# Second harmonic generation spectroscopy of excitons in ZnO

## Abstract

Nonlinear optics of semiconductors is an important field of fundamental and applied research, but surprisingly the role of excitons in the coherent processes leading to harmonics generation has remained essentially unexplored. Here we report results of a comprehensive experimental and theoretical study of the three-photon process of optical second harmonic generation (SHG) involving the exciton resonances of the noncentrosymmetric hexagonal wide-band-gap semiconductor ZnO in the photon energy range of eV. Resonant crystallographic SHG is observed for the , , , and excitons. We show that strong SHG signals at these exciton resonances are induced by the application of a magnetic field when the incident and the SHG light wave vectors are along the crystal -axis where the crystallographic SHG response vanishes. A microscopic theory of SHG generation through excitons is developed, which shows that the nonlinear interaction of coherent light with excitons has to be considered beyond the electric-dipole approximation. Depending on the particular symmetry of the exciton states SHG can originate from the electric- and magnetic-field-induced perturbations of the excitons due to the Stark effect, the spin as well as orbital Zeeman effects, or the magneto-Stark effect. The importance of each mechanism is analyzed and discussed by confronting experimental data and theoretical results for the dependencies of the SHG signals on photon energy, magnetic field, electric field, crystal temperature, and light polarization. Good agreement is obtained between experiment and theory proving the validity of our approach to the complex problem of nonlinear interaction of light with ZnO excitons. This general approach can be applied also to other semiconductors.

###### pacs:

71.35.Ji, 42.65.Ky, 78.20.Ls## I Introduction

Nonlinear optics has opened multifaceted possibilities for studying and tailoring light-matter interaction. Nowadays nonlinear optical phenomena and materials are a broad basis for fundamental and applied research Bloembergen (); Shen (); Boyd (); Bass (); Nikogosyan (). In linear optics, propagation, absorption, and emission of light are essentially single-photon processes. In contrast, in nonlinear optics the interaction of light with a medium is governed by multi-photon processes. Obviously the light-matter interaction becomes more intricate thereby. Linear and nonlinear optical experiments address different types of optical susceptibilities, still they all are determined by the features of the crystal structure of the material under study as well as the resulting charge and spin properties. Therefore they open versatile opportunities for in-depth analysis from different perspectives. In this sense, linear and nonlinear optics may be regarded as independent and simultaneously complementary to each other in material investigations.

In nonlinear optics, frequency conversion processes such as second, third, and higher order harmonics generation as well as the sum and difference frequency generation play a particularly important role Bloembergen (); Shen (); Boyd (). Among these phenomena, most prominent is the simplest three-photon process of optical second harmonic generation (SHG): first, the parity selection rules for the optical transitions between the contributing electronic states radically differ from those in linear optics and also from other, more complicated nonlinear phenomena; second, besides the parity selection rules, the time-reversal symmetry operation is a principally important factor in harmonics generation when the spin system becomes involved in applied magnetic fields or in magnetically ordered materials Fiebig1 (); Pisarev-pss (); Pisarev-JL ().

From the beginning of nonlinear optics, optical generation of second and higher order harmonics has been subject of active research in various semiconductors Garmire (); Garmire2 (). However, the majority of these studies, typically performed on bulk crystals or thin films, were limited to fixed excitation wavelengths when the fundamental and harmonics photon frequencies were in the transparency region, below the fundamental band gap. This approach is motivated by avoiding any absorption in the medium which would impede potential applications. There are only few examples where the SHG spectroscopic studies of semiconductors covered broad spectral ranges. Absolute values of the SHG coefficients for bulk zinc-blende ZnTe, ZnSe, and ZnS were measured at room temperature in the SHG spectral range eV Wagner (). Spectroscopic SHG in bulk GaAs was reported in the range eV, covering several electronic transitions at critical points Bergfeld (). The SHG spectral features found for these materials were in reasonable agreement with theoretical calculations and experimental data acquired by other techniques. For hexagonal ZnO, the material selected for the present study, SHG was reported for selected wavelengths below the band gap in numerous publications, see e.g. Miller (); Wang (); Wang2 (); Abe (). Further, also spectroscopic SHG studies in the range eV were reported for ZnO microcrystallite thin films Zhang (). The SHG output was found to increase significantly in the vicinity of the direct band gap. SHG over broad infrared spectral ranges was also applied to various chalcopyrites, semiconductors of practical importance, see e.g. Medvedkin (); Kumar () and references therein.

