hole band d ch* d ch = d 2  eˆx eˆx ( cos 2 θ p cos 2 φ p + sin 2 φ p ) + eˆy eˆy ( cos 2 θ p sin 2 φ p + cos 2 φ p ) − ( eˆx eˆy +eˆy eˆx ) sin 2 θ p sin φ p cos φ p − ( eˆx eˆz +eˆz eˆx ) sin θ p cos θ p cos φ p

, (8)

+eˆz eˆz sin 2 θ p − ( eˆy eˆz +eˆz eˆy ) sin θ p cos θ p sin φ p 

while for the light hole band

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1 d cl* d cl = d 2 eˆx eˆx (1 + 3sin 2 θ p cos 2 φ p ) + eˆy eˆy (1 + 3sin 2 θ p sin 2 φ p ) 3 + ( eˆx eˆy +eˆy eˆx ) 3sin 2 θ p sin φ p cos φ p + ( eˆx eˆz +eˆz eˆx ) 3 sin θ p cos θ p cos φ p . (9) +eˆz eˆz (1 + 3cos 2 θ p ) + ( eˆy eˆz +eˆz eˆy ) 3 sin θ p cos θ p sin φ p 

Where d ch ( d cl ) denotes the transition dipole of heavy (light) hole band to conduction band. And d is a constant which is related to the interband transition and determined by material. From Eqs. (8) and (9) we can see that the transition dipole is dependent on the direction of momentum p. In the following, we will see this is the root why the NIR laser polarization direction could affect the sideband intensity. By substituting Eqs. (8) and (9) back into Eq. (4), the integral over p could be performed in spherical coordinate frame. To investigate the effect of the polarization direction of the NIR laser on the HSG, we discuss P(t) in two different cases: the polarization direction of the NIR laser is parallel to the THz field, i.e. E0 = E0 eˆz , and the polarization direction of the NIR laser is perpendicular to the THz field, i.e. E0 = E0 eˆx . Since the interband polarization for any other polarization direction of NIR laser is a linear combination of the interband polarization for these two cases. For the former case, the polarization can be gotten by substituting Eqs. (8) and (9) back into Eq. (4). It is P ( t ) = +

id 2 E0 eˆz

( 2π )

3



4

 sin

2

id E0 eˆz

3 ( 2π )  4 3

2

θ p eiS θ (τ ) dτ p 2 sin θ p dpdθ p dφ p ch

iS 2 2  (1 + 3cos θ p ) e cl θ (τ ) dτ p sin θ p dpdθ p dφ p ,

(10)

where the subscript  denotes the polarization direction of the NIR laser is parallel to the THz field, and Scj = −  dt ′′  p 2 − 2 p cos θ p + e 2 A2 ( t ′′ )  2 μcj  − ( Eg  − Ω − iγ 2 )τ − Ωt. (11) t −τ t

Here the letter j indicates h and l, and μch is the reduced mass of pair of the conduction band electron and heavy hole, μcl represents the reduced mass of pair of the conduction band electron and light hole. While for the case of E0 = E0 eˆx , the polarization is given by P⊥ ( t ) = +

id 2 E0 eˆx

( 2π )

3

4

id 2 E0 eˆx

3 ( 2π )  4 3

 ( cos

2

θ p cos 2 φ p + sin 2 φ p ) eiS θ (τ ) dτ p 2 sin θ p dpdθ p dφ p

 (1 + 3sin

ch

2

θ p cos φ p ) e θ (τ ) dτ p sin θ p dpdθ p dφ p . 2

iScl

(12)

2

Where the subscript ⊥ denotes the polarization direction of the NIR laser is perpendicular to the THz field. According to Eqs. (10) and (12), it is very easy for one to verify that P ( t ) = P⊥ ( t ) when the THz field is zero. While when a THz field is applied, the THz field leads to an additional

{

item exp i 

t

t −τ

}

A ( t ′′ ) dt ′′2ep cos θ p 2μ  in the integrand of Eq. (4), which influences the

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results of integral over θ p to make P ( t ) ≠ P⊥ ( t ) , then the sideband intensity is not equal to each other in the two cases. And it could also be found, if the transition dipole does not depend on the direction of the momentum, P ( t ) is always equal to P⊥ ( t ) . Therefore the fundamental cause of the effect of the NIR laser polarization direction on the HSG is the momentum direction dependent transition dipole. To calculate the difference of the sideband intensity of the two situations, we study the Fourier transformation of P ( t ) and P⊥ ( t ) . After some algebraic operation, we could obtain that 1  P (Ω + 2 N ω0 ) = i − N DE p  μch   Q τ  +U ch μch Q

ch

ch

J − N (Φch ) sin 3 θ pθ (τ )dθ p dτ

γ 2  2iJ − N (Φch ) − J − N −1 (Φch ) + J − N +1 (Φch ) 

