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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 11694–11699
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Proposal for the momentum-resolved and time-resolved optical measurement of the current distribution in semiconductors

Jiang-Tao Liu, Fu-Hai Su, Xin-Hua Deng, and Hai Wang  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 11694-11699 (2012)
http://dx.doi.org/10.1364/OE.20.011694


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Abstract

The two-color optical coherence absorption spectrum (QUIC-AB) of semiconductors in the presence of a charge current is investigated. We find that the QUIC-AB depends strongly not only on the amplitude of the electron current but also on the direction of the electron current. Thus, the amplitude and the angular distribution of current in semiconductors can be detected directly in real time with the QUIC-AB.

© 2012 OSA

1. Introduction

Directly mapping the amplitude and the angular distribution of current in real time plays important roles both in basic physics and applied engineering. For example, the graphene transistor is promising to extend to THz frequency [1

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

, 2

2. Y. M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.Y. Chiu, A. Grill, and P. Avouris, “100-GHz transistors from wafer-scale epitaxial graphene,” Science 327, 662–662 (2010). [CrossRef] [PubMed]

], which require the ability to detect the ultrafast current with sub-picosecond time resolution. Real-time current detection in momentum space is also important in the studies of Bloch oscillator [3

3. T. Hyart, N. V. Alexeeva, J. Mattas, and K. N. Alekseev, “Terahertz bloch oscillator with a modulated bias,” Phys. Rev. Lett. 102, 140405 (2009). [CrossRef] [PubMed]

], quantum transport in nano-electronics [4

4. J. Rammer, “Quantum transport theory of electrons in solids: a single-particle approach,” Rev. Mod. Phys. 63, 781–817 (1991), and references therein. [CrossRef]

], spin Hall effect [5

5. M. I. Dyakonov and V. I. Perel, “Current-induced spin orientation of electrons in semiconductors,” Phys. Lett. A 35, 459–460 (1971). [CrossRef]

8

8. W. Yang, K. Chang, and S. C. Zhang, “Intrinsic spin Hall effect induced by quantum phase transition in HgCdTe quantumwells,” Phys. Rev. Lett. 100, 056602 (2008). [CrossRef] [PubMed]

], and scattering processes in semiconductors such as the scattering between carriers and impurity [9

9. J. Yan and M. S. Fuhrer, “Correlated charged impurity scattering in graphene,” Phys. Rev. Lett. 107, 206601 (2011). [CrossRef] [PubMed]

11

11. J. Smit, “The spontaneous Hall effect in ferromagnetics II,” Physica 24, 39–51 (1958). [CrossRef]

], lattice vibration, and crystal defects etc.

Some experimental techniques such as angle-resolved photoemission spectroscopy (ARPES) [12

12. S. Hüfner, Photoelectron Spectroscopy, (Springer-Verlag, 1995).

, 13

13. A. Damascelli, Z. Hussain, and Z. Shen, “Angle-resolved photoemission studies of the cuprate superconductors,” Rev. Mod. Phys. 75, 473–541 (2003). [CrossRef]

] have been used to observe the distribution of electrons in the momentum space. Especially, the time-resolved two-photon photoemission experiment (2PPE) was used to access the electron current dynamics at a metal surface on the femtosecond time scale [14

14. J. Güde, M. Rohleder, T. Meier, S. W. Koch, and U. Höer, “Time-resolved investigation of coherently controlled electric currents at a metal surface,” Science 318, 1287–1291 (2007). [CrossRef]

]. However, the photoemission spectroscopy normally is only appropriate for the mapping of the surface current. Khurgin proposed that the current induced second harmonic generation (SHG) can be used to map the current in semiconductor [15

15. J. B. Khurgin, “Current induced second harmonic generation in semiconductors,” Appl. Phys. Lett. 67, 1113–1115 (1995). [CrossRef]

] and have been successfully verified by experiments very recently [16

16. B. A. Ruzicka, L. K. Werake, G. W Xu, J. B. Khurgin, E. Ya. Sherman, J. Z. Wu, and H. Zhao, “Second-harmonic generation induced by electric currents in GaAs,” Phys. Rev. Lett. 108, 077403 (2012). [CrossRef] [PubMed]

]. However, this effect is quite small. For instance, a laser pulse with the intensity of 1010 W/cm2 only can generate 103 harmonic photons with the presence of 1 A/cm current density. Therefore, the current-induced SHG may be suppressed by the surface SHG and bulk SHG with small current density.

