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PHYSICAL REVIEW B 92, 085202 (2015)

Electric field effect on optical harmonic generation at the exciton resonances in GaAs D. Brunne,1 M. Lafrentz,1 V. V. Pavlov,2 R. V. Pisarev,2 A. V. Rodina,2 D. R. Yakovlev,1,2 and M. Bayer1,2 1

Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia (Received 18 March 2015; revised manuscript received 27 July 2015; published 11 August 2015) 2

An electric field applied to a semiconductor reduces its crystal symmetry and modifies its electronic structure which is expected to result in changes of the linear and nonlinear response to optical excitation. In GaAs, we observe experimentally strong electric field effects on the optical second (SHG) and third (THG) harmonic generation. The SHG signal for the laser-light k vector parallel to the [001] crystal axis is symmetry forbidden in the electric-dipole approximation, but can be induced by an applied electric field in the vicinity of the 1s exciton energy. Surprisingly, the THG signal, which is allowed in this geometry, is considerably reduced by the electric field. We develop a theory which provides good agreement with the experimental data. In particular, it shows that the optical nonlinearities for the 1s exciton resonance are modified in an electric field by the Stark effect, which mixes the 1s and 2p exciton states of opposite parity. This mixing acts in opposite way on the SHG and THG processes, as it leads to the appearance of forbidden SHG in (001)-oriented GaAs and decreases the crystallographic THG. DOI: 10.1103/PhysRevB.92.085202

PACS number(s): 71.35.Ji, 42.65.Ky, 78.20.Ls

I. INTRODUCTION

Optical frequency conversion is one of the most interesting and technically important processes in a large variety of coherent nonlinear optical phenomena [1–3]. Among the simplest examples are the second harmonic generation (SHG) and third harmonic generation (THG), which convert the optical frequency of fundamental laser light ω into 2ω and 3ω for the outgoing coherent light, respectively. These processes are widely used for fundamental studies of the nonlinear light-matter interaction, as well as for numerous technical applications in coherent light sources and detectors. Experimental techniques based on SHG and THG are successfully used in a wide spectral range from the near-infrared to extreme ultraviolet for imaging of ferroelectric and magnetic domains, domain walls, spin and current distributions, visualization of biological objects, etc. [4–8]. The application of optical harmonics generation for the spectroscopic investigation of semiconductors can provide important information about their electronic band structure. Various theoretical approaches have been used to examine optical nonlinearities in different semiconductors in a broad spectral range up to a few electronvolts. The analysis, which is typically restricted to the electric-dipole (ED) approximation, shows that the SHG is symmetry allowed in noncentrosymmetric semiconductors, while THG can be observed both in centrosymmetric and noncentrosymmetric semiconductors [9–16]. Also more sophisticated calculations based on nonlocal effects in second- and third-order processes [17] have been developed. From the very beginning of the field of nonlinear optics, experimental studies of SHG and THG processes in semiconductors have been mostly limited to single wavelengths and spectroscopic studies have remained scarce [9,18,19]. Only recently it was shown that nonlinear optics becomes much more rich when spectroscopy in a wide spectral range is applied. In particular, the SHG signals can strongly enhance at low temperatures in the vicinity of the band-edge exciton resonances [20]. Experiments performed in applied magnetic 1098-0121/2015/92(8)/085202(9)

fields on the diamagnetic semiconductors GaAs, CdTe, ZnO, on the diluted magnetic semiconductors (Cd,Mn)Te, and on the magnetic semiconductors EuTe, EuSe have shown that nonlinear optics goes far beyond the ED approximation and demonstrates various microscopic mechanisms of SHG related to the orbital quantization, Zeeman spin splitting, magnetic-dipole contribution, and magneto-Stark effect on excitons [21–26]. The action of an electric field on insulators and, in particular, on semiconductors may be regarded as one of the strongest perturbations of their electronic and ionic charge distributions, and consequently of their band structures. First of all, the applied electric field reduces the macroscopic crystal symmetry of a solid, thus inducing new types of nonlinear susceptibilities. It is important to emphasize here that higher-order optical susceptibilities involved in the harmonics generation are much more sensitive to the macroscopic and microscopic symmetry in comparison to the linear optical susceptibilities. The electric field can either modify the nonlinear susceptibilities or modify the phase-matching conditions which play a crucial role for the light propagation in a crystal [27]. These two factors contribute to the efficiency of the optical harmonic generation. The fundamental role of a dc electric field as a primary source of SHG was first demonstrated in the calcite crystal CaCO3 , which has a lattice with a center of inversion [28]. Mechanisms for the SHG generation by an electrical current were suggested theoretically [14,29] and demonstrated experimentally recently [30,31]. We note that these experiments were done using single-wavelength laser sources when the nonlinear response was due to the integrated contributions of excitons and band states. The role of an applied electric field becomes even more prominent when one proceeds to nonlinear optical effects in the vicinity of sharp resonances. It is well known that an electric field strongly modifies exciton states in a semiconductor via the Stark effect, which results in exciton polarization [32–34]. The Stark effect changes the binding energies of the exciton states and mixes states with different symmetries [35,36], which do not interact with each other in the absence of the electric field. Evidently, such modifications of the excitons should

