Circular photogalvanic effect in HgTe/CdHgTe quantum well structures
We describe the observation of the circular and linear photogalvanic effects in HgTe/CdHgTe quantum wells. The interband absorption of mid-infrared radiation as well as the intrasubband absorption of terahertz (THz) radiation in the QWs structures is shown to cause a dc electric current due to these effects. The photocurrent magnitude and direction varies with the radiation polarization state and crystallographic orientation of the substrate in a simple way that can be understood from a phenomenological theory. The observed dependences of the photocurrent on the radiation wavelength and temperature are discussed.
pacs:73.21.Fg, 72.25.Fe, 78.67.De, 73.63.Hs
High mobility HgTe quantum well (QW) structures are attracting rapidly growing attention due to the peculiar band structure properties. HgTe quantum wells exhibit an inverted band structure ordering for QWs exceeding a certain critical width, i.e., the ordering of electron- and hole-like states is interchanged. Buhmann09 (); Koenig07 () In this kind of structures the structural inversion asymmetry leads to a strong Rashba spin-orbit splitting with energies of the order of 30 meV. Gui04 () Furthermore the quantum spin Hall effect Koenig07 () has been observed in the bulk insulating regime and manifests itself in the formation of two spin polarized counter propagating one dimensional helical edge channels which results in a quantized conductance without magnetic field. The latter demonstrates that such inverted HgTe structures are an example for a topologically non-trivial insulator. Bernevig06 () Inverted band structure and strong spin-orbit interaction also give rise to unusual opto-electronic phenomena, e.g., a recently observed nonlinear magneto-gyrotropic photogalvanic effect. Diehl09 () Photogalvanic effects (PGE), in particular the circular photogalvanic effect (CPGE), in low dimensional structures have been proved be very powerful for the study of non-equilibrium processes in semiconductor QWs yielding information on their point-group symmetry, inversion asymmetry of QW structures, details of the band spin-splitting, processes of electron momentum, spin, and energy relaxation (for a recent review see Ref. IvchenkoGanichev_book, ). The CPGE is a photon helicity-dependent photocurrent, caused by a transformation of the photon angular momenta into a translational motion of charge carriers. Microscopically, the conversion can be due to spin-dependent mechanisms Ganichev01 (); GanichevPrettlreview (); Bieler05 () or orbital effects. Tarasenko07 (); Olbrich09 () The mechanisms may, as shown theoretically in Refs. Deyo09, ; Moore09, , contain contributions associated with the Berry curvature and side jumps giving rise to a macroscopic current quadratic in the amplitude of the electric field.
Here we report the observation and investigation of the circular photogalvanic effect in HgTe QW structures with different growth direction induced by terahertz (THz) as well as mid-infrared radiation. Our investigations show that the CPGE can be effectively generated in HgTe quantum wells with a strength of about an order of magnitude larger than that observed in GaAs, InAs and SiGe low dimensional structures. IvchenkoGanichev_book () This large photocurrent magnitude is of particular importance for an optimization of all-electric semiconductor room temperature detectors, which are based on photogalvanic effects and, which provide information about the polarization state of the laser radiation. JAP2008 () We present the phenomenological theory of the linear photogalvanic effect (LPGE) and CPGE in two-dimensional channels with point-group symmetry corresponding to (001)- and (013)-oriented structures and compare the results with experimental data on polarization dependences. We show the results of the band structure calculations and discuss possible mechanisms of infrared/THz radiation absorption.
