Optical Properties of Semiconductor Nanostructures

Optical Properties of Semiconductor Nanostructures under Intense Terahertz Radiation Tong-Yi Zang and Wei Zhao State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, No. 17 Xinxi Road, Xi’an 710119, People’s Republic of China ABSTRACT The optical absorption properties of semiconductors and their nanostructures under intense terahertz (THz) radiation are investigated theoretically. We derived the extended Semiconductor Bloch Equations (SBEs), which include the eects of the Coulomb interaction among the photoexcited carriers and the eects of the applied external classical (static and/or THz) electric and magnetic elds. We presented two formulations of the SBEs in wavevector space and real space, respectively, which are appropriate to analyzing respective semiconductor structures. Dierent semiconductor nanostructures (such as quantum wells, quantum wires, quantum rings, quantumdot-superlattice nanowires, and quantum-dot-superlattice nanorings) in dierent congurations of the applied external elds (with the THz electric eld applied along the heterostructure interface or the growth-direction) are considered. We showed that the driving of the THz eld can give rise to many intriguing phenomena, such as THz dynamical Franz-Keldysh eect, ac Stark eect, THz-sideband, replica of dark exciton states. We also suggested some potential application of these new phenomena in developing novel semiconductor optoelectronic devices. Keywords: optical absorption, semiconductor nanostructure, terahertz radiation, Semiconductor Bloch Equations

1. INTRODUCTION The eect of an applied static electric eld on the optical absorption coe!cient of semiconductor structures is known as Franz-Keldysh eect (FKE), i.e., a pronounced oscillations occurring in the optical absorption spectrum above the bandgap and nonvanishing exponential absorption tail below the bandgap.1, 2 If the static electric eld is replaced by a time-dependent electric elds, the eect is called dynamical Franz-Keldysh eect (DFKE), where the bandedge blueshift by the ponderomotive energy, Hsrq = h2 I02 @4p 2 (i.e., the average kinetic energy of a particle of mass p and charge h in an electric eld I0 cos w) and thus show above-band-gap transparency and increased absorption below the gap.3—6 Since the optical absorption spectrum near the bandgap is dominated by exciton eects, which manifest itself as strong resonances occurring at bound exciton states and enhanced continuum absorption, the DFKE is subsequently extended to include the exciton eects.7, 8 Semiconductor Bloch equations (SBEs) is one of the powerful tools for semiconductor optics, which properly describe the dependence on the light eld, include the valence conduction band continuum states, exciton eects, as well as band-lling dynamics.9 In this paper, we present two form of the SBEs in wave-vector space and in real space, and use them to calculate the optical absorption of an ultrashort interband probe pulse in semiconductor nanostructures driven by a cw-THz beam. This paper is organized as follows: After this introduction section, section 2 presents the fundamental theory about optical absorption and formulations of SBEs. The section 3 contains the numerical results obtained by applying the theories to some typical semiconductor structures, such as quantum well (subsection 3.1 and 3.2), cylindrical quantum wire (subsection 3.3) and cylindrical quantum-dot-superlattice wire (subsection 3.4), and nanoring (subsection 3.5). Finally, a brief summary is given in section 4. (Send correspondence to T. Y. Zhang) E-mail: [email protected], Telephone: 86 29 88888049

International Symposium on Photoelectronic Detection and Imaging 2009: Terahertz and High Energy Radiation Detection Technologies and Applications, edited by X.-C. Zhang, James M. Ryan, Cun-lin Zhang, Chuan-xiang Tang, Proc. of SPIE Vol. 7385, 738505 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.835684 Proc. of SPIE Vol. 7385 738505-1

2. THEORY The absorption coe!cient ($) of a dielectric medium is related to the optical dielectric function ($) by $ 00 ($)> q($)f

($) =

where q($) =

r h 1 2

0 ($) +

(1)

i p 02 ($) + 002 ($) is the index of refraction, f is the speed of light in free-space, and

0 ($) and 00 ($) are the real and imaginary parts of ($), respectively. The optical dielectric function is related to the optical susceptibility "($) via ($) = 1 + "($). While the optical susceptibility relates the excitation optical eld E(w) with its induced macroscopic polarization S (w), by S (w) = 0

Z

w

gw0 "(w; w0 )H(w0 )>

(2)

4

where 0 is the vacuum permittivity. For a system in equilibrium, the optical susceptibility depends only on the time dierence. By performing a Fourier transform of eq.(2) with the time dierence, "($) can be obtained as "($)

S˜ ($) > ˜ 0 E($)

(3)

˜ where E($) and S˜ ($) are the Fourier transform of the optical eld E(w) and the polarization function S (w). Thus, it is the macroscopic polarization that one has to nd. One of the powerful tools used to calculate the macroscopic polarization induced by a light eld is the SBEs, which originally developed in wavevector space but can be cast into a real-space form under some limited cases.

