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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 22563–22574
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Electric field-induced coherent control in GaAs: polarization dependence and electrical measurement [Invited]

J. K. Wahlstrand, H. Zhang, S. B. Choi, J. E. Sipe, and S. T. Cundiff  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 22563-22574 (2011)
http://dx.doi.org/10.1364/OE.19.022563


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Abstract

A static electric field enables coherent control of the photoexcited carrier density in a semiconductor through the interference of one- and two-photon absorption. An experiment using optical detection is described. The polarization dependence of the signal is consistent with a calculation using a 14-band k · p model for GaAs. We also describe an electrical measurement. A strong enhancement of the phase-dependent photocurrent through a metal-semiconductor-metal structure is observed when a bias of a few volts is applied. The dependence of the signal on bias and laser spot position is studied. The field-induced enhancement of the signal could increase the sensitivity of semiconductor-based carrier-envelope phase detectors, useful in stabilizing mode-locked lasers for use in frequency combs.

© 2011 OSA

1. Introduction

Recently, it was shown that an electric field enables coherent control of the carrier population injection rate via the interference of one- and two-photon pathways [12

12. 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]

], a process that ordinarily requires a medium without a center of inversion symmetry [13

13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

]. This field-induced quantum interference control (QUIC) process can be explained by a theory of the nonlinear optical Franz-Keldysh effect (FKE). Here we present the results of calculations using a 14-band k · p model, which can predict the dependence of the effect on the directions of the optical and dc fields with respect to the crystal orientation. We describe an all-optical study of the dependence of the signal on the polarization of the ω and 2ω beams. We then present results of an electrical measurement of field-induced QUIC, which bears directly on a potential application of the process to the detection of carrier-envelope phase evolution [8

8. T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004). [CrossRef] [PubMed]

]. We find that applying a bias of a few volts across electrodes patterned on a semiconductor surface enhances the sensitivity of the photocurrent to ϕ2ω – 2ϕω by more than a factor of 10.

2. Theory

For an optical field at ω and its second harmonic both incident on a crystal, with 2ω above the band gap, the rate of carrier injection ṅ = ṅ1 + ṅ2 + ṅI, where ṅ1 is the rate due to the absorption of one photon at energy 2h̄ω, ṅ2 is the rate due to the absorption of two photons at h̄ω, and ṅI is the rate due to the interference between one- and two-photon absorption (i.e. the QUIC process). For linearly polarized light, in the limit of long pulses we write the optical field Eωeiωt+ω + E2ωe−2iωt+2ω + c.c., where Eω and E2ω are real vectors with magnitude Eω and E2ω respectively, the interference term n˙I=ηIjkl(ω)E2ωjEωkEωlexp(iϕ2ω2iϕω)+c.c., where ηIjkl(ω) is a tensor that describes the efficiency of the population control process [13

13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

,14

14. R. D. R. Bhat and J. E. Sipe, “Calculations of two-color interband optical injection and control of carrier population, spin, current, and spin current in bulk semiconductors,” arXiv:cond-mat/0601277 (unpublished).

]. Unlike ṅ1 or ṅ2, ṅI depends on the parameter ϕ2ω – 2ϕω involving the relative phases of the two fields. In a crystal with center of inversion symmetry, in the absence of an external field, ηIjkl(ω)=0. If there is no center of inversion symmetry, then in general ηIjkl0 for 2ω above the band gap, and coherent control of population is possible even in the absence of an external field [13

13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

]. We refer to this as the χ(2)-enabled population control process. In GaAs, the only nonzero element is ηIxyz(ω) and all permutations of {x,y,z}, so one must have components of the optical field pointing along all three crystal directions. For this reason, it is forbidden for light normally incident on a (100) GaAs surface; one can have optical field components along ŷ and , but not . The χ(2)-enabled process has been observed in a (111) GaAs sample using optical [13

13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

] and electrical [15

15. J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett. 89, 241115 (2006). [CrossRef]

] detection.

When a DC field is applied, ηIjkl0 even in a crystal with center of inversion symmetry [12

12. 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]

]. We have extended a theoretical framework for one-photon absorption in the presence of a dc field [16

16. J. K. Wahlstrand and J. E. Sipe, “Independent-particle theory of the Franz-Keldysh effect including interband coupling: Application to calculation of electroabsorption in GaAs,” Phys. Rev. B 82, 075206 (2010). [CrossRef]

] to calculate two-photon electroabsorption processes including field-induced QUIC. For a model consisting of two parabolic bands, under the assumption that the valence band to conduction band matrix element for absorption Vcv is constant within the Brillouin zone, one can derive analytical expressions for ηIjkl(ω). We find [12

12. 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]

]
ηIzzz(ω)=e3EDC2|Vcvz|22h¯3ω4Ω(Ai2(2ωωgΩ)2Ω+2ωωgΩAi2(2ωωgΩ)[Ai(2ωωgΩ)]2ω)
where g is the band gap, Ω(e2EDC2/2μh¯)1/3 is the electro-optic frequency, μ is the reduced effective mass, and Ai(x) is the Airy function. The oscillations for 2ω > ωg are strongly damped at room temperature (as in the one-photon Franz-Keldysh effect [17

17. D. E. Aspnes, “Electric field effects on the dielectric constant of solids,” Phys. Rev. 153, 972–982 (1967). [CrossRef]

]). In the limit of strong damping, the fractional change in carrier injection rate due to QUIC is
n˙In˙=6e2EDCEω2E2ωω(4ωg7ω)16e2Eω4(2ωωg)2+3E2ω2μh¯ω4(2ωωg).
for 2ω > ωg. In this large-broadening limit, the effect is linear in the dc field EDC.

