1. Introduction

Difference-frequency mixing (DFM) is an effective method to produce monochromatic terahertz (THz) waves with a certain frequency in a wide frequency range. It has attracted much attention, because THz waves with a narrow linewidth are required to detect specific molecules with unique absorption frequencies in e.g. human breath or the atmosphere [1–5]. Previously, we reported a wide frequency tunability of THz waves by DFM using excitons in multiple quantum wells (MQWs) [6,7]. Herein, while the exciton optical nonlinearity enhances the THz-wave intensity around the optical transition energy of a heavy-hole (HH) exciton confined in an MQW, no enhancement was observed at the simultaneous excitation of HH and light-hole (LH) excitons. Furthermore, the overlap of the exciton states, for instance, the HH and LH excitons in different MQWs, can increase the DFM signal [8], but that requires a complicated sample structure.

An internal electric field plays an important role in breaking the centrosymmetry of the envelope functions in the process of THz-wave generation by DFM in semiconductors. Usually, as is well known, semiconductor crystals have a surface electric field caused by surface states [9,10], hence, the electron-hole dipole is aligned along the growth direction, as shown in Fig. 1(a). In our samples, the MQWs were embedded in a pin structure where excitons are longitudinally polarized, creating the second nonlinear polarization due to breaking the centrosymmetry. On the contrary, the direction of laser light polarization is usually parallel to the surface, decreasing the coupling efficiency between light and dipoles. Therefore, changing the dipole direction is one of the methods to enhance the THz signal as shown in Fig. 1(b). Indeed, applying a magnetic field enhances the THz signal generated by ultrashort pulses [11–16]. Meanwhile, the coupling efficiency between light and excitons should be enhanced in the in-plane direction of the exciton dipole, hence increasing the THz-wave intensity due to DFM. In the QW systems, the exciton binding energy is enhanced, so that the electron and hole may be tightly bounded. Nevertheless, in the case of GaAs/AlAs MQWs, the exciton binding energy is approximately 10 meV [17], which is less than the thermal energy at room temperature. Then, we considered that the excitons created in the MQW can be easily changed to the electron-hole pairs. In fact, previously, we reported the possibility of ultrafast dephasing of excitons in a GaAs/AlAs MQW [18]. From these points, we considered that the change in the dipole can be described with the same model. Therefore, this study deals with the effect of changing the exciton dipole direction by permanent magnets on the intensity of THz waves due to DFM, because the magnetic effects on the continuous THz wave generation is very important for aplications such as the THz spectroscopy. The excitation-power and excitation-energy dependences show an increase in the THz-wave intensity by the magnetic field. Furthermore, the results of our calculations indicate that a strong magnetic field is not required to increase the signal intensity. The change in the THz-wave intensity is discussed on the basis of the dependence of the exciton dipole alignment.

 

Fig. 1. (a) Direction of carrier movement. The electron precedes the hole because of the surface field. (b) Ideal carrier configuration to enhance exciton coupling with light.

Download Full Size | PDF

2. Experiment

We used an undoped GaAs/AlAs MQW embedded in a pin structure on a (001) $n^{+}$-doped GaAs substrate grown by molecular-beam epitaxy. Detailed information on the sample, including the exciton energies, and the measurement system configuration is available in our previous report [6]. The MQW consists of 30 alternating layers of GaAs and AlAs with a thickness of 7.35 nm each. The parameters of terahertz waves were measured at 297 K. A semiconductor laser with a photon energy of 1.506 eV, and a continuous-wave mode Ti:sapphire laser were used to vary the excitation energy. The two beams were combined on a half mirror in free space with the same polarization, and the combined beam was focused on the sample surface. The terahertz signal was detected by a pyroelectric sensor (THZ2I-BL-BNC, Gentech-EO) and amplified by a lock-in-amplifier at 5 Hz. To apply the magnetic field, commercial neodymium magnets manufactured by SIMOTECH Co., Ltd. [19] were attached to the sample, as shown in Fig. 2(a). A nominal magnetic field ranging from 0.2 to 0.5 T was applied by changing the magnets. The magnetic-field direction was parallel to the surface, while an iron plate was installed on the opposite side to suppress field spread. Because of the p-i-n structure, electrons created by the laser light moved toward the substrate. Therefore, the Lorentz force worked along the z-direction as shown in Fig. 2(b). The polarization component in the plane is created as the hole current direction is opposite to that of electrons.

 

Fig. 2. (a) Configuration of the magnet, sample, and iron plate. (b) Schematic of the directions of the magnetic field, Lorentz force, and electron velocity directions. It is noted that the direction of the current is opposite to the velocity.

Download Full Size | PDF

3. Results and discussion

Figure 3 demonstrates the signal intensities measured as a function of excitation power under the magnetic fields of 0.3 and 0.4 T. In these measurements, the lasers with the intensities ratio of 1:5 excited HH and LH excitons, respectively [6]. The intensity demonstrates the square dependence on the excitation power, which is a characteristic of the second-order optical nonlinear effects causing the emission of THz wave due to the DFM process [6,7]. because the amplitude of the terahertz wave is given by the product of those of two excitation sources. Furthermore, the intensity at 0.4 T is larger than that at 0.3 T due to the stronger magnetic field.