In semiconductors the optical properties in close vicinity of the band gap are largely determined by excitons, bound complexes of an electron and a hole. The exciton energy levels including their spin properties have been intensely studied using both linear and nonlinear optical methods such as absorption, reflection, photoluminescence, two-photon absorption, four-wave mixing, etc Garmire (); Garmire2 (). Surprisingly, the contributions of excitons to harmonics generation have remained essentially unexplored. Typically studies lack a microscopic theoretical explanation, with scarce exceptions Chang (); Leitsmann (); Pedersen (). Experimental observations were reported for forbidden SHG in resonance with the Wannier exciton in ZnSe thin films Minami (); Minami1 (); resonant SHG at the orthoexciton level in CuO MYShen (); Kono (); and second and third harmonic spectroscopy of excitons in a homoepitaxial GaN layer Schweitzer (). An early attempt to detect SHG signals in the spectral region of the exciton in ZnO and the excitons in CuCl was undertaken in Refs. Haueisen0 (); Haueisen (). The concept of Wannier excitons was used in the SHG study of CuCl, whereas Frenkel excitons were explored in SHG studies of a C molecular crystal Janner1 (); Janner2 ().

Detail insight into the role of excitons in harmonics generation can be taken if the studies are performed at low temperatures with high spectral resolution. External magnetic and electric fields can perturb and mix charge and spin states, providing novel mechanisms for nonlinear harmonics generation. For example, in diamagnetic materials GaAs and CdTe the orbital quantization is the origin of magnetic-field-induced SHG Pavlov1 (); Sanger2 (). In diluted magnetic semiconductors (Cd,Mn)Te, on the other hand, the giant Zeeman spin splitting was shown to be the source of magnetic-field-induced SHG Sanger1 (). Even for the centrosymmetric magnetic semiconductors EuTe and EuSe a magnetic field was found to induce SHG with high efficiency Kaminski (); Kaminski1 ().

This rudimentary state of the exciton SHG problem has motivated our spectroscopic research of SHG as the simplest frequency conversion process in the wide-band-gap semiconductor ZnO, characterized by a large exciton binding energy of meV and a rich exciton level structure Klingshirn1 (). This material has recently gained substantial renewed interest, partly because the large exciton binding energy could lead to lasing by exciton recombination even at room temperature. This and other potential ZnO applications are discussed in the comprehensive review by Özgür et al Alivov (). In order to get deeper insight, our study of SHG at excitons is performed in applied magnetic and electric fields. In conjunction with a detailed theoretical analysis we show that SHG spectroscopy allows us to work out the underlying microscopic mechanisms of the nonlinear process of simultaneous coherent two-photon excitation and subsequent one-photon emission involving excitons. Magnetic and electric fields can perturb the exciton states through the Stark, the magneto-Stark, and the Zeeman effects and may act therefore as sources of SHG carrying characteristic signatures for the chosen field geometry. Our findings open new opportunities for studying exciton complexes in detail and involving them in frequency conversion processes.

The paper is organized as follows. In Sec. II, we describe the crystallographic and electronic structures as well as the optical and magneto-optical properties of hexagonal ZnO. Also a symmetry analysis of the SHG polarization selection rules is given. The details of the experiment are presented in Sec. III, followed by Sec. IV where the experimental data are shown. In Sec. V the microscopic theory of the SHG is introduced and several mechanisms involving exciton states are suggested. The comparison of experiment and theory in Sec. VI allows assignment of particular signals in the SHG to a specific mechanism, which to out knowledge has been mission so far. The developed understanding can be applied also to other semiconductor materials.

## Ii Second harmonic generation in ZnO

### ii.1 Symmetry of electronic states, excitons and polaritons

ZnO crystallizes preferably in the wurtzite-type structure LB (); Alivov (), see Fig. 1(a), characterized by two interconnected sublattices of Zn and O ions with a strong ionic binding. The lattice constants of ZnO are Å and Å Jagadish (). The unit cell is formed by two Zn and two O ions () each of them being tetrahedrally surrounded by four ions of the other species. Wurtzite ZnO has a hexagonal crystal lattice, belonging to the point group and space group P Hahn (); Klingshirn3 (). The -direction of the used Cartesian -system is chosen parallel to the polar hexagonal -axis, the so-called -axis, which subsequently will be referred to as -axis following the Birss notation, and Birss (). From the optical point of view ZnO is a uniaxial material with the optical axis directed along the crystallographic -axis.

The electronic band structure of wurtzite ZnO is shown in Fig. 1(b). The valence band is formed by the orbitals of the O ions and the conduction band is formed by the orbitals of the Zn ions. The the -levels (and the antibonding orbitals) are split by the hexagonal crystal field into two and subbands. Including the spin through the spin-orbit interaction leads to a further splitting into three twofold degenerate valence band states . In all wurtzite-type semiconductors these bands are usually labeled from higher to lower energies as (), () and () bands. However, ZnO has an inverted valence band ordering (), () and () Klingshirn1 (). The selection rules for transitions from the upper valence bands () and () to the conduction band () are essentially the same, because the admixture of the character to the Bloch wave functions of the () valence band is small Lambrecht (). As a result, these transitions are allowed for , where is the electric field of the fundamental light wave. Transitions from the () valence band to the conduction band () are allowed for .