Fig. 1 Calculation of the spectral dependence of population QUIC using a 14-band model for EDC = 44 kV/cm. The arrows show the position of the half band gap h̄ωg/2 and the half split-off gap h̄ωSO/2. (a) Results for EDC along [001]. Black: Eω, E2ω along [001]; Red: Eω along [010], E2ω along [001]. (b) Results for EDC along [011]. Black: Eω, E2ω along [011]; Red: Eω along [01̄1], E2ω along [011].

Another process that causes a change in the rate of carrier injection that depends on ϕ2ω – 2ϕω is cascaded second harmonic generation [20

20. M. J. Stevens, R. D. R. Bhat, X. Y. Pan, H. M. van Driel, J. E. Sipe, and A. L. Smirl, “Enhanced coherent control of carrier and spin density in a zinc-blende semiconductor by cascaded second-harmonic generation,” J. Appl. Phys. 97, 093709 (2005). [CrossRef]

]. This is a two-step process in which second harmonic generation of ω generates a field at 2ω which then interferes optically with the light at 2ω. Like population control, this leads to a change in the injected carrier density that depends on ϕ2ω – 2ϕω, and thus would produce an identical signal in the experiment. In this case the second harmonic generation would be electric field-induced [21

21. C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967). [CrossRef]

]. In a bulk semiconductor like the one studied here, we expect that the direct process previously described in terms of the multi-photon Franz-Keldysh effect [12

12. 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]

] is dominant. In future experiments, it could be distinguished from the direct coherent control processes by using a layered structure [20

20. M. J. Stevens, R. D. R. Bhat, X. Y. Pan, H. M. van Driel, J. E. Sipe, and A. L. Smirl, “Enhanced coherent control of carrier and spin density in a zinc-blende semiconductor by cascaded second-harmonic generation,” J. Appl. Phys. 97, 093709 (2005). [CrossRef]

].

3. All-optical experiment

3.1. Technique

A detailed description of the experiment used to observe field-induced QUIC was given in [12

12. 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]

]. An optical parametric oscillator (OPO), pumped by a Ti:sapphire laser centered at 830 nm, produces pulses centered at 1550 nm. The residual light from the pump, after spatial filtering, is used as the probe beam. The sample is a 1 μm thick undoped GaAs epilayer grown by molecular beam epitaxy. After removal of the substrate, a thin insulating SiO2 layer was deposited on the surface, and Au electrodes were patterned on the surface of the insulating layer using photolithography. To avoid DC field nonuniformity caused by injection of carriers from the electrodes into the sample [25

25. S. E. Ralph and D. Grischkowsky, “Trap-enhanced electric fields in semi-insulators: the role of electrical and optical carrier injection,” Appl. Phys. Lett. 59, 1972–1974 (1991). [CrossRef]

] (discussed in more detail in Section 4.4.2), an effective DC field was applied using a radio frequency bias synchronized to the laser repetition rate [26

26. J. K. Wahlstrand, H. Zhang, and S. T. Cundiff, “Uniform-field transverse electroreflectance using a mode-locked laser and a radio-frequency bias,” Appl. Phys. Lett. 96, 101104 (2010). [CrossRef]

]. The characteristic feature of the two-color QUIC signal is its dependence on the phase parameter ϕ21 = ϕ2ω – 2ϕω. To detect only QUIC, we use a two-color interferometer, modulate ϕ21, and measure the modulation of the probe transmission using a lock-in amplifier. The second harmonic is generated with a β-barium borate (BBO) crystal. A two-color interferometer using a prism to separate the harmonics [27

27. R. L. Snider, J. K. Wahlstrand, H. Zhang, R. P. Mirin, and S. T. Cundiff, “Quantum interference control of photocurrent injection in Er-doped GaAs,” Appl. Phys. B 98, 333–336 (2010). [CrossRef]

] is used to control ϕ21.

3.2. Results

Experimental results are shown in Fig. 2. Typical experimental traces are shown in Fig. 2a for no applied bias and for the maximum achievable positive and negative bias. A negative effective bias is created by changing the phase of the radio frequency bias by π. The signal scales as P2ω1/2Pω as predicted by the theory, where Pω (P2ω) is the power of the ω (2ω) beam. The maximum observed modulation of the carrier density was 2%, which is within an order of magnitude of the theoretical prediction. Multiple reflection and phase walkoff inside the sample make quantitative comparison of the experimental results and theoretical calculations challenging.

Fig. 2 Field-induced QUIC measured using an optical probe. (a) Data traces showing the differential transmission as a function of ϕ2ω – 2ϕω for zero, negative, and positive external bias. (b) Polarization dependence of the signal for EDC || [110]: E2ω, Eω || EDC (black); E2ωEDC, Eω || EDC (blue); EωEDC, E2ω || EDC (red); and E2ω, Eω || EDC (magenta).

Waveplates in the two-color interferometer allow the polarization of the ω and 2ω beams to be varied. Traces for four polarization configurations and the DC field along [110] are shown in Fig. 2b. The signal is largest for all fields parallel and falls by more than a factor of five when any of the polarizations is rotated by 90°. This enhancement is consistent, within the noise, with the calculation shown in Fig. 1b, which predicts that the signal should be smaller by a large factor for EωEDC and zero when E2ωEDC. Polarization impurity causes a small component of the beam to be polarized parallel to the DC field, and we attribute the non-zero signal for E2ωEDC seen in Fig. 2b to that experimental imperfection, which is a particular problem for the 2ω beam because of the P2ω1/2 dependence. The polarization dependence for the DC field along [100] (not shown) is noisier but essentially the same as the data for the DC field along [110] in Fig. 2b. We have also observed a DC field induced modulation of the transmitted 2ω beam that depends on ϕ21, which was observed in the analogous process enabled by χ(2) [13

13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

].