 

Fig. 3. Signal intensity measured at various excitation power under the magnetic fields of 0.3 and 0.4 T indicated by the circles and squares, respectively. The dashed lines are least-square fits.

Download Full Size | PDF

Next, the magnetic-field effect is studied at different excitation energies, i.e., the energy of the Ti:sapphire laser (Fig. 4). The magnetic field of up to 0.4 T basically enhanced the signal intensity. In particular, at the LH exciton energies, the intensity ratio depends on the magnetic field . This increase is caused by the change in the exciton dipole alignment from the MQW growth direction to the in-plane direction.

 

Fig. 4. Increment of the signal intensity (the intensity ratio to that without magnetic field) at different field strength and various excitation energies (energy of Ti:sapphire laser). The dotted lines indicate the heavy-hole (HH) and light-hole (LH) exciton energies [6].

Download Full Size | PDF

On the contrary, the intensity rapidly decreases at 0.5 T; in particular, at the HH exciton energy, the intensity is maximal at 0.2 T and gradually decreases with the magnetic field. In the case of the LH-exciton energy, the maximum is at 0.4 T. To discuss the dependence of the signal intensity on the magnetic field strength, the carrier displacement caused by the Lorentz force was calculated. Here, the previous explanations related to the enhancement of the terahertz waves by magnetic fields have been developed under pulse excitation conditions. On the other hand, the basic idea related to the change in the direction of the dipole and carrier displacements is a Hall effect which is based on the continuous current. Then, the results measured with the continuous lasers are discussed on the basis of the Hall effect. When the electron and hole velocities become larger, the larger dipole will be generated because the electrons and holes can move longer during their lifetime. According to Fig. 2(b), the electron current, magnetic field, and the Lorentz force are aligned along the $x$, $y$, $z$ axes, respectively. Based on the explanation of the Hall effect, the electron velocities $v_{ex}$, $v_{ey}$, and $v_{ez}$ are described as follows:

 

Fig. 5. Calculated velocities of electrons with various lifetimes as a function of the magnetic field in the $x$- (a) and $z$-(b) directions. For holes, the velocities of heavy-hole (HH) and light-hole (LH) are plotted for the $x$- and $z$-directions in (c)-(f), respectively.

Download Full Size | PDF

The velocities in the $x$-direction, i.e., the growth direction, decrease due to the applied magnetic field. On the other hand, the velocities in the $z$-direction caused by the Lorentz force demonstrate a different behavior: while the dependences of electron and LH velocities possess a clear maxima; $v_{Hz}$ simply increases with the magnetic field. This difference is attributed to the effective masses: the heavier the mass, the stronger the magnetic field at the peak velocity.

The change in the separation length of electrons and holes caused by the magnetic field was calculated assuming that the electron and hole lifetimes are the same according to the continuous-wave excitation. Using the values of each velocity, $\Delta _{eh}=\sqrt {(v_{ex}\tau -v_{H (L) x}\tau )^{2}-(v_{ez}\tau -v_{H (L) z}\tau )^{2}}$ were calculated. The large separation may induces the exciton ionization, which reduces the signal intensity. The results for HH and LH excitons are shown by the solid and dotted curves in Fig. 6(a), respectively. Because the typical $\Delta _{eh}$ is below 1 nm, the decrease in the overlap integral corresponding to the oscillator strengths hardly affects the signal intensity. This suggests that the quantum-confined Stark effect has more impact on electron-hole spatial separation than the Lorentz force.

 

Fig. 6. (a) Calculated spatial separation between electron and hole $\Delta _{eh}$ caused by the Lorentz force for various lifetime values. The solid and dotted curves indicate the results for electron-heavy-hole (HH) and electron-light-hole (LH), respectively. (b) $z$-direction component of the spatial separation.

Download Full Size | PDF

In addition, separation in the $z$-direction $\Delta _z$ was calculated, as shown in Fig. 6(b). Interestingly, the value of $\Delta _z$ depends on the magnetic field. In particular, the peak appears in the weaker fields at longer lifetimes.

Returning to results in Fig. 4, the intensity at the HH exciton energy was maximal at 0.2 T. Only HH excitons were excited in that condition. Therefore, the dependence of $\Delta _z$, which can be considered as the coupling strength projection, on the magnetic field strongly affects the signal intensity. On the contrary, the intensity reached the maximum at 0.4 T in the case of excitation of both HH and LH excitons. This result is difficult to explain. However, we assume that $v_{Lz}$ is maximal when the coupling of HH and laser light is the strongest around 0.2 T. Namely, the signal may be enhanced not only due to the exciton-light coupling but also by carrier acceleration. Moreover, the intensity in the linear increase region in Fig. 3 is also changed. This is also attributed to the change in the velocity of z-direction because the current density depends on the velocity.

4. Conclusion

The magnetic-field effects on the DFM signal intensity in the THz range were studied under exciton-excitation conditions as a function of excitation intensity and energy. The intensity of the THz wave was enhanced by the magnetic field at weaker field strengths, whereas it decreased under a stronger field. According to the calculations, carrier lifetimes are strongly related to the intensity enhancement factor. In particular, a stronger magnetic field was not required around the HH exciton energy. These results indicate that the proposed simple and cost effective method enables the THz signal enhancement by using weak commercial magnets.