Correspondingly, three exciton series are formed in ZnO by a electron and a hole from one of the (), (), or () valence bands. These excitons have approximately the same binding energy of meV and a Bohr radius of nm. The exciton symmetry results from the direct product of the envelope function symmetry and the symmetry of conduction and valence band Bloch states, see e.g., Ref. Wheeler (). The energies of the resulting exciton states are split by the short-range exchange interaction. For the -symmetry excitons of the and series, the strongest state has symmetry. It is twofold degenerate and polarized perpendicular to the -axis, while for the exciton it has symmetry and is polarized parallel to the -axis. As a result, for light propagating along the -axis () both excitons are transversal, while the exciton is longitudinal and cannot be excited. For light with , one of the exciton states is transverse and the other is longitudinal, while the exciton is transversal. The resonances of the longitudinal excitons are shifted to higher energies by the long-range exchange interaction.

The strong light-matter interaction in ZnO leads to the formation of exciton-polaritons and their symmetries depend on the direction of the light propagation. The interaction of the transverse excitons with photons leads to the formation of two transverse, lower (LPB) and upper (UPB), polariton branches. Their dispersion relations can be obtained from the condition for the excitons and for the excitons. Here is the speed of light and is the photon frequency. and are the dielectric functions for the electric field of light polarized perpendicular and parallel to the -axis, respectively, including contributions of exciton resonances with energies close to . The energies of the UPB at coincide with the energies of the longitudinal excitons determined from , while the energies of the LPB at coincide with the energies of the transversal excitons. If is not parallel or perpendicular to the crystal -axis, one obtains the so called mixed-mode polaritons Ivchenko (); Wrzesinski (); Klingshirn2 ().

### ii.2 Polarization selection rules for SHG

Wurtzite ZnO belongs to the noncentrosymmetric point group and, consequently, the leading-order SHG is allowed in the electric-dipole (ED) approximation. The crystallographic SHG polarization can be written as

(1) |

where are the Cartesian indices, is the vacuum permittivity, is the nonlinear optical susceptibility, are the components of the fundamental light electric field . In the ED approximation and in absence of external fields a group theoretical analysis predicts the following nonzero components of the crystallographic nonlinear optical susceptibility for bulk ZnO : , , and Boyd (); Popov ().

Note, that Eq. (1) accounts only for the ED contributions on- or off-resonant with electronic band transitions at the fundamental and SHG photon frequencies and 2. However, more generally the SHG process can involve also electric-quadrupole (EQ) and magnetic-dipole (MD) contributions. They become important when the outgoing SHG is resonant with the exciton energy , for example. Taking into account higher order contributions and the feasibility of a resonance, the incoming fundamental electric field generates an effective polarization inside the semiconductor at the double frequency as Sionnest88 ():

(2) |

where the nonlinear optical susceptibility describes the spatial-dispersion phenomena entering in the EQ and MD approximation. is the exciton wave vector, is the refractive index at the fundamental energy , and is the wave vector of the incoming light.

Additional information on the exciton energy levels including their spin structure, as well as on their wave functions can be obtained by applying external fields. The symmetries of exciton states may be modified by electric or magnetic fields, enabling mixing of states. This opens the way for novel SHG mechanisms induced by the fields. In this case, the effective polarization inside the semiconductor can be written as

(3) |

where the nonlinear optical susceptibility accounts for phenomena induced by the external magnetic () and electric () fields. The nonlinear polarizations in Eqs. (1)-(3) are the sources of the outgoing SHG electric field with SHG intensity .

For a resonant SHG process, which involves the ground state of the unexcited crystal and an exciton state , the optical transition from to should be allowed both for the two-photon excitation and the one-photon emission process. Fulfillment of this condition strongly depends on the crystal symmetry and experimental geometry. With including excitonic effects, this situation becomes richer due to the different symmetries of the exciton states with , and type of envelope wave functions.

Further, external perturbations such as stress, electric or magnetic field can mix the exciton states, thereby reducing their symmetry. For linear optical spectroscopy on ZnO (e.g. one-photon absorption or emission) only exciton states are active, while states cannot be seen. In order to study exciton states, either nonlinear spectroscopy (e.g. two-photon absorption) or external perturbations, which mix and states have to be used. To study exciton states and their mixing, we performed detailed experimental and theoretical studies of ZnO, serving as a model system, by SHG spectroscopy with application of magnetic and electric fields.

