1. Introduction

When a semiconductor is illuminated with light, quantum interference of absorption pathways can cause effects that depend on a parameter involving the phases of the harmonically related frequency components, enabling control over the end state of the absorption process. Ballistic currents can be injected [1–4], the direction of which depends on the polarization of the light and the phase 2ϕ_ω – ϕ_2ω between incident frequencies at ω and 2ω. These optically driven currents have proven useful in studying ultrafast current relaxation processes [5, 6], and under certain conditions the process can inject a pure spin current [7]. By measuring the current injected in a semiconductor device, it has been shown that the phase evolution of the pulse train from a mode-locked laser can be measured [8, 9]. The signal in turn be used to lock the laser’s carrier-envelope offset frequency [10,11], an important application for optical frequency metrology.

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], a process that ordinarily requires a medium without a center of inversion symmetry [13]. 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]. We find that applying a bias of a few volts across electrodes patterned on a semiconductor surface enhances the sensitivity of the photocurrent to ϕ_2ω – 2ϕ_ω 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 2h̄ω above the band gap, the rate of carrier injection ṅ = ṅ₁ + ṅ₂ + ṅ_I, 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 ṅ_I 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 E_ωe^{−iωt+iϕ_ω} + E_2ωe^{−2iωt+iϕ_2ω} + c.c., where E_ω and E_2ω are real vectors with magnitude E_ω and E_2ω respectively, the interference term ${\dot{n}}_I={\eta }_I^{\mathit{jkl}}\left(\omega \right)E_{2\omega }^j{E}_{\omega }^k{E}_{\omega }^l\text{exp}\left(i{\varphi }_{2\omega }-2i{\varphi }_{\omega }\right)+c.c.$, where ${\eta }_I^{\mathit{jkl}}\left(\omega \right)$ is a tensor that describes the efficiency of the population control process [13,14]. Unlike ṅ₁ or ṅ₂, ṅ_I depends on the parameter ϕ_2ω – 2ϕ_ω involving the relative phases of the two fields. In a crystal with center of inversion symmetry, in the absence of an external field, ${\eta }_I^{\mathit{jkl}}\left(\omega \right)=0$. If there is no center of inversion symmetry, then in general ${\eta }_I^{\mathit{jkl}}\ne 0$ for 2h̄ω above the band gap, and coherent control of population is possible even in the absence of an external field [13]. We refer to this as the χ⁽²⁾-enabled population control process. In GaAs, the only nonzero element is ${\eta }_I^{\mathit{xyz}}\left(\omega \right)$ 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 ẑ, but not x̂. The χ⁽²⁾-enabled process has been observed in a (111) GaAs sample using optical [13] and electrical [15] detection.

When a DC field is applied, ${\eta }_I^{\mathit{jkl}}\ne 0$ even in a crystal with center of inversion symmetry [12]. We have extended a theoretical framework for one-photon absorption in the presence of a dc field [16] 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 V_cv is constant within the Brillouin zone, one can derive analytical expressions for ${\eta }_I^{\mathit{jkl}}\left(\omega \right)$. We find [12]

Models with more bands capture the polarization dependence of the effect because they account for the dependence of V_cv on k, nonparabolic band dispersion, and the presence of many possible intermediate bands for the two-photon absorption pathway. Calculations using a 14-band k·p model [18] are shown in Fig. 1 for two directions of the DC field. The efficiency of the field-induced process is predicted to be largest near the half band gap h̄ω_g/2. Note that the calculation assumes zero temperature, including the values of the band energy parameters, and neglects the electron-hole interaction. The effect is largest when E_DC is parallel to the optical fields. The strong polarization dependence of field-induced QUIC is related to the contribution of intraband matrix elements to the two-photon absorption pathway, which leads to an enhanced efficiency when E_ω || E_DC. This enhancement also plays a role in the polarization dependence of the two-photon Franz-Keldysh effect [19]. As in the one-photon FKE, the oscillation period depends weakly on the DC field direction because of band warping [16]. The χ⁽²⁾-enabled process (not shown) is affected weakly by the DC field, leading to oscillations in ${\eta }_I^{\mathit{jkl}}\left(\omega \right)$.

 

Fig. 1 Calculation of the spectral dependence of population QUIC using a 14-band model for E_DC = 44 kV/cm. The arrows show the position of the half band gap h̄ω_g/2 and the half split-off gap h̄ω_SO/2. (a) Results for E_DC along [001]. Black: E_ω, E_2ω along [001]; Red: E_ω along [010], E_2ω along [001]. (b) Results for E_DC along [011]. Black: E_ω, E_2ω along [011]; Red: E_ω along [01̄1], E_2ω along [011].