4. Electrical detection

In the electrical measurement scheme, we detect the change in the photocurrent read out by the same electrodes used to apply the bias to the sample. The signal is a combination of two processes: QUIC current injection [3

3. A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, “Observation of coherently controlled photocurrent in unbiased, bulk GaAs,” Phys. Rev. Lett. 78, 306–309 (1997). [CrossRef]

], which ballistically injects a photocurrent that depends on ϕ2ω – 2ϕω, and field-induced modulation of the carrier injection rate [12

12. 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]

], which modulates the carrier density and thus the photocurrent through the biased structure. For electrical detection, we use a low-temperature (LT) grown GaAs sample, which has a subpicosecond carrier lifetime [28

28. S. Gupta, M. Y. Frankel, J. A. Valdmanis, J. F. Whitaker, G. A. Mourou, F. W. Smith, and A. R. Calawa, “Subpi-cosecond carrier lifetime in GaAs grown by molecular beam epitaxy at low temperatures,” Appl. Phys. Lett. 59, 3276–3278 (1991). [CrossRef]

] and a high breakdown threshold [29

29. M. Ruff, D. Streb, S. U. Dankowski, S. Tautz, P. Kiesel, B. Knupfer, M. Kneissl, N. Linder, G. H. Dohler, and U. D. Keil, “Polarization dependence of the electroabsorption in low-temperature grown GaAs for above band-gap energies,” Appl. Phys. Lett. 68, 2968–2970 (1996). [CrossRef]

]. Because of the short lifetime, the resistance of the sample stays high even under intense illumination, resulting in a larger signal in QUIC current injection experiments [30

30. A. Haché, J. Sipe, and H. van Driel, “Quantum interference control of electrical currents in GaAs,” IEEE J. Quantum Electron. 34, 1144–1154 (1998). [CrossRef]

]. We have also used Er-doped GaAs [31

31. S. Gupta, S. Sethi, and P. K. Bhattacharya, “Picosecond carrier lifetime in erbium-doped-GaAs,” Appl. Phys. Lett. 62, 1128–1130 (1993). [CrossRef]

] – a material that has an ultrashort carrier lifetime like LT-GaAs, and produces a large QUIC signal [27

27. R. L. Snider, J. K. Wahlstrand, H. Zhang, R. P. Mirin, and S. T. Cundiff, “Quantum interference control of photocurrent injection in Er-doped GaAs,” Appl. Phys. B 98, 333–336 (2010). [CrossRef]

]. We do not show data using GaAs:Er here because it is essentially identical to the LT-GaAs data.

4.1. Technique

Reading out the signal using electrodes has been used previously in QUIC current injection [3

3. A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, “Observation of coherently controlled photocurrent in unbiased, bulk GaAs,” Phys. Rev. Lett. 78, 306–309 (1997). [CrossRef]

, 8

8. T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004). [CrossRef] [PubMed]

, 27

27. R. L. Snider, J. K. Wahlstrand, H. Zhang, R. P. Mirin, and S. T. Cundiff, “Quantum interference control of photocurrent injection in Er-doped GaAs,” Appl. Phys. B 98, 333–336 (2010). [CrossRef]

, 30

30. A. Haché, J. Sipe, and H. van Driel, “Quantum interference control of electrical currents in GaAs,” IEEE J. Quantum Electron. 34, 1144–1154 (1998). [CrossRef]

, 32

32. N. Laman, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference control of currents in CdSe with a single optical beam,” Appl. Phys. Lett. 75, 2581–2583 (1999). [CrossRef]

] and population control experiments [15

15. J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett. 89, 241115 (2006). [CrossRef]

]. The average current injected by QUIC in the absence of a field are < 500 pA. We use a circuit with separate transimpedence amplifiers for low frequency and high frequency components. This design keeps the DC bias across the sample constant while greatly amplifying signals at high frequencies. The gain of the DC transimpedence amplifier was 1 kΩ and the gain of the AC transimpedence amplifier is 1 MΩ, with a low frequency cutoff frequency of 50 Hz.

A mode-locked Er-doped fiber laser produces 150 fs pulses at 1550 nm at a repetition rate of 26 MHz. The output is stretched using dispersion compensating fiber, amplified, and then compressed using a grating compressor. The average power at the output of the compressor is approximately 80 mW. The light is doubled and the harmonics split using a prism-based interferometer. Unlike in the all-optical experiment, where a mirror mounted on a piezoelectric transducer is used to modulate ϕ21, here an acousto-optic modulator (AOM) provides the phase modulation. Inside the interferometer, the 775 nm light (the 2ω beam) is frequency shifted using an AOM by fAOM ≈ 78 MHz, and then recombined with the ω beam. The frequency shift of the 2ω beam is equivalent to a linear phase ramp. The phase parameter ϕ21 is typically modulated at approximately 20 kHz. The amplified signal is measured using a lock-in amplifier referenced to fdet = 3frepfAOM. The reference signal is generated using an RF phase detector.

The recombined ω and 2ω beams are focused on the sample surface using a microscope objective. A glass slide is placed in the beam to sample some of the light that reflects from the sample, allowing imaging of the sample surface and precise placement of the laser spot. At the sample, the maximum power available is typically 30 mW at 1550 nm and 400 μW at 775 nm. In all of the data presented using electrical detection, the optical fields are linearly polarized along the DC field direction.

4.2. Bias dependence

The bias dependence of the QUIC signal is shown in Fig. 3. Unlike in the all-optical measurement, shown in Fig. 3a, the bias dependence in the electrical measurement, shown for a 10 μm electrode spacing in Fig. 3b, is not monotonic. As the magnitude of the bias is increased from zero, the signal decreases to nearly zero first, then increases rapidly. The phase of the electrically detected signal, shown as a red dashed line in Fig. 3b, reverses at the same time that the signal dips to near zero. The signal is greatly enhanced at a bias of ±2 V. The maximum fractional modulation of the photocurrent by QUIC is 5 × 10−3, about a quarter as large as the signal observed in the all-optical experiment [12

12. 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]

], where an electric field of approximately 20 kV/cm was present. This is consistent with the electric field calculated from the voltage and electrode spacing of 10 μm.