## Iii Experiment

A hydrothermally grown hexagonal ZnO crystal of high optical quality with orientation and thickness of 500 m was chosen for this study. The SHG technique used for exciton spectroscopy was described in Ref. Sanger2 (). The linearly polarized fundamental light with photon energy was provided by a laser system with an optical parametric oscillator tunable in the spectral range of interest ( eV) and generating optical pulses of ns duration with energies up to mJ per pulse. The experiments were performed in the transmission geometry with the light wave vector either parallel or tilted to the -axis of the ZnO sample. The SHG signal at photon energies was spectrally selected by a monochromator and detected by a cooled charge-coupled-device camera. The experimental geometry is shown in Fig. 2. Here is the angle between the light wave vector and the -axis which gives the sample tilting. is the azimuthal angle of the fundamental light polarization, where coincides with the crystallographic -axis. Magnetic fields up to T generated by a split-coil superconducting solenoid were applied in the Voigt geometry () or the Faraday geometry (). External electric fields up to V/cm were applied via contacts perpendicular both to the magnetic field and the propagation direction of the light . The sample temperature was varied in the range K.

As we will show below in Secs. IV and Sec. V, decisive experiments for distinguishing different microscopic contributions to the SHG signal are measurements of the rotational anisotropy, i.e., the dependence of the SHG signal on the azimuthal angle. Such rotational anisotropies were measured for four different geometries:

1. , fundamental and SHG light polarizations are rotated synchronously such that they are parallel to each other.

2. , fundamental and SHG light polarization are rotated synchronously, such that the SHG light polarization is perpendicular to the fundamental light.

3. , SHG light polarization is fixed parallel to the magnetic field direction while the fundamental light polarization is rotated around .

4. , SHG light polarization is fixed perpendicular to the magnetic field direction while the fundamental light polarization is turned around .

The corresponding patterns of rotational anisotropies are modeled according to Eqs. (1-3). The results are discussed in Sec. VI.

## Iv Experimental results

### iv.1 Crystallographic SHG

It follows from the symmetry analysis of the selection rules in Sec. II.2, that for laser light propagating along the hexagonal -axis () no ED crystallographic SHG is allowed. This geometry addresses solely components of the susceptibility without index, which are all zero. Indeed, no SHG signals are found experimentally in the vicinity of the ZnO band gap for zero tilting angle , see Fig. 3(b).

For tilted geometry, , SHG is provided by the nonzero components with index, see Eq. (1). For parallel orientation of the linear polarizations of fundamental and SHG light (), strong SHG signals are found for (see Fig. 4). As one can see in Fig. 3(b), the SHG consists of a broad band in the spectral range below the exciton transitions. Further, several sharp lines show up in the exciton spectral range. Above the band gap the SHG signal vanishes. The SHG intensity shows pronounced rotational anisotropies summarized in Fig. 4. These anisotropies allow separation of the SHG signals from two-photon photoluminescence signals which are expected to be isotropic.

The exciton spectral range is shown in more detail in Fig. 3(a). Arrows mark the the reported energies of the different exciton states in hexagonal ZnO Fiebig2 (). Letters T and L mark the transversal and longitudinal excitons, respectively. A feature at eV has not been reported in literature so far. It will be referred to as the -line and discussed in more detail in Sec.VI. In the tilted geometry, both and exciton states are SHG-active. Using linear spectroscopy, only the exciton states are observed in absorption and reflection spectra due to their strong oscillator strength. By contrast, the SHG intensity in the range of states of and excitons is surprisingly much weaker than the intensity of their excited and states. This observation is a clear manifestation that nonlinear SHG spectroscopy addresses exciton properties inaccessible by linear spectroscopy.

Rotational anisotropy diagrams were measured for and at the following spectral positions: , , , , , , , , , and eV. Those anisotropies in Figs. 4(a) and 4(c) are representative for the off-resonant and exciton regions. Figs. 4(b) and 4(d) show rotational anisotropies specific for energies close to the -line and the exciton. The strongest signal for is found for all energies in the ()-plane meaning that the shape is dominated by the component. Indeed, the fitting procedure gives an value which is an order of magnitude larger than the other components. For the crossed geometry , the fitting procedure gives the same ratio of nonlinear components. On the other hand, for energies close to the exciton and the -line in Figs. 4(b) and 4(d), the SHG intensity has a pronounced feature along the -axis, which can be explained by a phase shift. The real and imaginary parts of the nonlinear components change signs in these specific regions, leading to a strong distortion in the crossed geometry .