× sin 3 θ p cos 2 θ pθ (τ )dθ p dτ

μcl  1 Q cl J − N (Φcl ) sin θ p (1 + 3cos 2 θ p )θ (τ )dθ p dτ 3 τ U μ + cl cl Q cl γ 2  2iJ − N (Φcl ) − J − N −1 (Φcl ) + J − N +1 (Φcl ) 

+

3

(13)

}

× sin θ p cos 2 θ p (1 + 3cos 2 θ p ) θ (τ )dθ p dτ ,

and P⊥ (Ω + 2 N ω0 )

μ  1 = i − N DE p  ch  Q ch J − N (Φch ) sin θ p ( cos 2 θ p + 1) θ (τ )dθ p dτ  2 τ U μ + ch ch Q ch γ 2  2iJ − N (Φch ) − J − N −1 (Φch ) + J − N +1 (Φch )  2 × sin θ p cos 2 θ p ( cos 2 θ p + 1) θ (τ )dθ p dτ

(14)

μcl  1 Q cl J − N (Φcl ) sin θ p ( 2 + 3sin 2 θ p ) θ (τ )dθ p dτ 6 τ U μ + cl cl Q cl γ 2  2iJ − N (Φcl ) − J − N −1 (Φcl ) + J − N +1 (Φcl )  +

6

}

× sin θ p cos 2 θ p ( 2 + 3sin 2 θ p ) θ (τ ) dθ p dτ .

Where

πμcj  i ( Λcj + Nω0 )τ d2  ) − cos 2 θ p γ  ,Q cj = e ,D = , 2  2iτ (2π ) 2  4   (15) Eg U cj U cj sin(ω0τ 2) e2 F 2 2 2 Λcj = Ω − cos θ p γ , U cj = ,γ = . − + iγ 2 + ω0τ 2    4μcj ω02

Φcj =

U cj



τγ cos(

ω0τ

Here the letter j indicates h and l. The HSG is then determined by

χ 2λN =

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Pλ ( Ω + 2 N )

ε 0 E0

( λ =⊥,  ) .

(16)


According to Eqs. (13)-(16), we calculate the sideband intensity numerically and plot it in 

Fig. 1. In our calculation, the parameters that we have used are: γ 2 = 2.5meV , d = 6.7e A , and the effective mass of electrons of conduction band (mc = 0.067me), heavy hole band (mh = 0.45me) and light hole band (ml = 0.082me), which yields μch = 0.0787me and μcl = 0.3663me . And the photon energy of the THz field is chosen to be 2.4meV and its strength be 11kV cm−1, which is same as the THz field utilized by B. Zaks et al. [12] in their experiment; and the NIR laser detuning is Ω − Eg = 0 .

Fig. 1. The sideband intensity when the polarization direction of the NIR laser is parallel and 

perpendicular to the THz field. We have chosen γ 2 = 2.5meV , d = 6.7e A , the effective mass of electrons of conduction band (mc = 0.067me), heavy hole band (mh = 0.45me) and light hole band (ml = 0.082me), the photon energy of the THz field is 2.4meV and its strength is 11kV cm−1, and the NIR laser detuning is Ω − Eg = 0 .

From Fig. 1 we can see, for sidebands of order 2N = 16 and 2 N ≥ 22 , the intensity for the case of polarization direction of the NIR laser perpendicular to the THz field is greater than that for polarization direction of the NIR laser parallel to the THz field. For sidebands of other orders, however, the situation is reverse. We could also find when the NIR laser polarization direction is perpendicular to the THz field, the sideband plateau is smoother than that when the NIR laser polarization direction is parallel to the THz field; and the plateau in the former case is greater than the latter one about two orders of magnitude, therefore in experiments and applications of the HSG in GaAs, the polarization direction of the NIR laser perpendicular to the THz field is a better option. Comparison with the experiment data of B. Zaks et al. [12], in their work sidebands of orders −6 ≤ 2 N ≤ 18 are observed and the intensity of the second order is about four orders of magnitude greater than that of the 18th order sideband, which is good agreement with our result. Summary

In summary, we have studied the effect of the polarization direction of the linear polarized NIR laser on the HSG in bulk GaAs driven by intense THz field. Our result shows, under the condition of same excitation intensity, the sideband intensity when the polarization direction of the NIR laser is perpendicular to the THz field is not same to that when the polarization direction of the NIR laser is parallel to the THz field. And we find that the transition dipole is dependent on the direction of momentum is the reason why the polarization direction of the

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NIR laser could affect the sideband intensity. In this letter, our calculations are performed in bulk material, while for the case of quantum well, the quantum confinement breaks the degeneracy between light- and heavy-hole subbands, and there is also effect of quantum confinement on the interband transition dipole, therefore the effect of the NIR laser polarization on the HSG in quantum well is worthy of further investigation. Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (Grant NO. 11274401).

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