Ii Samples and experimental technique
The experiments were carried out on -type HgCdTe/HgTe/HgCdTe quantum well structures of (001) and (013) crystallographic orientation and various quantum well widths . Structures with (001) surface orientation were grown by molecular beam epitaxy (MBE) on a CdZnTe substrate and have either 8 nm or 22 nm. Structures with (013) surface orientation were grown on a GaAs substrate by a modified MBE method MBE () and have = 21 nm. Samples with sheet density of electrons from cm to cm and mobility at 4.2 K in the range between and cm/Vs at K were studied in the temperature range from 4.2 K up to room temperature. The samples were closely to square shaped of size mm. While for (001) QWs all edges were obtained by cleaving, for (013) samples two edges were prepared by cleaving and the others were cut normal to them. We fabricated two pairs of ohmic contacts prepared by thermal In-bonding and centered in the middle of sample edges (see insets in Fig. 1). In this way we enabled investigations of photocurrents for (001)-grown QWs in and crystallographic directions and in (013) oriented samples in and directions. The geometry of the experiment is sketched in the insets in Fig. 1. The photocurrent was measured in unbiased structures via the voltage drop across a load resistor. The voltage was recorded with a storage oscilloscope. For low-temperature measurements the samples were mounted in an optical cryostat with -cut crystal quartz windows which allowed us to study photocurrents in the temperature range from 4.2 K up to room temperature.
The photocurrents were induced by direct band-to-band optical transitions or indirect intrasubband (Drude-like) optical transitions within the lowest size-quantized subband, applying mid-infrared (MIR) or terahertz (THz) radiation, respectively. We note, that in some samples transitions between size-quantized subbands may contribute to the absorption of MIR radiation. As a radiation source in the MIR range between 6 m and 15 m (corresponding to photon energies from meV to 82.7 meV) we used a frequency tunable free electron laser “FELIX” at FOM-Rijnhuizen in the Netherlands. Knippels99p1578 () The output pulses of light from FELIX were chosen to be 6 ps long, separated by 40 ns, in a train (or “macropulse”) of 7 s duration. The macropulses had a repetition rate of 5 Hz. The extraction of the absolute photocurrent magnitude in response to a such short pulses is not an easy task. To calibrate the data we used ns long pulses of a transversely excited atmospheric pressure (TEA) CO laser with a fixed operating wavelength of m (corresponding photon energy meV) and power 2 kW. The wavelength dependence of the current from FELIX was scaled to coincide with the CO laser result. For the measurements in the terahertz range we used a molecular gas laser, which was pumped optically by a TEA CO laser. GanichevPrettl () With NH as the active laser gas, we obtain 100 ns pulses of linearly polarized radiation at wavelengths , 148 and 280 m (corresponding photon energies are 13.7, 8.4 and 4.4 meV). We also used DO and CHF as laser gases to obtain radiation with and 496 m ( and meV), respectively. The peak power of THz radiation used in our experiments was in the range from 3 kW to 30 kW. Radiation was applied with an angle of incidence varying from to +30 to the QW normal in the () plane, see inset to Fig. 1. To investigate the circular photogalvanic effect we use elliptically polarized light. The polarization of the laser beam was modified from linear to elliptical (and circular) by means of ZnSe Fresnel rhombus for the mid-infrared radiation and crystal quartz -plates for the THz radiation. The radiation helicity of the incident light varies from -1 () to +1 () according to , where is the angle between the initial plane of the laser radiation polarization and the optical axis of the quarter-wave polarizer. The light polarization states for some characteristic angles are sketched on the top of Fig. 1.
Iii Experimental results
Irradiating (001)-grown HgTe/HgCdTe quantum well structures with polarized light at oblique incidence we detected a photocurrent signal measured across one of the contact pairs. Figure 1 (left panel) shows the dependence of the photocurrent on the angle measured at room temperature in the direction normal to the plane of incidence () and obtained at two angles of incidence . The current reversed its direction when switching the sign of the radiation helicity. For the shown data, the polarization dependence of the photocurrent is well described by . footnote11 () We emphasize that by inverting the incidence angle the photocurrent changes its sign (at ). In the longitudinal geometry, when the photocurrent is picked up in the plane of incidence, no helicity dependent photocurrent was observed. Below, in Sec. IV, we will demonstrate that these polarization and angle of incidence dependences follow exactly from a phenomenological theory. The effect is observed in a wide spectral and temperature range, from 6 m to 496 m and from 4.2 K to 300 K.