2.1. SBEs in k -space In k -space, the P macroscopic interband polarization S (w) is the sum of the microscopic interband polarization, S (w) = 2gfy k sk (w), where gfy is the interband dipole matrix element and the factor 2 is due to the spin ® degeneracy. In the two-band model, the microscopic interband polarization is dened as sk (w) = k k , where k ( k ) is the annihilation operator of an electron (a hole) with wavevector n (n) in the conduction (valence) band. Under Hartree-Fock approximation, from the Heisenberg equations of motion for the electron and hole we can D operators, D derive the E E well-known SBEs for microscopic polarization sk (w) and carrier population qhk = †k k and qkk = †k k . In the presence of an in-plane driving eld, the two-band SBEs for a quantum well are given by,10—15 ¡ ¢ ¢ C l¡ h C sk = hhk + hkk sk l qhk + qkk 1 U>k F · uk sk + sk |vfdww > Cw ~ ~ Cw C h q = 2 Im [ U>k sk ] Cw k C k q = 2 Im [ U>k sk ] Cw k

h F · uk qhk + ~ h F · uk qkk + ~

C h q |vfdww > Cw k C k q |vfdww > Cw k

where ~ is the reduced Planck constant, h is the absolute value of the electron charge, hk = (Hk (h>k)

(4) P

q Y|kq| qq )@~

is the electron and hole band are the electron and hole energies renormalised by Coulomb interactions, Hk P 2 dispersions, U>k = (gfy E(w) + q6=k Y|kq| Sq )@~ is the generalized Rabi frequency, Yq = 2h0 q1 is the Fourier transform of ideal two-dimensional Coulomb potential, the eect of nite width of realistic quantum well can be accounted for by introducing a form factor into the Coulomb potential. The scattering terms in (4) represent the higher-order Coulomb correlations between carriers and carrier-phonon collisions, F = FW K} (w) + Fgf , with FW K} (w) = F0 cos( W K} w ) the in-plane driving THz eld, and F0 , W K} , the amplitude, frequency, and phase, respectively.

Proc. of SPIE Vol. 7385 738505-2

In the presence of an growth-direction driving eld for a quantum well, we have to include the subband mixing eect. Then the two-band is extended to multiband SBEs. In the low excitation limit, the population equations is decoupled with the polarization equations and can be neglected. In such a case, the multiband SBEs are given by,16, 17 gspo k gw

where pq = h} pq = h

¢ l X ps l¡ o p Hj spo Y (k q) ssv Hhk + Hkk k + q ~ ~ sv>q ov ¢ sv l X ¡ ln pq po +lE(w) po + sk pq sqo k sq vf sk > ~ q

=

R O@2

O@2

(5)

g}*p (})}*q (}) are the intersubband dipole matrix elements with *p (}) the

envelop function of electrons and holes in the quantum well level of index p, Yovps (q) = matrix elements with the Coulomb overlap integrals Z O@2 ps t|}h }k | s iov = g}h g}k *oh (}h )*p *k (}k )*vh (}h )= k (}k )h

ps h2 iov 20 q

are the Coulomb

O@2

2.2. SBEs in real-space For some realistic one-dimensional structures, like V-groove and T-shaped quantum wires, it is very di!cult to employ the SBEs in k -space to dealing the optical properties of these structures. For these realistic structures, one has to take advantage of real-space method. This is due that the expansion in a problem-adapted basis leads to four-dimensional integrals for the Coulomb interaction, while an expansion in a basis where the Coulomb potential is not a problem will probably show a very slow convergence with basis size. In real space, in the optical polarization can be calculated from the time-dependent electron-hole envelope wave function (> }h > }k ; w) by Z O@2 S (w) = gfy g} (0> }> }; w)> (6) O@2