Fig. 3 Bias dependence of field-induced QUIC. (a) All-optical measurement, showing a linear dependence of the signal on bias for the DC field along two directions with respect to the crystal structure: EDC || [001] (blue) and EDC || [011] (red). (b) Electrical measurement, showing non-monotonic dependence of the signal magnitude (black line). The phase of the signal (dashed red line) flips by nearly π when the magnitude dips near zero.

One is tempted to interpret the signal as having two components which are nearly π out of phase: a bias-independent signal, and a signal that depends on the magnitude of the bias voltage, but not its sign. As argued later, current injection via QUIC [3

3. A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, “Observation of coherently controlled photocurrent in unbiased, bulk GaAs,” Phys. Rev. Lett. 78, 306–309 (1997). [CrossRef]

] is essentially bias-independent. The bias-dependent component’s even symmetry is consistent with field-induced QUIC. While the field-induced QUIC signal at room temperature is linear in the electric field and thus the bias, the direction of the photocurrent is the same sign as the bias, resulting in an overall dependence that is even. In an electrical measurement of population control enabled by χ(2) [15

15. J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett. 89, 241115 (2006). [CrossRef]

], the signal was found to be odd in the bias voltage, which is consistent with a coherent control process that is independent of bias coupled with a detection scheme that is odd in the bias. However, the relative phase dependence of current injection and population control has been calculated to be π/2 except when 2ω is very near the band edge [35

35. R. D. R. Bhat and J. E. Sipe, “Excitonic effects on the two-color coherent control of interband transitions in bulk semiconductors,” Phys. Rev. B 72, 075205 (2005). [CrossRef]

] and this has been verified experimentally in CdTe [36

36. C. Sames, J. Menard, M. Betz, A. L. Smirl, and H. M. van Driel, “All-optical coherently controlled terahertz ac charge currents from excitons in semiconductors,” Phys. Rev. B 79, 045208 (2009). [CrossRef]

]. Like χ(2)-enabled population control, electric field-induced population control has a cos(ϕ2ω – 2ϕω) dependence. For the photon energy used a π/2 phase shift is expected. As discussed in Section 4.4 an interpretion in terms of contributions from current injection and population control is likely oversimplified.

4.3. Dependence on position of laser spot

In the absence of a bias, the QUIC signal is enhanced when the laser spot is near an electrode edge [15

15. J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett. 89, 241115 (2006). [CrossRef]

, 34

34. P. Roos, Q. Quraishi, S. Cundiff, R. Bhat, and J. Sipe, “Characterization of quantum interference control of injected currents in LT-GaAs for carrier-envelope phase measurements,” Opt. Express 11, 2081–2090 (2003). [CrossRef] [PubMed]

]. When a bias is applied, the QUIC signal also depends on the position of the laser spot, as shown in Fig. 4. Figure 4a shows the bias dependence for three different laser spot positions on a sample with 10 μm electrode spacing. This electrode spacing produces the optimal signal-to-noise ratio at zero bias. The spot size in this data is approximately 6 μm. When the laser spot is centered between the electrodes (the data in Fig. 3b and the green curve in Fig. 4a), the bias dependence is symmetric. But when the laser spot is focused on one side (red curve) of the structure or the other (blue curve), the enhancement of the signal is greater for either positive or negative bias.

Fig. 4 Dependence of signal on laser spot position. (a) Bias dependence of QUIC signal for 10 μm electrode spacing. Curves are shown for three different laser spot positions, separated by 2 μm, as shown in the inset. (b) Map of the QUIC signal magnitude as a function of bias and the position of the laser spot for a sample with 150 μm electrode spacing. The edges of the electrodes are at 40 and 190 μm.

The connection between the bias and spatial dependence is somewhat clearer in a sample with wider spaced electrodes. Results are shown for a sample with electrodes spaced by 150 μm in Fig. 4b. The ϕ21-dependent photocurrent is shown as a function of bias voltage and the position of the laser spot. Regardless of the bias, the signal is strongest when the laser spot, which in this case is approximately 12 μm wide, is focused near the electrode edges. Depending on which electrode the laser spot is near, we observe an enhancement of the signal for positive or negative bias. For a bias of the opposite sign, the signal magnitude is independent of bias.

The bias dependence when the laser spot is positioned near an electrode is shown in Fig. 5. Data for two laser spots are shown, separated by 12 μm. For the spot closer to the electrode (solid lines), a dip in the amplitude (black) is observed at −2.5 V, and as the bias is swept through that voltage, the phase (red) changes by approximately 0.63π. The data for the spot farther from the electrode (dashed lines) shows no discernible dip in the amplitude (black), and the phase (red) changes by approximately π/2. The data for the laser spot focused near the other electrode displays the same behavior, but at positive rather than negative bias. For a wide electrode spacing, the observed phase shift is closer to the value of π/2 predicted by the theory [35

35. R. D. R. Bhat and J. E. Sipe, “Excitonic effects on the two-color coherent control of interband transitions in bulk semiconductors,” Phys. Rev. B 72, 075205 (2005). [CrossRef]

] if we identify the bias-independent signal as current injection and the rest as field-induced population control.

Fig. 5 Bias dependence of QUIC signal in a sample with 150 μm electrode spacing, for two laser spots spaced by 12 μm: nearer the electrode (solid) and farther away (dashed). The amplitude (black) and phase (red) of the QUIC signal are shown.

4.4. Discussion

The symmetry of the signal with respect to bias is consistent with a combination of current injection and field-induced population control, but the phase shift observed using a narrow-spaced sample is nearly π, larger than the phase shift of π/2 predicted by the theory of current injection and population control. In this section we discuss effects that might explain this behavior.