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Another process that causes a change in the rate of carrier injection that depends on ϕ_2ω – 2ϕ_ω is cascaded second harmonic generation [20]. 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 ϕ_2ω – 2ϕ_ω, and thus would produce an identical signal in the experiment. In this case the second harmonic generation would be electric field-induced [21]. 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] is dominant. In future experiments, it could be distinguished from the direct coherent control processes by using a layered structure [20].

3. All-optical experiment

The cleanest way to measure a change in the photoexcited carrier density due to QUIC is to measure the change in transmission of a probe pulse that passes through the sample after a two-color pump pulse. The absorption coefficient depends on the density of excited carriers [22], and the change in transmission is proportional to small changes in the carrier density. This technique was used in previous measurements of population QUIC enabled by χ⁽²⁾ [13, 20, 23, 24].

3.1. Technique

A detailed description of the experiment used to observe field-induced QUIC was given in [12]. 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] (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]. The characteristic feature of the two-color QUIC signal is its dependence on the phase parameter ϕ₂₁ = ϕ_2ω – 2ϕ_ω. 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] 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 $P_{2\omega }^{1/2}{P}_{\omega }$ as predicted by the theory, where P_ω (P_2ω) 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 ϕ_2ω – 2ϕ_ω for zero, negative, and positive external bias. (b) Polarization dependence of the signal for E_DC || [110]: E_2ω, E_ω || E_DC (black); E_2ω ⊥ E_DC, E_ω || E_DC (blue); E_ω ⊥ E_DC, E_2ω || E_DC (red); and E_2ω, E_ω || E_DC (magenta).

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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 E_ω ⊥ E_DC and zero when E_2ω ⊥ E_DC. 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 E_2ω ⊥ E_DC seen in Fig. 2b to that experimental imperfection, which is a particular problem for the 2ω beam because of the $P_{2\omega }^{1/2}$ 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].

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], which ballistically injects a photocurrent that depends on ϕ_2ω – 2ϕ_ω, and field-induced modulation of the carrier injection rate [12], 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] and a high breakdown threshold [29]. 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]. We have also used Er-doped GaAs [31] – a material that has an ultrashort carrier lifetime like LT-GaAs, and produces a large QUIC signal [27]. 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, 8, 27, 30, 32] and population control experiments [15]. 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 f_AOM ≈ 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 f_det = 3f_rep – f_AOM. The reference signal is generated using an RF phase detector.

The AOM technique has many advantages over implementations using a moving mirror to dither the phase. The phase modulation ramps constantly over 2π, and the modulation frequency can be varied over a wide range simply by varying f_AOM, with no complications arising from resonances in the piezoelectric transducer. The bandwidth limit caused by the capacitance of the electrode structure has been found to be one of the limiting factors in carrier-envelope phase detectors [33], and a two-pulse setup using an AOM could be used to optimize a carrier-envelope detection system with no need for a carrier-envelope phase stabilized source. Also, due to periodic tilting of the mirror on the piezoelectric transducer as it shakes, a DC offset has been encountered in previous experiments [34]; this artifact does not arise when an AOM is used to modulate the phase.

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], 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: E_DC || [001] (blue) and E_DC || [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.

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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] 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], 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 2h̄ω is very near the band edge [35] and this has been verified experimentally in CdTe [36]. Like χ⁽²⁾-enabled population control, electric field-induced population control has a cos(ϕ_2ω – 2ϕ_ω) 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, 34]. 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.

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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] 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.

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We have observed the same general trends when using an OPO and tuning h̄ω from 0.72 eV to 0.83 eV, showing that this effect is not localized to any particular photon energy. We have also observed the same bias dependence in an Er-doped GaAs sample [27]. With a widely spaced electrode structure and higher voltage, we have also observed field-induced QUIC in a semi-insulating (SI) GaAs sample. In SI-GaAs, no signal could be seen at zero bias. In all experiments, we observe the strongest signal near the electrode edges.

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 ϕ_2ω – 2ϕ_ω, 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]. A dependence of the current injection efficiency on E_DC 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 [25,37]. 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]. By examining the spacing of the Franz-Keldysh oscillations in the spectrum, one can determine the field, assuming one knows the effective mass [38]. 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]. 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]. 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], 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].

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], 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, 42–44] 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.

We have also observed a strong enhancement in the sensitivity of a semiconductor device to the phase between harmonically related fields when an external bias is applied. While we cannot currently explain all features of the data, we have observed evidence that electric fields in metal-semiconductor-metal structures contribute strongly to the QUIC process. In principle, this enhancement could be used to improve carrier-envelope phase detection devices based on semiconductors [33], though the effect predicted by the theory is largest when 2h̄ω is just above the band edge [12]. In implementations used so far using a Ti:sapphire laser [10, 11], 2h̄ω is much larger than the band edge in GaAs, but a different semiconductor could in principle be used to better match the photon energy to the band gap.