We note, that our experiments do not allow us to measure the absolute values of the nonlinear susceptibilities. Therefore, it is difficult to compare the relative nonlinearities for SHG signals which are widely separated on the photon energy scale. To measure the absolute values of the nonlinear susceptibilities one has to take into account the complex linear refraction indices for both the fundamental and the SHG photon energies Haueisen (). We also note, that the expected quadratic increase of the SHG intensity with the fundamental power has been confirmed for the off-resonant and on-resonant crystallographic SHG signals, as well as for the magnetic-field-induced SHG signals, described in the next subsection.

### iv.2 Magnetic-field-induced SHG

For investigating magnetic-field-induced SHG (MFISH) we chose the experimental geometry with , where the crystallographic SHG signal vanishes. A magnetic field T was applied perpendicular to the -axis in the geometry . SHG signals are observed only in resonance with excitons, see Fig. 5. Two strong lines are seen in the spectral range of the exciton states and three much weaker lines in the range of the excitons. Note, that this behavior differs from the observations in GaAs and CdTe, where the exciton line always dominates in the MFISH spectrum Pavlov1 (); Sanger2 (). In ZnO, the weak SHG intensity observed at the excitons can be related to the strong absorption of SHG light due to the large absolute values of the complex dielectric function Cobet ().

A magnetic field is an axial vector of even parity and, therefore, it is not supposed to mix wave functions of opposite parities. In spite of this restriction, a strong magnetic-field-induced contribution to SHG was found in ZnO. An in-depth analysis of the SHG microscopic mechanisms is required for understanding these experimental findings. Such analysis based on the theoretical model of Sec. V will be given in Sec. VI.

#### SHG on excitons

Let us consider the experimental observations in the exciton region in magnetic field more thoroughly. The corresponding SHG spectra for different fields up to T are shown in Fig. 6(a). Three lines corresponding to the paraexciton, the middle polariton branch comment (), and the paraexciton are clearly seen at the strongest field with the first line at eV being the most intense. The integrated intensity of the line shows a dependence, see Fig. 7(a). Its temperature dependence measured at T shows a rapid decrease; see Fig. 14(a). For excitons with strong binding energies it is expected that the diamagnetic shift of their states for the magnetic field strengths used here is very small, as illustrated in Fig. 6(b).

Figure 6(c) shows the rotational anisotropies of the SHG intensities in a magnetic field of T for the parallel and the perpendicular detection geometries. The rotational anisotropies show twofold symmetry patterns, have the same amplitudes and are rotated relative to each other by , where the strongest signal is found for in the parallel geometry. These patterns clearly differ from the crystallographic ones in Fig. 4, highlighting the difference of involved SHG mechanisms. The fits are done according to Eqs. (9) and (11) in Sec. V.

#### SHG at excitons

Figure 8(a) shows the magnetic-field-induced SHG spectra at the exciton photon energies. A double peak structure with lines at and eV appears with increasing magnetic field, corresponding to the energies of the excitons Dinges1 (); Fiebig3 (). In strong magnetic fields exceeding T the doublet structure splits further into at least four peaks. In fact, more states can be distinguished in high magnetic fields when the signals in different polarization geometries are analyzed, see Fig. 8(b). The energy shifts of these lines are plotted as a fan chart diagram in Fig. 9, where the SHG peak intensities are represented by the symbol sizes.

The magnetic field dependence of the integrated SHG intensity of the states shows a quadratic increase at low fields T, similar to the exciton, and then tends to saturate for T, see Fig. 7(b). This behavior, which was not observed for the states, gives a strong hint that the mechanisms responsible for the magnetic-field-induced SHG differ for the and the excitons in ZnO. The temperature dependence of the SHG intensity in Fig. 14(a) shows a similar but slightly faster decrease than that of the excitons.

SHG rotational anisotropies detected in different geometries at T for the states are shown in Figs. 10 and 11. They have different patterns depending on the exciton state involved, see, e.g., Figs. 10(b) and 11(c). The anisotropies differ strongly in the amplitude ratios for different geometries. In the perpendicular geometry the shapes are quite different, compare Figs. 11(b) and 11(f), leading to the assumption that the responsible SHG mechanisms are different and vary with the photon energy.

#### Magnetic-field-induced versus crystallographic SHG

It is instructive to compare the intensities of the crystallographic and the magnetic-field-induced SHG signals. This comparison is presented in Fig. 12, where all four panels, recorded with the tilting angle , have the same intensity scale. In absence of magnetic field the strongest crystallographic SHG signal is found for and , while it vanishes for and ; see Fig. 4(c). Fig. 12(a) demonstrates, that even for a tilted sample no signal is observed for , , but in magnetic field strong signals appear for this configuration, see Figs. 10(a) and 10(b). Thus, for a tilted sample Fig. 12(b) shows the intensity of a pure MFISH signal for the states, which is even more intense than the strongest crystallographic signal in Fig. 12(c) observed for the with and , see Fig. 4(c). Consequently, we can conclude that the susceptibilities of MFISH and crystallographic SHG have comparable values. On the other hand, the state is not strongly modified by the MFISH contributions, compare amplitudes in Figs. 12(c) and 12(d).