4.4.1. Bias dependence of current injection

Any process that depends on ϕ2ω – 2ϕω, affects the photocurrent, and depends on an applied bias could in principle contribute to the signal observed. In the absence of an electric field, a current is injected due to quantum interference between one- and two-photon absorption [2

2. R. Atanasov, A. Haché, J. L. P. Hughes, H. M. van Driel, and J. E. Sipe, “Coherent control of photocurrent generation in bulk semiconductors,” Phys. Rev. Lett. 76, 1703–1706 (1996). [CrossRef] [PubMed]

]. A dependence of the current injection efficiency on EDC would contribute to a change in the current directly injected by this process. In a crystal with center of inversion symmetry, the current injection efficiency cannot change to first order in an applied dc field, but it could change to second-order in the field. From the point of view of nonlinear optics, this field-induced current injection process would be associated with the nonlinear susceptibility χ(5). When the crystal lacks a center of inversion symmetry, there could in principle be a change in the current injection linear in the field that is associated with χ(4), but note that this would be odd in the DC electric field and thus would not explain the experimental results. Because it is a higher-order process than field-induced population control (which is associated with χ(3)), we consider it unlikely to be as large. It is also not likely that the current injection reverses sign in a DC field, as would be required to account for the bias dependence observed in the experiment.

4.4.2. Electric field distribution

When a bias is applied across metal-semiconductor-metal structures such as the one used in the experiment, the electric field distribution is very different from what would be expected for a perfect dielectric. The field is strongly enhanced near the anode because of the interaction between electrically injected carriers and carrier-trapping impurities that are always present in semi-insulating semiconductors [25

25. S. E. Ralph and D. Grischkowsky, “Trap-enhanced electric fields in semi-insulators: the role of electrical and optical carrier injection,” Appl. Phys. Lett. 59, 1972–1974 (1991). [CrossRef]

,37

37. D. S. Kim and D. S. Citrin, “Efficient terahertz generation using trap-enhanced fields in semi-insulating photo-conductors by spatially broadened excitation,” J. Appl. Phys. 101, 053105 (2007). [CrossRef]

]. The field distribution can become even more distorted when a laser spot also injects carriers.

It is possible to map the electric field by examining the electroreflectance signal as a function of the position of the laser spot on the sample [26

26. J. K. Wahlstrand, H. Zhang, and S. T. Cundiff, “Uniform-field transverse electroreflectance using a mode-locked laser and a radio-frequency bias,” Appl. Phys. Lett. 96, 101104 (2010). [CrossRef]

]. By examining the spacing of the Franz-Keldysh oscillations in the spectrum, one can determine the field, assuming one knows the effective mass [38

38. H. Shen and M. Dutta, “Franz–Keldysh oscillations in modulation spectroscopy,” J. Appl. Phys. 78, 2151–2176 (1995). [CrossRef]

]. It might be wondered whether an interpretation based on the Franz-Keldysh effect is valid, considering that we use LT-GaAs and Er-doped GaAs samples, which have a subpicosecond carrier lifetime. But the characteristic Franz-Keldysh lineshape in electroabsorption has been observed in LT-GaAs [29

29. M. Ruff, D. Streb, S. U. Dankowski, S. Tautz, P. Kiesel, B. Knupfer, M. Kneissl, N. Linder, G. H. Dohler, and U. D. Keil, “Polarization dependence of the electroabsorption in low-temperature grown GaAs for above band-gap energies,” Appl. Phys. Lett. 68, 2968–2970 (1996). [CrossRef]

]. The lifetime certainly restricts the number of Franz-Keldysh oscillations one can observe, but the coherence time required for the Franz-Keldysh lineshape at the band edge to appear is less than 100 fs. In measurements of the field using electroreflectance, we have found that the electric field strength and distribution depends strongly on the optical power used, but in all cases, it is enhanced near the electrode. The measured enhancement of the electric field near one electrode is consistent with the QUIC experiments, where the bias-induced signal appears when the laser spot is near an electrode.

Another potential source of an electric field is the photo-Dember effect. This is the generation of an electric field when carriers are photo-excited at a surface by a pulse. Because of their higher mobility, electrons diffuse much more quickly than holes, and the separation of electrons and holes generates a transient electric field [39

39. H. Dember, “Über eine photoelektronische Kraft in Kupferoxydul-Kristallen,” Z. Phys. 32, 554 (1931).

]. The same effect occurs near the edge of an opaque electrode, in which case a component of the DC field is directed parallel to the surface of the sample. These fields are strong enough (> 10 kV/cm) to generate THz radiation [40

40. G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010). [CrossRef] [PubMed]

], so they are certainly strong enough to enable QUIC. An intriguing possibility is that, at zero external bias, the photo-Dember effect causes an electric field to be present when the laser is focused near the electrode, and that this is the source of the enhancement of the QUIC signal when the laser spot is near the electrode [34

34. P. Roos, Q. Quraishi, S. Cundiff, R. Bhat, and J. Sipe, “Characterization of quantum interference control of injected currents in LT-GaAs for carrier-envelope phase measurements,” Opt. Express 11, 2081–2090 (2003). [CrossRef] [PubMed]

].

4.4.3. Other issues

Many questions remain about the interpretation of the electrically measured experiment, particularly for the case of narrowly spaced electrodes. The theory predicts a π/2 phase shift between current injection and population control enabled by a DC field, and we have observed the phase shift between zero and non-zero bias to vary depending on the electrode spacing and the position of the laser spot with respect to the electrode. We note that resonances and complex impedences have been observed in metal-semiconductor contacts [41

41. J. Werner, K. Ploog, and H. J. Queisser, “Interface-state measurements at Schottky contacts: a new admittance technique,” Phys. Rev. Lett. 57, 1080–1083 (1986). [CrossRef] [PubMed]

], and we have examined the possibility that capacitances in the metal-semiconductor contact could complicate the transmission of the photocurrent into the electrode. We measured the QUIC signal using a phase modulation frequency from a few Hz up to 100 kHz and found no dependence of the signal on frequency. Ballistic current injection is a promising, untapped technique for the study of transport processes across metal-semiconductor interfaces, but the work here shows that the presence of an electric field complicates the physics considerably. Better models of the carrier dynamics in these structures would be useful, not only to explain this experiment, but to improve photoconductive THz emitters [40

40. G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010). [CrossRef] [PubMed]

, 42

42. J. H. Kim, A. Polley, and S. E. Ralph, “Efficient photoconductive terahertz source using line excitation,” Opt. Lett. 30, 2490–2492 (2005). [CrossRef] [PubMed]

44

44. H. Zhang, J. K. Wahlstrand, S. B. Choi, and S. T. Cundiff, “Contactless photoconductive terahertz generation,” Opt. Lett. 36, 223–225 (2011). [CrossRef] [PubMed]

] based on transverse electrode structures.