Figure 2 shows the wavelength dependence of the circular (photon-helicity-dependent) photocurrent, . To extract from the total signal we used the fact that it changes direction upon switching the helicity. Taking the difference of right- and left-handed radiation induced photocurrents we define = = = . Figure 2 demonstrates that in the mid-infrared range both samples exhibit similar spectral behavior: the signal does not depend on the wavelength in the range from 6 to 12 m and then rises rapidly, so that from 12 to 15 m the photocurrent strength increases by factor of 4. While the initial lack of spectral dependence of can be attributed to the spectral behavior of the interband absorption in this range, the signal rise at longer wavelength remains unclear. One of the possible explanations might be attributed to intersubband resonance absorption. However, the wavelength of resonant transitions between the two lowest subbands should depend strongly on the quantum well width, whereas Fig. 2 shows that the signal increase occurs at the same wavelengths for samples with very different quantum well widths. This observation excludes resonant intersubband transitions as an origin of such spectral behavior, unless there is a accidental coincidence of the absorption between the lowest subbands in the narrow well and the absorption into higher subbands in the wider well. This is possible in principle in HgTe-based QWs which have a complex band structure. One can also imagine that, in spite of the fact that direct optical transitions dominate in the absorption of mid-infrared radiation, the weaker free carrier absorption (Drude-like) may contribute substantially to the photocurrent generation yielding larger signals at lower frequencies. This assumption, however, is not supported by the measurements carried out in the THz range, where current is excited by radiation with photon energy smaller than the energy gap and is due to free carrier absorption. The inset in Fig. 2 demonstrates that, as expected for free carrier absorption, the signal rises significantly with increasing wavelength as expected for Drude absorption. We emphasize that in the whole THz range the photocurrent magnitude is substantially smaller than that excited by mid-infrared radiation. This fact shows that the photocurrent in the mid-infrared can not be attributed to indirect transitions.
Helicity dependent photocurrent was also detected in (013)-grown samples (see Figs. 1 and 3). However, in contrast to (001)-oriented QWs, in these structures the current is observed even at normal incidence. Depending on the sample type, radiation wavelength and/or sample temperature the currents in both and directions can be well fitted simply by (as for the circumstances of Fig. 1) or by more complex dependence on the angle (see Fig. 3) given by
Here, , , , and are fitting parameters, described in more detail below in Sec. IV. While the first term in the right-hand side of the Eq. (1) describes the CPGE, Ganichev01 () the other terms yield the LPGE. GanichevPrettl (); bookIvchenko (); SturmanFridkin_book () We note that even if B, C and D are non-zero, = = = is proportional to A and hence is a measure of the CPGE. Measuring the CPGE and LPGE contributions as a function of the angle of incidence we observed that all currents for the (013)-grown samples reach a maximal value at normal incidence. Below we focus on CPGE.
Experimentally, we observed a major difference between the photocurrent excited in (013)-oriented QWs and that in (001)-grown QWs. In the higher symmetry structures, (001)-grown QWs, can only be excited by oblique incidence, its direction is always restricted to the perpendicular to the plane of incidence for the used experimental geometry, and only the photocurrent magnitude and sign are functions of angle, temperature and wavelength. In the (013)-grown structure, can be produced by normal incidence and can flow in any direction as shown in Figs. 1, 4 and 5, where it can be seen that the photocurrent components and behave differently and even change their relative sign depending on the wavelength and temperature. The spectral behavior of the circular photocurrent is shown in Fig. 4 for one of the pairs of contacts. As for (001)-grown QWs, we observed that for some of the mid-infrared spectral range (in this case m) the signal is almost independent of the wavelength. Unlike (001)-grown QWs, it becomes larger at shorter . Applying THz radiation we observed the photocurrent caused by free carrier absorption which, like for (001)-grown QWs, is substantially smaller than that in mid-infrared range. The wavelength and the temperature dependence of and components are shown in the inset in Fig. 4 and in Fig. 5, respectively. We emphasize that the lack of restriction on the photocurrent direction was detected for (013)-grown QWs only, which are characterized by the point symmetry group C, the group of the lowest symmetry comprising only the identity.