gfy

is the complex conjugate of gfy , i.e. the projection on the direction of the optical polarization of where the interband dipole matrix element, O is the length of a quantum wire or the width of a quantum well. The electron—hole envelope wave function depends on three arguments: the electron and hole coordinates }h , }k in the growth direction and the in-plane relative coordinate . For example, using the eective mass theory for the envelope function, the electron—hole envelope function in a quantum wire obeys the SBEs in real space as18 : l~

l~

C (}> w) Cw

Cq(}> w) Cw

ˆ hi i (}> w) + gfy H(w)[2q(}> w) O(})] = K Z 1 g} 0 [Y hhhh (} 0 ) + Y kkkk (} 0 )]q(} 0 w) (} } 0 > w) O Z 2 + g} 0 Y hkkh (} 0 )q(} } 0 > w) (} 0 > w)> O

= (}> w)H (w)gfy (}> w)H(w)gfy Z 1 g} 0 Y hkkh (} 0 )[ (} } 0 > w) (} 0 > w) (} } 0 > w) (} 0 > w)]> O

(7)

where is the Dirac’s delta function, } = }h }k is the relative coordinate of the electron and hole in the axial direction, is the phenomenological dephasing constant of the optical polarization resulted from various ˆ hi i in (7) is the eective electron-hole Hamiltonian expressed in real space, which scatterings in the system. K form is determined by the considered system. For example, for a quantum dot superlattice nanowire, the eective Hamiltonian for an electron-hole pair is19


2 2 2 2 ˆ = ~ C ~ C + Xh (}h ) + Xk (}k ) + hI (}h }k ) YFrxo (}h }k )> K 2ph }h2 2pk }k2

(8)

where ph and pk are masses of the electron and hole respectively, and Xh (}h ) and Xk (}k ) denote the electron and hole conning potentials along the quantum-dot-superlattice nanowire respectively due to the discontinuities of conduction band and valence band for alternating materials, YFrxo is the eective Coulomb interaction between electron and hole averaged over the lateral states. The innite potential at the lateral wall of the thin nanowire is adequate for free standing nanowires. This implies the Bessel function of zero order for the envelope of the lateral ground state wave function. Considering the fundamental electron and hole subbands only, the quasi 1-D Coulomb interaction YFrxo (}h }k ) is obtained from the average of the 3-D Coulomb interaction with the lateral ground state19—22 : YFrxo (}h }k ) =

Z

0

U

Z

U 0

Z

0

2

Z

0

2

h k M02

¡ 0 h ¢ U

M02

¡ 0 k ¢

U p gh gk gh gk (}k }h )2 + 2h + 2k 2h k cos(k h )

(9)

where = h2 @[43 %U4 M14 (0 )], % the dielectric constant, U the radius of the nanowire, h(k) and h(k) are the polar angle and radial coordinate of electron (hole) respectively, M0 , M1 the Bessel functions and 0 is the zero of the M0 .

3. NUMERICAL RESULTS OF OPTICAL ABSORPTION FOR SOME SEMICONDUCTOR STRUCTURES We have applied the two forms of SBEs to investigating optical absorption spectra for dierent semiconductor structures. In this section, the numerical results presented in Subsections 1 and 6 are simulated using the k -space SBEs, while those in Subsections 2 to 5 are simulated using the real-space SBEs.

3.1. Quantum well with THz wave polarized in in-plane direction11 The dependence of the optical absorption spectrum of a two-dimensional semiconductor on the intensity of the in-plane driving THz eld with frequency of 2.5 THz (10.3 meV) and 3.5 THz (14.42 meV) is shown in Figs 1(a) and (b), respectively. The two selected frequencies are near resonant with the 1s-2p transition but be tuned below or above the 1s-2p transition. Compared to the eld-free absorption, the THz eld induces the 2p replica in the vicinity of the 1s peak, leads to the appearance of the 1s sideband and shifts of the main exciton peak. As the intensity increases, the 1s peak redshifts and reaches a maximum and reverses becoming blueshifts For the frequency of 10.3 meV, but blueshifts for all THz intensities for the frequency of 14.42 meV. This is due to the interplay between the DFKE and ac Stark eect. Apart from the shifts, the 1s peak broadens as the eld strength increases due to the THz-induced oscillations in the relative position of the bound electron and hole.