5. Conclusion

In summary, we have shown the results of calculations and an all-optical experiment on the polarization dependence of field-induced quantum interference control in bulk GaAs. The experimental results, which show the largest signal when the optical fields are parallel to the applied DC field, are consistent within error with theoretical predictions using a 14-band k · p model.

Acknowledgments

J.E.S. thanks the National Sciences and Engineering Research Council of Canada for financial support. We thank A. D. Bristow and M. J. Stevens for helpful suggestions and Robert Snider, Jessica Pipis, and Ryan Smith for technical contributions. We also thank R. Mirin for providing the LT-GaAs and GaAs:Er samples.

References and links

1.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995). [CrossRef] [PubMed]

2.

R. Atanasov, A. Haché, J. L. P. Hughes, H. M. van Driel, and J. E. Sipe, “Coherent control of photocurrent generation in bulk semiconductors,” Phys. Rev. Lett. 76, 1703–1706 (1996). [CrossRef] [PubMed]

3.

A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, “Observation of coherently controlled photocurrent in unbiased, bulk GaAs,” Phys. Rev. Lett. 78, 306–309 (1997). [CrossRef]

4.

L. Costa, M. Betz, M. Spasenovic, A. D. Bristow, and H. M. van Driel, “All-optical injection of ballistic electrical currents in unbiased silicon,” Nat. Phys. 3, 632–635 (2007). [CrossRef]

5.

Y. Kerachian, P. A. Marsden, H. M. van Driel, and A. L. Smirl, “Dynamics of charge current gratings generated in GaAs by ultrafast quantum interference control,” Phys. Rev. B 75, 125205 (2007). [CrossRef]

6.

H. Zhao, E. J. Loren, A. L. Smirl, and H. M. van Driel, “Dynamics of charge currents ballistically injected in GaAs by quantum interference,” J. Appl. Phys. 103, 053510 (2008). [CrossRef]

7.

M. J. Stevens, A. L. Smirl, R. D. R. Bhat, A. 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]

8.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004). [CrossRef] [PubMed]

9.

P. Roos, X. Li, J. Pipis, and S. Cundiff, “Solid-state carrier-envelope-phase noise measurements with intrinsically balanced detection,” Opt. Express 12, 4255–4260 (2004). [CrossRef] [PubMed]

10.

P. A. Roos, X. Li, R. P. Smith, J. A. Pipis, T. M. Fortier, and S. T. Cundiff, “Solid-state carrier-envelope phase stabilization via quantum interference control of injected photocurrents,” Opt. Lett. 30, 735–737 (2005). [CrossRef] [PubMed]

11.

R. P. Smith, P. A. Roos, J. K. Wahlstrand, J. A. Pipis, M. B. Rivas, and S. T. Cundiff, “Optical frequency metrology of an iodine-stabilized He-Ne laser using the frequency comb of a quantum-interference-stabilized mode-locked laser,” J. Res. Natl. Inst. Stand. Technol. 112, 289–296 (2007).

12.

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]

13.

J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. 83, 4192–4195 (1999). [CrossRef]

14.

R. D. R. Bhat and J. E. Sipe, “Calculations of two-color interband optical injection and control of carrier population, spin, current, and spin current in bulk semiconductors,” arXiv:cond-mat/0601277 (unpublished).

15.

J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett. 89, 241115 (2006). [CrossRef]

16.

J. K. Wahlstrand and J. E. Sipe, “Independent-particle theory of the Franz-Keldysh effect including interband coupling: Application to calculation of electroabsorption in GaAs,” Phys. Rev. B 82, 075206 (2010). [CrossRef]

17.

D. E. Aspnes, “Electric field effects on the dielectric constant of solids,” Phys. Rev. 153, 972–982 (1967). [CrossRef]

18.

P. Pfeffer and W. Zawadzki, “Five-level k · p model for the conduction and valence bands of GaAs and InP,” Phys. Rev. B 53, 12813–12828 (1996). [CrossRef]

19.

J. K. Wahlstrand, S. T. Cundiff, and J. E. Sipe, “Polarization dependence of the two-photon Franz-Keldysh effect,” Phys. Rev. B 83, 233201 (2011). [CrossRef]

20.

M. J. Stevens, R. D. R. Bhat, X. Y. Pan, H. M. van Driel, J. E. Sipe, and A. L. Smirl, “Enhanced coherent control of carrier and spin density in a zinc-blende semiconductor by cascaded second-harmonic generation,” J. Appl. Phys. 97, 093709 (2005). [CrossRef]

21.

C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967). [CrossRef]

22.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, “Room-temperature optical nonlinearities in GaAs,” Phys. Rev. Lett. 57, 2446 (1986). [CrossRef] [PubMed]

23.

J. M. Fraser and H. M. van Driel, “Quantum interference control of free-carrier density in GaAs,” Phys. Rev. B 68, 085208 (2003). [CrossRef]

24.

M. J. Stevens, R. D. R. Bhat, J. E. Sipe, H. M. van Driel, and A. L. Smirl, “Coherent control of carrier population and spin in (111)-GaAs,” Phys. Status Solidi B 238, 568–574 (2003). [CrossRef]

25.