### iv.3 Temperature dependence

Figure 13 compares the crystallographic SHG intensities recorded at K and K. While the off-resonant contribution has comparable intensity, the exciton SHG signals strongly decrease with rising temperature. A closer look at the detailed evolutions of the peak intensities shows that all state and the -line intensities decrease slower with temperature than the states, compare the results shown in Fig. 14(a). At the same time, the full width at half maximum (FWHM) of the line increases much faster than those of the state and the -line, see Fig. 14(b). The magnetic field influences the temperature dependence only slightly: the magnetic-field-induced signals show a similar behavior as the crystallographic ones; the closed and open dots in Figure 14(a) give the temperature dependencies for zero field and T, respectively. We conclude that the signal decays are rather independent of the SHG generating mechanism. An explanation based on our theoretical model will be discussed in Sec. VI.

### iv.4 Joint action of magnetic and electric field

An applied electric field modifies the wave functions of the exciton states and, therefore, offers another promising option for SHG spectroscopy. The electric field is a polar vector of odd parity, in contrast to the even parity magnetic field, so that it can mix exciton wave functions of opposite parity.

The two spectra in Fig. 15(b) demonstrate the effect of an applied electric field on SHG spectra for the excitons in ZnO at zero magnetic field. In the absence of electric field the residual SHG amplitude (about 1% of those previously discussed) originates from small strain caused by the sample holder with electrical contacts. Application of an electric field perpendicular to the -axis leads to an increase in the SHG signal. The electric field effect is much more pronounced in combination with an applied magnetic field , see Fig. 15(a). Here the SHG amplitude is initially gained by applying a magnetic field of T and then tuned by adding an electric field of V/cm. The SHG signal increases for positive electric fields and decreases for negative fields. The rotational anisotropies of these signals are not changed by the electric field. The variation of the integral SHG intensity with electric field strength is shown in the inset of Fig. 15(a). Such a behavior, that a weak effect (here induced by the electric field) is enhanced when combined with a stronger effect (by the magnetic field) is known in SHG spectroscopy when these effects interfere with each other, see e.g. Eq. (3) in Ref. Kaminski ().

Resistivity measurements have shown, that the incident laser beam reduces the sample resistivity enormously (by several orders of magnitude), when twice the fundamental photon energy comes close to that of the exciton states, see Fig. 15(c). The resistivity is not instantaneously restored when the laser is switched off.

## V Theory of SHG at exciton resonances

### v.1 General consideration

Theoretical studies of SHG were performed for many model semiconductors, see for example Sipe1 (); Sipe2 (); Sipe3 (); Sipe4 (); Sipe5 (); Sipe6 (); Sipe7 (); Rashkeev1 (); Rashkeev2 (); Rashkeev3 (); Rashkeev4 (). These publications analyze the generation of the second and higher harmonics by band theory or first-principle calculations, while exciton contributions have remained essentially unexplored. The complex experimentally observed exciton SHG signals for ZnO in external fields as reported here demands development of a corresponding microscopic theory.

In this section we focus on SHG effects in resonance with exciton states. This requires the analysis of the wave function symmetries for different exciton states and their modifications in external magnetic and electric fields. We present a theoretical analysis for the excitons in ZnO with wurtzite-type crystal structure. The developed theoretical approach, however, can be readily applied to other semiconductors. In particular, most of the suggested mechanisms of magnetic- and electric-field-induced SHG at the exciton resonances should prevail also in other materials.

To analyze SHG in close vicinity of an exciton resonance we write the nonlinear optical susceptibilities introduced in Eq. (3) for each exciton energy in general form as

(4) |

Here denotes the unperturbed ground state with zero energy, describes intermediate virtual states, describes the exciton state, and is the exciton damping constant. The summation in Eq. (4) is carried out over all intermediate states satisfying the symmetry selection rules for the two-photon transition from the ground to the exciton state described by the matrix element .

To account for the effects of the external electric and magnetic fields we consider the geometry where the crystallographic SHG signals are suppressed, namely , , and . In this case the incoming field is described by so that the outgoing polarization can be written as .