To achieve high time resolution, ultrafast optical measurement techniques must be used. But the traditional absorption spectrum can not be used to detect the charge current with uniform spatial charge distributions because the optical transitions are the same at k and −k. Therefore, an asymmetric optical transition process in k space is essential. The optical quantum-interference process QUIC between one- and two-photon absorption is such a process, which has been used to inject charge current and spin current [17

17. J. B. Khurgin, “Generation of the theraherz radiation using χ(3) in semiconductor,” J. Nonlinear Opt. Phys. Mater. 4, 163–189 (1995). [CrossRef]

23

23. M. Sheik-Bahae, “Quantum interference control of current in semiconductors: universal scaling and polarization effects,” Phys. Rev. B 60, R11257–R11260 (1999). [CrossRef]

]. And as a detection technique, the Faraday rotation in QUIC has also been proposed to detect the amplitude and direction of the spin current [24

24. J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B 78, 113304 (2008). [CrossRef]

]. Based on these studies, in this paper we suggest a different scheme by utilizing a momentum-resolved two-color optical coherence absorption spectrum (QUIC-AB) for the direct detection of charge current. Our proposed current-detection method is based on the principle of two-color quantum interference that create an asymmetric optical transition process in k space, which is able to offer the information of amplitude and distribution of charge current in real time.

2. Theory

In the two-color quantum interference control (QUIC), the optical transition rate depends strongly on the direction of the wave vector k; e.g., the transition rate of QUIC can be enhanced at +k but disappears at −k. Thus, if the band states near +k are already filled with electrons [see Fig. 1(a)], the optical absorption of QUIC is reduced due to the state filling effects. However, if the electrons are located at −k, these electrons have no effect on the QUIC-AB because the optical transition at −k is forbidden.

Fig. 1 (a) Schematic illustration of the transition of two parallel-linearly polarized beams in semiconductor materials. (b) Illustration of the detection of the angular distribution of pure charge current by using two-color optical coherence absorption spectrum.

To reveal the relation between the electron distribution function and the QUIC-AB, we calculate the transition rate of QUIC using the Volkow-type wave function [22

22. J. T. Liu, F. H. Su, and H. Wang, “Model of the optical Stark effect in semiconductor quantum wells: evidence for asymmetric dressed exciton bands,” Phys. Rev. B 80, 113302 (2009). [CrossRef]

24

24. J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B 78, 113304 (2008). [CrossRef]

]. In optical QUIC, the total vector potential of ω and 2ω laser pulse is given by A = a1A1 cos(ωt +φ1) + a2A2 cos(2ωt + φ2). The transition rate of QUIC can be written as
W(k)=2π(e2h¯mc)2δ[ωcv(k)2ω]×(Wkn+Wki)[fv(k)fc(k)],
(1)
where
Wkn=(ka1)2|pvca1|2η12A12+|pvca2|2A22,Wki=ka1η1A1A2[(pvca1)*(pvca2)eiΔφ+c.c.],
where m is the electron mass, e the electron charge magnitude, c the speed of light in vacuum, ωcv the optical transition frequency, fv(k) (fc(k)) the distribution function of the valence (conduction) electrons, η1=eA12ωcmcv, 1/mcv = 1/mc − 1/mv, mc (mv) the effective mass of electrons (holes), pvc = 〈c|p|v〉 the dipole transition matrix element, and Δφ = 2φ1φ2 the relative phase of the ω and 2ω laser beams. Wkn and Wki describe the single-photon and two-photon transition and the quantum interference between the two-color photon excitations. Thus, the optical absorption coefficient is given by α(ω)=h¯ωW(ω)u(c/n0), where u=n02A22ω22πc2 is the electromagnetic energy density, and n0 the refractive index of the semiconductors.