S. E. Ralph and D. Grischkowsky, “Trap-enhanced electric fields in semi-insulators: the role of electrical and optical carrier injection,” Appl. Phys. Lett. 59, 1972–1974 (1991). [CrossRef]

26.

J. K. Wahlstrand, H. Zhang, and S. T. Cundiff, “Uniform-field transverse electroreflectance using a mode-locked laser and a radio-frequency bias,” Appl. Phys. Lett. 96, 101104 (2010). [CrossRef]

27.

R. L. Snider, J. K. Wahlstrand, H. Zhang, R. P. Mirin, and S. T. Cundiff, “Quantum interference control of photocurrent injection in Er-doped GaAs,” Appl. Phys. B 98, 333–336 (2010). [CrossRef]

28.

S. Gupta, M. Y. Frankel, J. A. Valdmanis, J. F. Whitaker, G. A. Mourou, F. W. Smith, and A. R. Calawa, “Subpi-cosecond carrier lifetime in GaAs grown by molecular beam epitaxy at low temperatures,” Appl. Phys. Lett. 59, 3276–3278 (1991). [CrossRef]

29.

M. Ruff, D. Streb, S. U. Dankowski, S. Tautz, P. Kiesel, B. Knupfer, M. Kneissl, N. Linder, G. H. Dohler, and U. D. Keil, “Polarization dependence of the electroabsorption in low-temperature grown GaAs for above band-gap energies,” Appl. Phys. Lett. 68, 2968–2970 (1996). [CrossRef]

30.

A. Haché, J. Sipe, and H. van Driel, “Quantum interference control of electrical currents in GaAs,” IEEE J. Quantum Electron. 34, 1144–1154 (1998). [CrossRef]

31.

S. Gupta, S. Sethi, and P. K. Bhattacharya, “Picosecond carrier lifetime in erbium-doped-GaAs,” Appl. Phys. Lett. 62, 1128–1130 (1993). [CrossRef]

32.

N. Laman, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference control of currents in CdSe with a single optical beam,” Appl. Phys. Lett. 75, 2581–2583 (1999). [CrossRef]

33.

P. A. Roos, X. Li, J. A. Pipis, T. M. Fortier, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Characterization of carrier-envelope phase-sensitive photocurrent injection in a semiconductor,” J. Opt. Soc. Am. B 22, 362–368 (2005). [CrossRef]

34.

P. Roos, Q. Quraishi, S. Cundiff, R. Bhat, and J. Sipe, “Characterization of quantum interference control of injected currents in LT-GaAs for carrier-envelope phase measurements,” Opt. Express 11, 2081–2090 (2003). [CrossRef] [PubMed]

35.

R. D. R. Bhat and J. E. Sipe, “Excitonic effects on the two-color coherent control of interband transitions in bulk semiconductors,” Phys. Rev. B 72, 075205 (2005). [CrossRef]

36.

C. Sames, J. Menard, M. Betz, A. L. Smirl, and H. M. van Driel, “All-optical coherently controlled terahertz ac charge currents from excitons in semiconductors,” Phys. Rev. B 79, 045208 (2009). [CrossRef]

37.

D. S. Kim and D. S. Citrin, “Efficient terahertz generation using trap-enhanced fields in semi-insulating photo-conductors by spatially broadened excitation,” J. Appl. Phys. 101, 053105 (2007). [CrossRef]

38.

H. Shen and M. Dutta, “Franz–Keldysh oscillations in modulation spectroscopy,” J. Appl. Phys. 78, 2151–2176 (1995). [CrossRef]

39.

H. Dember, “Über eine photoelektronische Kraft in Kupferoxydul-Kristallen,” Z. Phys. 32, 554 (1931).

40.

G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010). [CrossRef] [PubMed]

41.

J. Werner, K. Ploog, and H. J. Queisser, “Interface-state measurements at Schottky contacts: a new admittance technique,” Phys. Rev. Lett. 57, 1080–1083 (1986). [CrossRef] [PubMed]

42.

J. H. Kim, A. Polley, and S. E. Ralph, “Efficient photoconductive terahertz source using line excitation,” Opt. Lett. 30, 2490–2492 (2005). [CrossRef] [PubMed]

43.

A. Dreyhaupt, S. Winnerl, M. Helm, and T. Dekorsy, “Optimum excitation conditions for the generation of high-electric-field terahertz radiation from an oscillator-driven photoconductive device,” Opt. Lett. 31, 1546–1548 (2006). [CrossRef] [PubMed]

44.

H. Zhang, J. K. Wahlstrand, S. B. Choi, and S. T. Cundiff, “Contactless photoconductive terahertz generation,” Opt. Lett. 36, 223–225 (2011). [CrossRef] [PubMed]

OCIS Codes
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW
(270.4180) Quantum optics : Multiphoton processes

ToC Category:
Nonlinear Absorption and Dispersion

History
Original Manuscript: September 6, 2011
Manuscript Accepted: September 15, 2011
Published: October 25, 2011

Virtual Issues
Nonlinear Optics (2011) Optical Materials Express

Citation
J. K. Wahlstrand, H. Zhang, S. B. Choi, J. E. Sipe, and S. T. Cundiff, "Electric field-induced coherent control in GaAs: polarization dependence and electrical measurement [Invited]," Opt. Express 19, 22563-22574 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22563