Fig. 1. (Color online) Absorption spectra for THz frequencies 2.5 THz (10.3 meV) (a) and 3.5 THz (14.42 meV) (b). In each case, the amplitudes of the THz elds are, from top to bottom, 1.0, . . . , 7.0 kV/cm, respectively. Figure 2 shows the dependence of the optical absorption coe!cient on the frequency of the THz eld with strength of 6 kV/cm. As the frequency increases from 2.5 to 4.0 THz, the 2p replica anti-crosses with the 1s exciton peak from 3 meV above to 5 meV below the 1s exciton peak. The main 1s exciton peak broadens when the frequency is tuned above 1s-2p transition. This broadening is due to the breaking up of 1s exciton as THz photon energy is approximately equivalent to the 1s exciton binding energy. Though it is di!cult presently to lock the arrival of the probe pulse to a particular phase, the investigation of the phase dependence of the absorption should shed light on the fundamental physics of carrier-photon interaction in quantum wells. The dependence of the nonlinear optical features on the phase of the THz eld at the time of the optical pulse arrival is shown in Fig. 3. Varying the phase of the THz eld causes a signicant change in the shape of the sideband absorption characteristics. The characteristics evolve from absorption peaks to gain peaks. The steep transition from absorption to gain occurs at the position of sideband. Remarkably, the sidebands below the main exciton peak are much more pronounced than those above the main exciton peak. In contrast, the 2p replica have very weak phase dependence. The shape and position of the exciton peaks do not change.

Fig. 2. (Color online) (a) Absorption spectra for THz frequencies (from bottom to top) 2.5, 2.6, . . . , 4.0 THz, respectively. The 2p replica anticrosses the 1s exciton peak when the THz frequency increases from below to above the 1s-2p transition energy. (b) Absorption spectra for amplitudes of the THz elds 6.0 kV/cm and THz frequency 2.5 THz. From bottom to top, the phase of the THz eld is 0@4, @4, . . . , 7@4, respectively.

3.2. Quantum well under a THz eld along the growth direction23 The optical absorption of a QW under a growth-direction THz eld is shown in Fig. 3. Fig. 3(a) shows the optical absorption spectra for photon energies of 5, 11, and 17 meV, respectively, with the eld strength xed at 30 kV/cm. The cases with a 30 kV/cm dc bias eld (solid black line) and without an external eld (dashed black line) are also shown for comparison. It can be seen from Fig. 3 that without external elds, excitonic peaks corresponding to transitions with the same subband index for conduction and valence bands, i.e. e1-h1 and e2-h2, are shown up, while a dc eld induces a Stark redshift of the e1-h1 exciton peak and reduces the height, and gives rise to additional corresponding to transitions with dierent subband index for conduction and valence bands. The THz eld leads to further peaks corresponding to replicas and sidebands of the exciton transitions. Furthermore, the net shift of the main exciton e1-h1 peak can be either a redshift (red line) or a blueshift (green and blue lines). This is resulted from the competition between ac Stark eect and DFKE. The dependence of the optical absorption spectra on the eld strength is shown in Fig.3(b). We can see a stronger THz eld leads to more replicas.


Fig. 3. (a) the optical absorption spectra of the quantum well under 30 kV/cm strength THz eld with photon energies of 5, 11, and 17 meV, respectively, and for a 30 kV/cm dc bias eld (solid black line) and without an external eld (dashed black line). (b) the optical absorption spectra of the quantum well under a THz eld with photon energy 11 meV but with dierent strength of 15, 30, and 50 kV/cm.

3.3. Cylindrical quantum wire under an axial direction THz eld22 The optical absorption spectra of a cylindrical quantum wire with length 4200 nm and radius 4.2 nm under an THz electric eld is shown in Fig. 4(a). The frequency of the THz eld is 5 THz, nearly resonant with the 1s exciton. With the eld strength increasing, the main exciton peak splits, and the two peaks separate farther, and broaden, even become a band, and other peaks and oscillations emerge gradually. This splitting resembles the Autler-Townes splitting for three level atoms.24 When eld strength is lower (higher) than 9 kV/cm, the peak at high (low) energy side is dominant over the one at low (high) energy side. Furthermore, the 2s resonance and the band-edge blueshift due to dynamical FK eect. There appear two-photon replica in the continuum. With the THz eld increasing, the two-photon replica redshift initially and turns into blueshift and broadens. This is because the ac-Stark eect and DFKE oppose each other. In very high eld, the continuum becomes oscillating, and some other peaks corresponding to multi-photon replicas occur. Figure 4(b) shows the spatiotemporal evolution of the polarization wave-packet for some dierent values of the THz eld strength. The polarization wave-packet is distorted by the eld and spreads to two directions following the alternating changing of the eld. And there is a sharp drop for high eld strength which indicates the ionization of the excitons immediately after they are generated.