From Eq. (1), only the interference term Wki depends on both the direction and the magnitude of electron vector k. In the current paper, we focus on the detection of pure electron current. Thus, two parallel linearly polarized probe laser beams (i.e., a1||a2) are used [see Fig. 1(b)]. In this case, the interference term Wki reaches the maximum for Δφ = 0° but disappears for Δφ = 90°. The normal single-photon and two-photon transition term Wkn is invariant for different Δφ. If we measure the difference in absorption between Δφ = 0° and Δφ = 90°, the normal single-photon and two-photon transition term Wkn can be eliminated. Thus we can define the absorption coefficient of QUIC-AB as
αQUIC(ω)=αΔφ=0(ω)αΔφ=90°(ω).
(2)
Compared with the absorption of the fundamental radiation of the ω and 2ω light beams, using this definition, the background induced by other absorption process (e.g., the intraband absorption, the heating effect, and the electro-optic effect) can also be eliminated, and therefore the precision of measurement can be improved. It worth noted that a constant electric field may change the QUIC absorption in a semiconductor due to the nonlinear optical Franz-Keldysh effect [25

25. J. K. Wahlstrand, H. Zhang, S. B. Choi, S. Kannan, D. S. Dessau, J. E. Sipe, and S. T. Cundiff, “Optical coherent control induced by an electric field in a semiconductor: a new manifestation of the Franz–Keldysh effect,” Phys. Rev. Lett. 106, 247404 (2011). [CrossRef] [PubMed]

]. But this effect is quit small, e.g., the amplitude of the differential transmission is smaller than 10−6 even with a 100 V/cm electric field.

Using Eq. (1), the absorption coefficient of QUIC-AB in the presence of a charge current can be written as
αQUIC(ω)=π2e2h¯cn0m2ωA2η1A12ζh,lP2×k[fv(k)fc(k)]δ[ωcv(k)2ω]kcosθ,
(3)
where P2 = |〈S|Px|X|〉|2, |S〉 and |X〉 are the Kohn-Luttinger amplitudes, and θ denotes the angle between the electron wave vector k and the polarization direction of laser beam al, ζh,l is a coefficient dependent on the mixing of the light- and heavy- hole states. For bulk GaAs, the hole states can be obtained from the multiband Luttinger-Kohn (LK) effective-mass Hamiltonian [24

24. J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B 78, 113304 (2008). [CrossRef]

, 26

26. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955). [CrossRef]

, 27

27. J. M. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: general theory,” Phys. Rev. 102, 1030–1041 (1956). [CrossRef]

]. Thus we can find that ζh=1ch2[(Rh+E+)2(|M|2+|L|2)/3] for the transition from the heavy valence band to the conduction band, and ζl=1ch2[|M|2+|L|2+(RhE)2/3] for the transition from the light valence band to the conduction band [24

24. J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B 78, 113304 (2008). [CrossRef]

], where E±=±|M|2+|L|2+Rh2 is the energy difference between the hh and lh bands, Rh = −γ2(sin2 θe − 2 cos2 θe), M=3γ2sin2θee2iφe, L=i23γ3sinθecosθeeiφe, γ2 and γ3 are the Luttinger parameters, θe is the polar angle of wave vector k,

Thus, the difference in absorption between that with and that without an electron current can be written as (ω,Δφ)=eα(ω,Δφ)Leα0(ω,Δφ)Leα0(ω,Δφ)L[α0(ω,Δφ)α(ω,Δφ)]L, where L is the thickness of the sample, and α(ω, Δφ) [α0 (ω, Δφ)] is the corresponding absorption coefficient with [without] a electron current. Thus, the differential absorptivity of QUIC-AB is
Δ𝒜QUIC(ω)=Δφ=0(ω)Δφ=90°(ω)[αQUIC0(ω)αQUIC(ω)]L,
(4)
where αQUIC(αQUIC0) is the corresponding QUIC-AB absorption coefficient with (without) a electron current.

When valence states are fully occupied or the holes in the valence-band follow a uniform angular distribution in k-space (i.e., fv(k) ≡ fv(k)), from Eq. (3) we can get αQUIC0αQUIC12k[fc(k)fc(k)]δωcv,2ωkcosθ, which is linearly proportional to the current density along the optical pulse polarization direction Jeal=ene(k)ve(k)alh¯emek[fc(k)fc(k)]kcosθ. Therefore, using the QUIC-AB, the direction and amplitude of the electron current can be detected directly.