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References

  1. E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett.74, 3596–3599 (1995). [CrossRef] [PubMed]
  2. R. Atanasov, A. Haché, J. L. P. Hughes, H. M. van Driel, and J. E. Sipe, “Coherent control of photocurrent generation in bulk semiconductors,” Phys. Rev. Lett.76, 1703–1706 (1996). [CrossRef] [PubMed]
  3. A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, “Observation of coherently controlled photocurrent in unbiased, bulk GaAs,” Phys. Rev. Lett.78, 306–309 (1997). [CrossRef]
  4. L. Costa, M. Betz, M. Spasenovic, A. D. Bristow, and H. M. van Driel, “All-optical injection of ballistic electrical currents in unbiased silicon,” Nat. Phys.3, 632–635 (2007). [CrossRef]
  5. Y. Kerachian, P. A. Marsden, H. M. van Driel, and A. L. Smirl, “Dynamics of charge current gratings generated in GaAs by ultrafast quantum interference control,” Phys. Rev. B75, 125205 (2007). [CrossRef]
  6. H. Zhao, E. J. Loren, A. L. Smirl, and H. M. van Driel, “Dynamics of charge currents ballistically injected in GaAs by quantum interference,” J. Appl. Phys.103, 053510 (2008). [CrossRef]
  7. M. J. Stevens, A. L. Smirl, R. D. R. Bhat, A. 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]
  8. T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett.92, 147403 (2004). [CrossRef] [PubMed]
  9. P. Roos, X. Li, J. Pipis, and S. Cundiff, “Solid-state carrier-envelope-phase noise measurements with intrinsically balanced detection,” Opt. Express12, 4255–4260 (2004). [CrossRef] [PubMed]
  10. P. A. Roos, X. Li, R. P. Smith, J. A. Pipis, T. M. Fortier, and S. T. Cundiff, “Solid-state carrier-envelope phase stabilization via quantum interference control of injected photocurrents,” Opt. Lett.30, 735–737 (2005). [CrossRef] [PubMed]
  11. R. P. Smith, P. A. Roos, J. K. Wahlstrand, J. A. Pipis, M. B. Rivas, and S. T. Cundiff, “Optical frequency metrology of an iodine-stabilized He-Ne laser using the frequency comb of a quantum-interference-stabilized mode-locked laser,” J. Res. Natl. Inst. Stand. Technol.112, 289–296 (2007).
  12. 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]
  13. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett.83, 4192–4195 (1999). [CrossRef]
  14. R. D. R. Bhat and J. E. Sipe, “Calculations of two-color interband optical injection and control of carrier population, spin, current, and spin current in bulk semiconductors,” arXiv:cond-mat/0601277 (unpublished).
  15. J. K. Wahlstrand, J. A. Pipis, P. A. Roos, S. T. Cundiff, and R. P. Smith, “Electrical measurement of carrier population modulation by two-color coherent control,” Appl. Phys. Lett.89, 241115 (2006). [CrossRef]
  16. J. K. Wahlstrand and J. E. Sipe, “Independent-particle theory of the Franz-Keldysh effect including interband coupling: Application to calculation of electroabsorption in GaAs,” Phys. Rev. B82, 075206 (2010). [CrossRef]
  17. D. E. Aspnes, “Electric field effects on the dielectric constant of solids,” Phys. Rev.153, 972–982 (1967). [CrossRef]
  18. P. Pfeffer and W. Zawadzki, “Five-level k · p model for the conduction and valence bands of GaAs and InP,” Phys. Rev. B53, 12813–12828 (1996). [CrossRef]
  19. J. K. Wahlstrand, S. T. Cundiff, and J. E. Sipe, “Polarization dependence of the two-photon Franz-Keldysh effect,” Phys. Rev. B83, 233201 (2011). [CrossRef]
  20. M. J. Stevens, R. D. R. Bhat, X. Y. Pan, H. M. van Driel, J. E. Sipe, and A. L. Smirl, “Enhanced coherent control of carrier and spin density in a zinc-blende semiconductor by cascaded second-harmonic generation,” J. Appl. Phys.97, 093709 (2005). [CrossRef]
  21. C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett.18, 167–170 (1967). [CrossRef]
  22. Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, “Room-temperature optical nonlinearities in GaAs,” Phys. Rev. Lett.57, 2446 (1986). [CrossRef] [PubMed]
  23. J. M. Fraser and H. M. van Driel, “Quantum interference control of free-carrier density in GaAs,” Phys. Rev. B68, 085208 (2003). [CrossRef]
  24. M. J. Stevens, R. D. R. Bhat, J. E. Sipe, H. M. van Driel, and A. L. Smirl, “Coherent control of carrier population and spin in (111)-GaAs,” Phys. Status Solidi B238, 568–574 (2003). [CrossRef]
  25. S. E. Ralph and D. Grischkowsky, “Trap-enhanced electric fields in semi-insulators: the role of electrical and optical carrier injection,” Appl. Phys. Lett.59, 1972–1974 (1991). [CrossRef]
  26. J. K. Wahlstrand, H. Zhang, and S. T. Cundiff, “Uniform-field transverse electroreflectance using a mode-locked laser and a radio-frequency bias,” Appl. Phys. Lett.96, 101104 (2010). [CrossRef]
  27. R. L. Snider, J. K. Wahlstrand, H. Zhang, R. P. Mirin, and S. T. Cundiff, “Quantum interference control of photocurrent injection in Er-doped GaAs,” Appl. Phys. B98, 333–336 (2010). [CrossRef]
  28. S. Gupta, M. Y. Frankel, J. A. Valdmanis, J. F. Whitaker, G. A. Mourou, F. W. Smith, and A. R. Calawa, “Subpi-cosecond carrier lifetime in GaAs grown by molecular beam epitaxy at low temperatures,” Appl. Phys. Lett.59, 3276–3278 (1991). [CrossRef]
  29. M. Ruff, D. Streb, S. U. Dankowski, S. Tautz, P. Kiesel, B. Knupfer, M. Kneissl, N. Linder, G. H. Dohler, and U. D. Keil, “Polarization dependence of the electroabsorption in low-temperature grown GaAs for above band-gap energies,” Appl. Phys. Lett.68, 2968–2970 (1996). [CrossRef]
  30. A. Haché, J. Sipe, and H. van Driel, “Quantum interference control of electrical currents in GaAs,” IEEE J. Quantum Electron.34, 1144–1154 (1998). [CrossRef]
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