The perturbation caused by the photon field is described by , where and are the charge and mass of free electron and is the momentum operator. Then the perturbation is given by

(5) |

where are the projections of the momentum operator on the light polarization components, or , respectively. For one should substitute by in Eq. (5) and everywhere below, as well as by . We are interested in the lowest order effects in , that is zero-order independent of , if it exists, or first-order, linear in . Therefore, we keep only two terms in the expansion of and consider the matrix elements of the form . The first term corresponds to the electric-dipole (ED) approximation for which the matrix elements can be replaced Park () by the matrix elements of the dipole operator . The operator includes the electric-quadrupole (EQ) and the magnetic-dipole (MD) contributions where the matrix elements can be replaced Park () by the sum of matrix elements of the electric-quadrupole operator and the magnetic-dipole operator . Here is the orbital momentum operator. Depending on the perturbation and involved in the two-photon absorption and one-photon emission, respectively, we denote the resulting three-photon SHG process as , where the are either (ED) or (EQ+MD). We emphasize, that the presence of EQ or MD transition for one of the steps either excitation or emission, leads to a linear dependence of the susceptibility on .

In ZnO, the direct ED transitions between the valence and conduction band states are allowed. The strongest one-photon process for are the excitation of the () states or the emission from them. The respective matrix elements can be written as or , where the index denotes ”allowed” transition within the ED approximation according to the notation of Elliot elliot (). In contrast, the one-photon ED ”forbidden” transitions to the excitons in noncentrosymmetric wurtzite semiconductors like ZnO may occur because the valence and conduction band states do not have pure even or odd parities. These transitions are much weaker compared to the exciton transitions and can be described by the matrix elements or , where the index denotes the ”forbidden” character within the ED approximation elliot (). In the used geometry such ”forbidden” transitions are possible only for the states, and not for the state. Alternatively, the one-photon emission from all three , , and states may occur due to magnetic-dipole transitions described by the matrix element .

The strongest two-photon process in ZnO is the excitation of the exciton states. It is ED allowed, exploiting intermediate virtual states in the valence or conduction band. Such process involves a transition between valence and conduction band states and another transition between and envelopes in the same energy band. The relevant two-photon matrix element is . On the other hand, the direct two-photon absorption by the exciton states in non-centrosymmetric semiconductors may occur within the ED approximation via the intermediate virtual states in remote bands vs (). In this case the two-photon matrix element is . However, such processes are much weaker than those for the excitons ig (). Alternatively, the states can be excited in the two-photon process when the first transition is a MD transition (or ED transition of ”forbidden” character) to the states and the second one is a ED transition of ”forbidden” character between the and the envelopes. In this case the two-photon matrix element is (or ).

Important information on the symmetry of the exciton states involved in the SHG process is provided by the SHG rotational anisotropies. According to Eq. (3) the SHG intensity is given by

(6) |

if ; and by

(7) |

for the geometry. Here is the angle between and the -axis. Note that . For parallel polarization one obtains

(8) |

In a hexagonal 6mm crystal the relation is fulfilled Birss () so that

(9) |

For perpendicular polarization one finds

(10) |

and if the relation is fulfilled, then

(11) |

Below we will show, that a magnetic field applied perpendicular to the hexagonal -axis may reduce the symmetry of an exciton state and, consequently, it violates the relation .

In the following subsections we will proceed with the analysis of different specific mechanisms of the field-induced mixing of exciton states and derive the corresponding nonlinear optical susceptibilities. The results of this analysis are summarized in Table 1. Relations between , , and allow one to model the rotational anisotropies for each particular mechanism. We note that the admixture of exciton states in applied fields may lead to the dependence of the wave function and the respective energy on the exciton wave vector , as well as on and . These complex perturbations may lead to a nonlinear dependence of the susceptibilities on , and , via . They act in addition to those arising from the second term in the expansion of according to Eq. (5).

### v.2 SHG and exciton Stark effect ()

Let us first consider the SHG signals induced by an external electric field which mixes the and exciton states of opposite parity due to the Stark effect for the and excitons. However, it does not affect their spin states. Two polariton branches can be formed for each of the mixed exciton states. The new energies for the transversal lower polariton branch (LPB) and for the transverse upper polariton branch (UPB) are given in the Appendix. The resulting wave functions of the mixed states in Eq. (21) are constructed from the and components. In this process, all matrix elements for excitation and emission in Eq. (4) become allowed in the ED approximation . We denote the corresponding SHG as as shown in the first row of Table 1. The corresponding SHG signals can be observed only when the incoming light has a nonzero component responsible for excitation of the state. Therefore, for this process and the resulting electric-field-induced susceptibilities are proportional to the product of the wave function admixture coefficients in Eqs. (22) and (23). They can be written as

(12) |

where is the exciton Bohr radius. One sees that these electric-field-induced susceptibilities do not depend on the absolute value of , however, the direction of parallel to the -axis is important. If the electric field perturbation energy is much smaller than the zero-field splitting of the exciton states, , then the susceptibilities depend linearly on . However, for larger fields a saturation is expected because for the susceptibilities become independent of .