3. Numerical results

A numerical simulation is shown in Fig. 2. In this example, an electric field is applied in the plane of the n-doped GaAs sheet and along the −x direction. Thus, the distribution of the electrons can be described by f(k) = fe(kke) [15

15. J. B. Khurgin, “Current induced second harmonic generation in semiconductors,” Appl. Phys. Lett. 67, 1113–1115 (1995). [CrossRef]

,16

16. B. A. Ruzicka, L. K. Werake, G. W Xu, J. B. Khurgin, E. Ya. Sherman, J. Z. Wu, and H. Zhao, “Second-harmonic generation induced by electric currents in GaAs,” Phys. Rev. Lett. 108, 077403 (2012). [CrossRef] [PubMed]

,28

28. J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1979), 216–219.

], where fe(k) = {exp[(EkEF)/kbT]+ 1}−1 is the equilibrium Fermi function, EF the Fermi energy, ke = −eτED/h̄ is the electric field induced shifts in k space in the relaxation approximation, ED and τ is the applied electric field strength and the electron relaxation time, respectively. Thus the whole Fermi surface has been shifted in the +x direction, the occupation number of electrons at +k is larger than that at −k [see Fig. 2(a)], and a DC current is flowing in the −x direction. The corresponding QUIC-AB is shown in Fig. 2(b) for a 1 μm GaAs sheet with EF = 50 meV, τe = 0.1 ps, ED = 5 V/cm, and T = 77 K. Since the electron occupation number at +kx is larger than that at −kx, the absorption of probe laser beam decreases when the polarization of ω and 2ω optical pulse is in the x direction with φ1 = φ2 = 0 and Δφ = 0 (i.e., θal = 0°), a negative QUIC-AB is achieved. But when the polarization of ω and 2ω optical pulse is in the x direction with φ1 = φ2 = π and Δφ = π (i.e., θal = 180°), the absorption of probe laser beam increased because of the low electron occupation number at −kx, and the QUIC-AB is positive. When the sheet current density is about 1 A/cm, the correspond differential absorption is Δ𝒜QUIC/Jex1.4×103. Since the resolution of the differential transmission ΔT/T in the experiment can be high as 10−6 by the phase control technique [21

21. L. K. Werake, B. A. Ruzicka, and H. Zhao, “Observation of intrinsic inverse spin Hall effect,” Phys. Rev. Lett. 106, 107205 (2011). [CrossRef] [PubMed]

], the QUIC-AB can resolve the current difference smaller than 0.001 A/cm. From Fig. 2(b) we can also find that absorption peak appears at E2ω = 1562 meV, which is slight lager than the Femi energy EF at 77 K, and the full width at half maximum (FWHM) of the absorption peak is about 35 meV. When the tuning E2ωEF is about 30 meV, the correspond differential absorption Δ𝒜QUIC/Jex2×104.

Fig. 2 (a) Polar contour plot of the electron distribution function fe(k,θ) − fe(k,θ + π) under an electric field. (b) Polar contour plot of QUIC-AB Δ𝒜QUIC/Jex×106 as a function of the energy E2ω and the angle θal between the electron wave vector and the polarization direction of laser beam.

We now turn to the discussions for the influence on the QUIC-AB from the temperature, applied electric field strength, and relaxation time τe as shown in Fig. 3. In k-space, the electrons forming the current are located at the Fermi energy surface. Therefore, the energy spreading of the inject current is about 2kBT, where kB is the Boltzmann constant. Thus, the energy dispersion of the inject current in k-space will increase linearly with the temperature. But the optical absorption depends both on the optical transition line width and the energy of the electrons. If the optical transition energy of the corresponding electron current is far from the center energy of laser beam, these electrons have little contribution to the QUIC-AB. The current resolution of QUIC-AB will decrease with temperature increase [see Fig. 1(a)]. The applied electric field and relaxation time τe have small effect on the current resolution of QUIC-AB because electric field induced shifts ke is usually quite small. Only with an strong applied electric field and slow relaxation, the the current resolution of QUIC-AB decreases rapidly [see Fig. 3(b)].