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result in pronounced effects on SHG and THG signals. An instructive example is the study of the SHG in ZnO when both magnetic and electric fields were applied to the sample [26]. In view of this study, being an unique example up to now, we may claim that the electric-field-induced optical harmonic generation on exciton resonances in semiconductors is still largely unexplored both experimentally and theoretically. GaAs is a prototype semiconductor with the noncentrosymmetric zinc-blende structure (point-group symmetry ¯ 43m). This material as bulk crystal and as the basis for heterostructures has been widely used for basic research and optoelectronic applications. Its nonlinear optical properties have been studied theoretically and experimentally. The optical susceptibilities responsible for SHG and THG have been calculated for a wide spectral range [9–12,14,15]. SHG in GaAs has been addressed in various experiments [18,21,23,37–39]. In this paper, we report on a spectroscopic study of the electric field effects on SHG and THG at the exciton resonances in GaAs. We find that the electric field acts differently on the SHG and THG intensities, representing therefore a tool for their tuning. We propose a microscopic model which explains the observed effects by mixing exciton states of opposite parity. The paper is organized as follows. Section II is devoted to the symmetry analysis of second and third optical harmonics generation under applied electric field. Section III briefly describes the experimental details and is followed by Sec. IV where the experimental results are presented. Section V describes the theory of the electric field effects on SHG and THG and is followed by Sec. VI which is devoted to the discussion of the experimental and theoretical results. II. SYMMETRY ANALYSIS OF OPTICAL HARMONICS GENERATION IN ELECTRIC FIELD

changing the symmetry of the system, i.e., without allowing new components. In this case, modifications related to the electric field depend only on the E amplitude, but not on its direction. Second, the electric field E reduces the symmetry due to the fact that it is a polar vector. Correspondingly, new nonlinear contributions become allowed, which we will call induced ones. Obviously the induced nonlinear susceptibility may depend on the electric field amplitude, i.e., χ ind (E). Then the nonlinear SHG polarization in the electric field can be written as Pi2ω = 0 χijcrk (E)Ejω Ekω + 0 χijindkl (E)Ejω Ekω el .

Here the susceptibility tensor χijindkl accounts for the new nonlinear contributions in the external electric field. The susceptibility χijindkl is a polar fourth-rank tensor and, for the zinc-blende lattice structure, it has three nonvanishing components [40,41], yxxy = zyyz = xzzx = zxxz = yzzy = xyyx, yxyx = zyzy = xxzz = zxzx = yyzz = xxyy,

Pi2ω = 0 χijcrk Ejω Ekω ,

Note that for such crystals, the crystallographic SHG signal is symmetry forbidden for the experimental geometry with kω [001] z, i.e., χijcrk (E) ≡ 0 with i,j,k = x,y. In this case, the SHG signal is absent at zero electric field and an electric field effect is solely contributed by the induced susceptibility χijindkl with i,j,k,l = x,y. We will exploit this property for our experimental studies. The crystallographic and electric-field-induced THG polarization can be written in the ED approximation as Pi3ω = 0 χijcrkl (E)Ejω Ekω Elω + 0 χijindklm (E)Ejω Ekω Elω em . (5) Here the crystallographic susceptibility χijcrkl is a polar fourthrank tensor with two nonvanishing components, yxxy = zyyz = xzzx = zxxz = yzzy = xyyx, yyyy = xxxx = zzzz.