Iv Phenomenological theory and discussion
In order to describe the observed dependences of photocurrent on the light polarization and the angle of incidence we derive here phenomenological equations for the linear and circular photogalvanic effects in two-dimensional HgTe-based structures. The photogalvanic current density can be written as a function of the electric component of the radiation field and the propagation direction in the following form IvchenkoGanichev_book (); bookIvchenko ()
where the first term on the right hand side is proportional to the radiation helicity and represents the CPGE, while the second term corresponds to the linear photogalvanic effect, GanichevPrettl (); bookIvchenko () which may be superimposed on the CPGE. The indices , , , run over the coordinate axes. The second rank pseudotensor and the third rank tensor , symmetric in the last two indices, are material parameters. We note that while in the theoretical consideration the current density is used, in the experiments the electric current is measured which is proportional to .
The HgTe QWs grown along direction correspond to the C point group. This group includes a rotating axis C parallel to the -direction and two mirror planes, and , coinciding with the () and () planes, respectively (Fig. 6). It follows from Neumann’s Principle and Eq. (2) that the circular photocurrent can only occur along those axes where for all symmetry operations components of transform in the same way as components of the pseudovector describing the radiation helicity. Let us consider it for circularly polarized radiation propagating along -direction, i.e., for . The reflection in each mirror plane transforms the current component and the pseudovector component in the same way: , for the plane (see Fig. 6c) and , for the plane . Therefore, the photocurrent is possible. Similar arguments hold for . For any other relative directions of and , a linear coupling of the current and the radiation helicity is forbidden. For instance, the reflections in both and planes reverse the direction of while C does nothing. There is no polar vector component that transforms in the same way. It indicates that cannot give rise to a photocurrent in (001)-grown QW structures. Thus, CPGE can only be generated at oblique incidence and in the direction normal to the plane of incidence, () or (), as observed in the experiments. For QW structures of the C point group, these arguments can be used to show that the non-zero components of the second rank pseudotensor and the third rank tensor are the following: , , , and . These four components are linearly independent. Optical excitation of such structures at oblique incidence in the -plane with elliptically polarized light generates electric current whose - and -components depend on the angles and as follows
Here, is the electric field amplitude of the incident radiation, and are the Fresnel amplitude transmission coefficients from vacuum to the structure for the - and -polarized light, respectively, BornWolf () is the refraction angle related to the incidence angle by , and is the refractive index.
We note that the linear photogalvanic effect described by Eq. (3) and second term on the right hand side of Eq. (4) was detected for several wavelengths. It can be excited by linearly polarized radiation, which was also checked experimentally (the data are not shown here).
The (013)-oriented QWs belong to the trivial point group C lacking any symmetry operation except the identity. Hence, symmetry does not impose any restriction on the relation between radiation electric field and photocurrent components. All components of the pseudotensor and the tensor may be different from zero. Phenomenologically, for the C-symmetry group, the lateral photogalvanic current for the excitation along the QW normal with elliptically polarized light is given by
Exactly this polarization dependence is encapsulated in Eq. (1) and observed in experiment as can be seen in Fig. 1 and, in particular, in Fig. 3, where all terms given by Eq. (IV) contribute substantially. Equations (IV) and (IV) show that, in (013)-oriented QWs, the CPGE photocurrent direction is arbitrary and not forced to a definite crystallographic axis.
The fact that and as well as their ratio exhibit a non-trivial variation with the radiation wavelength (Fig. 4) or sample temperature (Fig. 5) is also in agreement with Eqs. (IV) and (IV) yielding for circularly polarized light . Indeed all the components of the tensor are linearly independent in structures of the C point-group symmetry, so the ratio between them can change by varying the experimental conditions. There is therefore no preferential direction of the circular photocurrent forced by the symmetry arguments. The same is valid, and observed experimentally, for the linear photogalvanic photocurrent. Symmetry analysis of Eq. (2) for QWs of the C point group shows that current excited by linearly polarized light in the geometry of normal incidence is described by
where is the angle between the light polarization plane and the axis. We have checked experimentally (not shown) that the photocurrent induced under linearly polarized excitation is well fitted by these equations.