Fig. 3 (a) The maximum QUIC-AB Δ𝒜QUIC/Jex×106 as a function of temperature. (b) The maximum QUIC-AB Δ𝒜QUIC/Jex×106 as a function of electron relaxation time τe with different applied electric field, E = 5 V/cm (black solid line), E = 10 V/cm (red dashed line), E = 30 V/cm, (blue dotted line).

Finally, we discuss the experimental realization of our theory predication. Benefitted from ultrafast laser technology, the time resolution of the pump-probe detection can be down to the femto-second. However, the pump and probe pulse length should not be too short. A short pulse with a too large energy range will lower the energy resolution and reduce the interference term Wki. As the probe pulse also injects a pure charge current, a relatively low energy density 2ω probe pulse must be chosen. In the detection of QUIC-AB, there is no spin or charge accumulation in real space. Thus, laser beams need not be focused, and the disturbance of edge states can be avoided.

4. Conclusion

In conclusion, the momentum-resolved and time-resolved QUIC-AB is investigated. The QUIC-AB can be used to detect both the amplitude and the direction of the charge current in real time. The QUIC-AB may play an important role in investigating the ultrafast transport process or ultrafast electronics and spintronics.

Acknowledgment

We would like to thank Hui Zhao for fruitful discussion. This work was supported by the NSFC Grant Nos. 10904059, 10904097, and 11004199.

References and links

1.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

2.

Y. M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.Y. Chiu, A. Grill, and P. Avouris, “100-GHz transistors from wafer-scale epitaxial graphene,” Science 327, 662–662 (2010). [CrossRef] [PubMed]

3.

T. Hyart, N. V. Alexeeva, J. Mattas, and K. N. Alekseev, “Terahertz bloch oscillator with a modulated bias,” Phys. Rev. Lett. 102, 140405 (2009). [CrossRef] [PubMed]

4.

J. Rammer, “Quantum transport theory of electrons in solids: a single-particle approach,” Rev. Mod. Phys. 63, 781–817 (1991), and references therein. [CrossRef]

5.

M. I. Dyakonov and V. I. Perel, “Current-induced spin orientation of electrons in semiconductors,” Phys. Lett. A 35, 459–460 (1971). [CrossRef]

6.

J. E. Hirsch, “Spin Hall effect,” Phys. Rev. Lett. 83, 1834–1837 (1999). [CrossRef]

7.

S. Zhang, “Spin Hall effect in the presence of spin diffusion,” Phys. Rev. Lett. 85, 393–396 (2000). [CrossRef] [PubMed]

8.

W. Yang, K. Chang, and S. C. Zhang, “Intrinsic spin Hall effect induced by quantum phase transition in HgCdTe quantumwells,” Phys. Rev. Lett. 100, 056602 (2008). [CrossRef] [PubMed]

9.

J. Yan and M. S. Fuhrer, “Correlated charged impurity scattering in graphene,” Phys. Rev. Lett. 107, 206601 (2011). [CrossRef] [PubMed]

10.

J. H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams, and M. Ishigami, “Charged-impurity scattering in graphene,” Nature Phys. 4, 377–381 (2008). [CrossRef]

11.

J. Smit, “The spontaneous Hall effect in ferromagnetics II,” Physica 24, 39–51 (1958). [CrossRef]

12.

S. Hüfner, Photoelectron Spectroscopy, (Springer-Verlag, 1995).

13.

A. Damascelli, Z. Hussain, and Z. Shen, “Angle-resolved photoemission studies of the cuprate superconductors,” Rev. Mod. Phys. 75, 473–541 (2003). [CrossRef]

14.

J. Güde, M. Rohleder, T. Meier, S. W. Koch, and U. Höer, “Time-resolved investigation of coherently controlled electric currents at a metal surface,” Science 318, 1287–1291 (2007). [CrossRef]

15.

J. B. Khurgin, “Current induced second harmonic generation in semiconductors,” Appl. Phys. Lett. 67, 1113–1115 (1995). [CrossRef]

16.