(1)

where i,j,k are the Cartesian indices, 0 is the vacuum permittivity, and Eω (r,t) = Eω exp[i(kω r − ωt)] is the incoming electric field at the laser fundamental frequency ω and the photon wave vector kω . The crystallographic nonlinear optical susceptibility χijcrk is a polar third-rank tensor with the following nonvanishing components in zinc-blende-type semiconductors [40]: yxz = xyz = xzy = yzx = zxy = zyx.

(2)

The nonlinear polarization P2ω is the source of the outgoing SHG electric field E2ω (r,t) ∝ P2ω exp[i(k2ω r − 2ωt)] with SHG intensity I 2ω ∝ |P2ω |2 , here k2ω = 2kω . Crystallographic SHG in GaAs has been reported in several papers; see, for example, Refs. [18,23], and references therein. The external electric field E = Ee, where E is the field amplitude and e is the unit vector along the electric field direction, applied to a crystal contributes in two ways to SHG processes. First, it changes the values of the χ cr components, without

(4)

yyyy = xxxx = zzzz.

A. Symmetry analysis of SHG and THG

In the electric-dipole approximation, the crystallographic SHG polarization P2ω can be written as

(3)

(6)

We note that some of these components formally coincide with those for the SHG in the applied electric field (see above). The induced nonlinear susceptibility χijindklm is a polar fifth-rank tensor. In crystals with the zinc-blende structure, such as GaAs, it has four nonvanishing components of the following types [41]: yxxzx = zxyyy = xyzzz = yxzzz = zxxyx = xyyzy, yxyzy = zxyzz = xxyzx, yxxxz = zyyyx = xzzzy = yzzzx = zxxxy = xyyyz,

(7)

yxyyz = zyzzx = xxxzy = yyyzx = zxzzy = xxxyz. Contrary to SHG, the THG intensity at zero electric field cannot be suppressed by a proper choice of the experimental geometry, i.e., even in the kω [001] geometry, finite THG intensity is provided by the crystallographic susceptibility χijcrkl . However, the induced THG signal is symmetry forbidden for the geometry kω [001] z and E ⊥ kω , which will also be exploited in our experiment, i.e., χijindklm ≡ 0 for i,j,k,l,m = x,y. As we will show below, the main effect of the electric field in GaAs comes from the modification of the crystallographic THG susceptibility. The outgoing THG electric field is E3ω (r,t) ∝ P3ω exp[i(k3ω r − 3ωt)] and, accordingly, the THG intensity is I 3ω ∝ |P3ω |2 , here k3ω = 3kω .

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Next we recall the symmetry properties of the exciton states in GaAs, which is a direct band-gap semiconductor with well-known electronic band structure. Excitons are formed from the electronic states in the vicinity of the band gap. These are the highest p-type valence-band states with 8 symmetry and the lowest conduction-band states of s type having 6 symmetry. The electron-hole optical transition that is allowed in the electric-dipole approximation has the symmetry 8 ⊗ 6 = 3 + 4 + 5 . For excitons, the symmetry of the envelope functions has to be taken into account in addition. The s states (the ground state of the exciton is 1s) are described by the same symmetry as the electron-hole transitions, but the states with higher angular momentum such as the p states have modified symmetry. As a result, the excitons may provide new mechanisms for SHG and THG processes, especially when external electric or magnetic fields mix exciton states of different parities. Some of these mechanisms induced by a magnetic field have been recently considered for the ZnO excitons [26].

2

0°

E E

(001)

3

EII

[110]

2

3

E┴

2

k 0°

0°

k y x

z

[110]

FIG. 1. (Color online) Sketch of experimental geometry in the SHG and THG studies of (001)-oriented GaAs epilayer.