V Band structure and optical transitions
HgTe, which is a semimetal, forms type-III QW with HgCdTe barrier as indicated in the cartoon insets of Fig. 7. Depending on the quantum confinement, i.e. the well width, and on the temperature, a normal (Fig. 7a) or inverted (Fig. 7b) band structure can be realized with positive or negative gap between - and -subbands (see Ref. Koenig2008, ).
We have calculated the band structure of HgTe/HgCdTe QWs grown along  direction using an envelope function approximation Burt1999 () based on eight-band Hamiltonian. The influence of the induced free carriers has been included by self-consistent solution of the Poisson and Schrödinger equations. The total single-particle wave function is given by:
where are the envelope functions; are the Bloch functions at the point of the Brillouin zone, which form a complete basis set, and are assumed to be the same in HgTe- and HgCdTe-layers; is the wave vector in the plane of the QW. For narrow gap structures it is important to include the coupling between the conduction and valence bands, and the spin-orbit coupling of the valence bands. Therefore, the conduction and valence Bloch functions () (, , ) were used as a basis set. Kane1957 () In this case the following system of eight coupled differential equations has to be solved in order to find the envelope functions and the energies:
Here, indices and run over the two conduction and six valence bands, is the respective band-edge potential; indices and run over , , ; and are components of the in-plane wave vector, is the operator ; is the momentum matrix element describing the interaction between two bands of the chosen basis set , describes the coupling between the bands in this basis set and remote bands in the second-order perturbation theory, is the self-consistently calculated Hartree-potential, and are the strain-induced terms of the Bir-Pikus Hamiltonian. Bir74 () A detailed description of the model as well as the band structure parameters employed in the calculations can be found in Ref. Novik2005, . Our calculations show that mid-infrared radiation with a photon energy of the order of 100 meV used in the experiments causes direct interband optical transitions in all our samples (see Fig. 7).
To summarize, our experiments show that helicity-driven photogalvanic currents can effectively be generated in HgTe quantum wells. The photocurrent strength, e.g., for CPGE current, of about an order of magnitude larger than that observed in GaAs, InAs and SiGe low dimensional structures.IvchenkoGanichev_book () Our results reveal that photogalvanic measurements open a rich field for investigation of microscopic properties of this novel and promising material. The microscopic theory of these phenomena in HgTe-based QWs, in particular with the inverted bands ordering, is a task for future, which is complicated by the band structure, with almost flat valence bands and strong spin slitting in comparison with other materials. All known mechanisms of the CPGE are enhanced in HgTe-based structures: spin-related mechanisms IvchenkoGanichev_book (); Ganichev01 (); GanichevPrettlreview (); Bieler05 () due to large spin-orbit coupling and orbital mechanisms Tarasenko07 (); Olbrich09 () due to very small band gap. We can suggest that the photocurrent induced by the free-carrier absorption of THz radiation is dominated by orbital mechanisms. For the direct optical transitions, spin-related mechanisms of the current formation can play an important role. To distinguish between spin and orbital mechanism additional research is required. In this respect future experiments on time resolved photogalvanics under short-pulsed circularly polarized photoexcitation, with the pulse duration being comparable with the free-carrier momentum and spin relaxation times, would be desirable and informative. Such experiments would also reveal a great deal about the momentum, energy and spin relaxation of nonequilibrium photoexcited carriers. The experiments will also clarify the importance of the recently proposed Berry-phase based effects which should be enlarged in narrow gap materials.
We thank E.L. Ivchenko for helpful discussions. The financial support of the DFG via programs WE2476/9-1 and AS327/2-1, the Linkage Grant of IB of BMBF at DLR, the RFBR, the President Grant for young scientists (MD-1717.2009.2), and Foundation ”Dynasty”-ICFPM is gratefully acknowledged.
* (for NQVinh) Present address ITST, Department of Physics, University of California, Santa Barbara CA 93106-4170.
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