B. A. Ruzicka, L. K. Werake, G. W Xu, J. B. Khurgin, E. Ya. Sherman, J. Z. Wu, and H. Zhao, “Second-harmonic generation induced by electric currents in GaAs,” Phys. Rev. Lett. 108, 077403 (2012). [CrossRef] [PubMed]

17.

J. B. Khurgin, “Generation of the theraherz radiation using χ(3) in semiconductor,” J. Nonlinear Opt. Phys. Mater. 4, 163–189 (1995). [CrossRef]

18.

R. D. R. Bhat and J. E. Sipe, “Optically injected spin currents in semiconductors,” Phys. Rev. Lett. 85, 5432–5435 (2000). [CrossRef]

19.

M. J. Stevens, A. L. Smirl, R. D. R. Bhat, Ali Najmaie, J. E. Sipe, and H. M. van Driel, “Quantum interference control of ballistic pure spin currents in semiconductors,” Phys. Rev. Lett. 90, 136603 (2003). [CrossRef] [PubMed]

20.

J. Hübner, W. W. Rühle, M. Klude, D. Hommel, R. D. R. Bhat, J. E. Sipe, and H. M. van Driel, “Direct observation of optically injected spin-polarized currents in semiconductors,” Phys. Rev. Lett. 90, 216601 (2003). [CrossRef] [PubMed]

21.

L. K. Werake, B. A. Ruzicka, and H. Zhao, “Observation of intrinsic inverse spin Hall effect,” Phys. Rev. Lett. 106, 107205 (2011). [CrossRef] [PubMed]

22.

J. T. Liu, F. H. Su, and H. Wang, “Model of the optical Stark effect in semiconductor quantum wells: evidence for asymmetric dressed exciton bands,” Phys. Rev. B 80, 113302 (2009). [CrossRef]

23.

M. Sheik-Bahae, “Quantum interference control of current in semiconductors: universal scaling and polarization effects,” Phys. Rev. B 60, R11257–R11260 (1999). [CrossRef]

24.

J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B 78, 113304 (2008). [CrossRef]

25.

J. K. Wahlstrand, H. Zhang, S. B. Choi, S. Kannan, D. S. Dessau, J. E. Sipe, and S. T. Cundiff, “Optical coherent control induced by an electric field in a semiconductor: a new manifestation of the Franz–Keldysh effect,” Phys. Rev. Lett. 106, 247404 (2011). [CrossRef] [PubMed]

26.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955). [CrossRef]

27.

J. M. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: general theory,” Phys. Rev. 102, 1030–1041 (1956). [CrossRef]

28.

J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1979), 216–219.

OCIS Codes
(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter
(300.6470) Spectroscopy : Spectroscopy, semiconductors
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 11, 2012
Revised Manuscript: May 3, 2012
Manuscript Accepted: May 3, 2012
Published: May 8, 2012

Citation
Jiang-Tao Liu, Fu-Hai Su, Xin-Hua Deng, and Hai Wang, "Proposal for the momentum-resolved and time-resolved optical measurement of the current distribution in semiconductors," Opt. Express 20, 11694-11699 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11694