1 .5

1 0.5

Fac=21

0

0

400

400 300 1.5

200 t (fs) 100

300

80 −80

0 z (nm)

1.5

200 t (fs) 100

80 −80

0 z (nm)

1

1 0.5

Fac=30

0.5

Fac=39

F =48 ac

0

0

400

400 300 200 t (fs) 100

80 −80

0 z (nm)

300 200 t (fs) 100

80 −80

0 z (nm)

Fig. 4. (Color online) Optical absorption spectra of the cylindrical quantum wire under THz electric elds. The eld strength Idf varies from 3 to 48 kV/cm at the step of 3 kV/cm from top to bottom with frequency W K} of the THz eld xed at 5 THz. (b) The spatiotemporal evolution of the polarization wave-packet when increasing the eld strength with frequency W K} =5 THz. The eight panels correspond to dierent eld strengths Igf =21, 30, 39, and 48 kV/cm, respectively.

3.4. Cylindrical quantum-dot-superlattice nanowire under an axial direction THz eld19 The eects of an external THz eld applied along the quantum-dot-superlattice nanowire on the interband optical absorption spectra are shown in Fig. 5. Fig. 5(a) shows the absorption spectra for dierent frequencies of the THz eld with the eld strength xed at Idf = 30 kV/cm. The main exciton peak decreases, broadens and becomes asymmetric. It has small gain and small peaks appears in the spectra. These peaks are replicas of the excitonic states dressed by THz photons. The excitonic absorption spectra for dierent eld strengths Idf with frequency xed at ~ =20 meV are shown in Fig. 5(b). The oscillator strength is completely transferred from the minibands to the exciton. Increasing eld strength the main excitonic peak decreases, broadens and splits up. The asymmetric splitting is the well known Autler-Townes splitting and is due to the coherent coupling of the lowest and higher exciton states. Furthermore, when the eld strength increase, there exists one-photon and two-photon gain spectra.

Fig. 5. (Color online) Optical absorption in the QDSLW under a THz eld Idf along the wire. (a) Dierent frequencies ~ =5, 11, and 20 meV, with eld strength xed at Idf =30 kV/cm; (b) Dierent eld strengths


Idf =20, 30, 50, and 60 kV/cm, with frequency xed at ~ =20 meV.

3.5. Quantum ring under a lateral THz eld25 Figure 6 shows the optical absorption in a nanoring with radius 50 nm under a lateral ac eld and dierent magnetic ux through the ring. The photon energy of the ac electric eld in Figs. 6(a) and 6(b) is 0.256 meV and 10.22 meV, respectively, but with the same strength 5 kV/cm. We see that the shape of the exciton peak does not change noticeable with magnetic ux in the lower frequency ac eld, but there is a slightly blue shift of the exciton peak with the ux increasing. However, in the higher frequency ac eld, the exciton peak splits remarkably and a prominent peak appears bellow the main exciton peak, which is the replica of the dark exciton state just below the bandgap.

Fig. 6. (Color online) Optical absorption of the nanoring under a lateral THz eld and a perpendicular magnetic eld. (a) In the presence of a lateral THz eld with Idf = 5 kV/cm and ~ = 0=256 meV, and dierent magnetic ux. (b) In the presence of the lateral THz eld with I = 5 kV/cm and ~ = 10=22 meV, and dierent magnetic ux.

4. CONCLUSION To summary, we have presented the extended SBEs in two forms as in k -space and in real-space, respectively, in this paper. We have applied them to deal with the optical absorption in some dierent semiconductor structures driven by a THz electric eld. And we reviewed some main results we have obtained in recent years. It is shown that an applied THz could give rise to dynamical FKE and ac Stark eect in the excitonic optical absorption. The THz eld can also lead to Autler-Townes splitting and replicas of the resonant exciton transitions. These eects may nd use in future multiplexing or optical switch applications.

ACKNOWLEDGMENTS This work is supported by the National Basic Research Program of China (973 Program) (2007CB310405) and the National Natural Science Foundation of China (60777017 and 10834015). The authors gratefully acknowledge the support of K. C. Wong Education Foundation, Hong Kong.

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