### v.3 SHG and magnetic field effects on excitons ()

The effect of an applied magnetic field B on excitons shows more facets than an electric field. We will discuss several mechanisms acting when the magnetic field is applied along the x-axis, :

1. The spin Zeeman effect, which mixes the exciton spin states through a perturbation , where is the corresponding Pauli matrix.

2. The orbital Zeeman effect, which affects the states having nonzero envelope orbital momentum and mixes the and states by a perturbation .

3. The magneto-Stark effect Samoilovich (); Thomas0 (); Thomas (); Gross (); Lafrentz (). This effect arises from the oppositely directed Lorentz forces acting on electron and hole in a magnetic field during the exciton center-of-mass motion. The resulting perturbation of the exciton wave function is equivalent to the effect of an effective electric field acting on the exciton at rest:

(13) |

Here denotes the exciton translational mass. In the given geometry the effective electron and hole masses for motion parallel to the hexagonal -axis and have to be used: and .

The diamagnetic shift of the exciton energy occurs for all states and is state-dependent Wheeler (). It does not directly lead to a state mixing, but is can enhance mixing by other mechanisms due to favorable energy shifts, bringing states closer to each other.

It is important, that the external magnetic field can mix exciton states of different symmetry allowing two-photon resonant excitation and one-photon resonant emission at a given energy and thus leading to SHG signals. The Zeeman spin mixing may induce SHG signals for one particular envelope exciton state. The orbital Zeeman effect and the magneto-Stark effect mix states with different envelope functions. The strength of this mixing depends on the energy separation of these states at zero field. At a given exciton energy the SHG signal might be induced by several mixing mechanisms acting simultaneously. Below we analyze these mechanisms in detail for each particular exciton state.

#### Magnetic-field-induced SHG for -type excitons due to spin Zeeman effect

The spin states of the -type excitons depend on the symmetries of the conduction and valence bands. For excitons formed from the conduction band of symmetry and the valence band of symmetry the resulting exciton states are of , and symmetry, split from each other by the electron-hole exchange interaction. Examples of such states are the and excitons in ZnO, or the and excitons in GaN. The dipole-allowed state can occur for a one-photon process and forms two polariton branches in the given geometry. The state can become excited by a two-photon process if one of the involved photons is due to the quadrupole perturbation or due to the involvement of intermediate virtual states in remote bands vs (). As a result, the exciton states of the and excitons might be observed in the SHG spectrum due to the Zeeman spin mixing of the and states. The energies of the new mixed polariton states are given by Eq. (26) in the Appendix. The resulting wave functions of the mixed states described by Eq. (27) are constituted by both and components.

The two-photon excitation of the state might occur through an electric-dipole/elecric-dipole () or an electric-dipole/magnetic-dipole () process as discussed above. For the sake of clarity we consider the second case in detail. It is represented by the process in the third row of Table 1. Then the SHG process involves a matrix element for the two-photon excitation with for one of the photons and for the second photon. The subsequent one-photon ED emission with is allowed through the part of the exciton wave function, so that . The resulting magnetic-field-induced nonzero susceptibilities are given by

(14) |

Here is the exciton -factor, is the lattice constant, and is the zero-field energy of the LPB or UPB exciton-polariton, respectively. The linear dependence on enters through the matrix element of the magnetic-dipole excitation with . If the exciton Zeeman splitting is much smaller than the zero-field splitting of the corresponding exciton state, then the susceptibilities depend linear on so that the SHG intensity follows a dependence.

For -symmetry states, for which the SHG process is allowed by the Zeeman spin effect, calculations show that and . The intensity of the SHG signal polarized perpendicular to the magnetic field does not depend on the excitation polarization direction, while the signal polarized parallel to the magnetic field vanishes: . Simultaneously, for the parallel and the crossed geometries SHG signals of the same amplitude are predicted. Their anisotropies are described by and .

For excitons formed by the conduction band of symmetry and the valence band of symmetry the resulting exciton states are of and symmetry. Examples are the excitons in ZnO and the excitons in GaN. The Zeeman effect mixes the dipole-allowed states and the dark state. In addition, one has to take into account the exchange interaction between the components of the and excitons which may lead to SHG from the exciton states with the same properties as described above for the excitons. In addition, the exciton states can be excited via the process.

#### Magnetic-field-induced SHG for mixed excitons

Let us now consider the effect of the effective electric field on the exciton states. The mixing of and