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References

  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004). [CrossRef] [PubMed]
  2. Y. M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.Y. Chiu, A. Grill, and P. Avouris, “100-GHz transistors from wafer-scale epitaxial graphene,” Science327, 662–662 (2010). [CrossRef] [PubMed]
  3. T. Hyart, N. V. Alexeeva, J. Mattas, and K. N. Alekseev, “Terahertz bloch oscillator with a modulated bias,” Phys. Rev. Lett.102, 140405 (2009). [CrossRef] [PubMed]
  4. J. Rammer, “Quantum transport theory of electrons in solids: a single-particle approach,” Rev. Mod. Phys.63, 781–817 (1991), and references therein. [CrossRef]
  5. M. I. Dyakonov and V. I. Perel, “Current-induced spin orientation of electrons in semiconductors,” Phys. Lett. A35, 459–460 (1971). [CrossRef]
  6. J. E. Hirsch, “Spin Hall effect,” Phys. Rev. Lett.83, 1834–1837 (1999). [CrossRef]
  7. S. Zhang, “Spin Hall effect in the presence of spin diffusion,” Phys. Rev. Lett.85, 393–396 (2000). [CrossRef] [PubMed]
  8. W. Yang, K. Chang, and S. C. Zhang, “Intrinsic spin Hall effect induced by quantum phase transition in HgCdTe quantumwells,” Phys. Rev. Lett.100, 056602 (2008). [CrossRef] [PubMed]
  9. J. Yan and M. S. Fuhrer, “Correlated charged impurity scattering in graphene,” Phys. Rev. Lett.107, 206601 (2011). [CrossRef] [PubMed]
  10. J. H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams, and M. Ishigami, “Charged-impurity scattering in graphene,” Nature Phys.4, 377–381 (2008). [CrossRef]
  11. J. Smit, “The spontaneous Hall effect in ferromagnetics II,” Physica24, 39–51 (1958). [CrossRef]
  12. S. Hüfner, Photoelectron Spectroscopy, (Springer-Verlag, 1995).
  13. A. Damascelli, Z. Hussain, and Z. Shen, “Angle-resolved photoemission studies of the cuprate superconductors,” Rev. Mod. Phys.75, 473–541 (2003). [CrossRef]
  14. J. Güde, M. Rohleder, T. Meier, S. W. Koch, and U. Höer, “Time-resolved investigation of coherently controlled electric currents at a metal surface,” Science318, 1287–1291 (2007). [CrossRef]
  15. J. B. Khurgin, “Current induced second harmonic generation in semiconductors,” Appl. Phys. Lett.67, 1113–1115 (1995). [CrossRef]
  16. B. A. Ruzicka, L. K. Werake, G. W Xu, J. B. Khurgin, E. Ya. Sherman, J. Z. Wu, and H. Zhao, “Second-harmonic generation induced by electric currents in GaAs,” Phys. Rev. Lett.108, 077403 (2012). [CrossRef] [PubMed]
  17. J. B. Khurgin, “Generation of the theraherz radiation using χ(3) in semiconductor,” J. Nonlinear Opt. Phys. Mater.4, 163–189 (1995). [CrossRef]
  18. R. D. R. Bhat and J. E. Sipe, “Optically injected spin currents in semiconductors,” Phys. Rev. Lett.85, 5432–5435 (2000). [CrossRef]
  19. M. J. Stevens, A. L. Smirl, R. D. R. Bhat, Ali Najmaie, J. E. Sipe, and H. M. van Driel, “Quantum interference control of ballistic pure spin currents in semiconductors,” Phys. Rev. Lett.90, 136603 (2003). [CrossRef] [PubMed]
  20. J. Hübner, W. W. Rühle, M. Klude, D. Hommel, R. D. R. Bhat, J. E. Sipe, and H. M. van Driel, “Direct observation of optically injected spin-polarized currents in semiconductors,” Phys. Rev. Lett.90, 216601 (2003). [CrossRef] [PubMed]
  21. L. K. Werake, B. A. Ruzicka, and H. Zhao, “Observation of intrinsic inverse spin Hall effect,” Phys. Rev. Lett.106, 107205 (2011). [CrossRef] [PubMed]
  22. J. T. Liu, F. H. Su, and H. Wang, “Model of the optical Stark effect in semiconductor quantum wells: evidence for asymmetric dressed exciton bands,” Phys. Rev. B80, 113302 (2009). [CrossRef]
  23. M. Sheik-Bahae, “Quantum interference control of current in semiconductors: universal scaling and polarization effects,” Phys. Rev. B60, R11257–R11260 (1999). [CrossRef]
  24. J. T. Liu and K. Chang, “Proposal for the direct optical detection of pure spin currents in semiconductors,” Phys. Rev. B78, 113304 (2008). [CrossRef]
  25. J. K. Wahlstrand, H. Zhang, S. B. Choi, S. Kannan, D. S. Dessau, J. E. Sipe, and S. T. Cundiff, “Optical coherent control induced by an electric field in a semiconductor: a new manifestation of the Franz–Keldysh effect,” Phys. Rev. Lett.106, 247404 (2011). [CrossRef] [PubMed]
  26. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev.97, 869–883 (1955). [CrossRef]
  27. J. M. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: general theory,” Phys. Rev.102, 1030–1041 (1956). [CrossRef]
  28. J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1979), 216–219.

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