Taking into account the crystallographic contribution from Eq. (5), the anisotropy of the ED THG polarization can be written as P3ω ∝ −A − 3B + (A − B) cos 4ϕ,

B. Rotational anisotropies of SHG and THG

The rotational anisotropies of SHG and THG are important properties which allow distinguishing different contributions to the nonlinear susceptibilities. Taking into account Eq. (1), the anisotropy of the ED crystallographic SHG polarization can be written as P2ω ∝ cos θ sin ϕ(sin ϑ sin θ sin ϕ

P⊥2ω ∝ cos θ cos ϕ(sin ϑ sin θ sin ϕ + cos ϑ cos ϕ) × (cos ϑ sin θ sin ϕ − sin ϑ cos ϕ) + (sin ϑ sin θ cos ϕ − cos ϑ sin ϕ) × cos θ sin ϕ(cos ϑ sin θ sin ϕ − sin ϑ cos ϕ) + (cos ϑ sin θ cos ϕ + sin ϑ sin ϕ) × cos θ sin ϕ(sin ϑ sin θ sin ϕ + cos ϑ cos ϕ).

(9)

As we noted, the crystallographic SHG signal is symmetry forbidden for the experimental geometry with kω [001] z, when ϑ = θ = 0◦ . Taking into account Eq. (3), the anisotropy of the EDinduced SHG polarization can be written as P2ω ∝ [3c − a − b + 2(a + b − c) cos2 ϕ] cos ϕ,

(10)

for the parallel geometry E2ω Eω , and P⊥2ω ∝ [a − b + c + 2(a + b − c) cos2 ϕ] sin ϕ,

(11)

for the crossed geometry E2ω ⊥ Eω , where the coefficients a = yxxy, b = yxyx, and c = yyyy are related to the three nonvanishing components of χijindkl .

(12)

for the parallel geometry E3ω Eω , and P⊥3ω ∝ (A − B) sin 4ϕ,

(13)

for the crossed geometry E3ω ⊥ Eω , where the coefficients A = yxxy and B = yyyy are related to the two nonvanishing components of χijcrkl . The equations given in this section will be used for fitting experimental data.

+ cos ϑ cos ϕ)(cos ϑ sin θ sin ϕ − sin ϑ cos ϕ), (8) for the parallel geometry E2ω Eω , where ϕ is the polarization rotation angle, ϑ is the rotation angle of the sample around the x axis, and θ is the rotation angle of the sample around the y axis; see Fig. 1. For the crossed geometry E2ω ⊥ Eω , one can write

3

III. EXPERIMENT

The investigated structure was a 10-μm-thick GaAs layer grown by gas phase epitaxy on a semi-insulating GaAs substrate with (001) orientation. The sample has a low defect density of 1014 cm−3 , which is confirmed by the good optical quality and pronounced magnetoexciton resonances in SHG (see Ref. [23]). The sample size is 5 × 7 mm. The electric field was applied via contacts placed on the narrow sides of the sample and was oriented along the [110] direction. The separation between contacts was 7 mm. Voltages up to V = 2 kV were applied at a temperature of T = 5 K. At this temperature, the sample resistivity was 30 M and it decreased down to 20 M when the sample was heated to room temperature T = 293 K. The maximum external electric field applied to the sample was E = 2.9 kV/cm, where E is the field inside the crystal and = 12.9 is the dielectric constant of GaAs [42]. SHG and THG spectra of (001)-oriented GaAs were recorded in transmission geometry using 8 ns light pulses with a 10 Hz repetition rate, generated by an optical parametric oscillator pumped by the third harmonic of a solid-state Nd:YAG laser. The SHG and THG signals were spectrally isolated by means of bandpass optical filters as well as a monochromator and detected by a liquid-nitrogen-cooled charge-coupled-device (CCD) camera. Both SHG and THG were detected in the vicinity of the GaAs band gap Eg = 1.519 eV at low temperature T = 5 K. The fundamental laser energies were tuned from 0.754 to 0.762 eV for SHG and from 0.503 to 0.508 eV for THG. Since nonlinear multiphoton optical transitions are symmetry sensitive, the incoming

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fundamental and outgoing SHG and THG polarizations play an important role for the rotational anisotropy patterns of the SHG and THG intensities, which will be used for the theoretical description of the different nonlinear optical contributions. The incoming fundamental light polarization was controlled by a half-wave plate oriented at an azimuthal angle ϕ. For ϕ = 0◦ , the incoming fundamental light is polarized along the y axis, and for ϕ = 90◦ , the light is polarized along the x axis. More details are given in Ref. [26]. IV. EXPERIMENTAL RESULTS A. Electric-field-induced SHG

In order to study the effects of an electric field on the SHG signals, which may be weak in comparison with the intensity of the crystallographic SHG signal, we use the experimental approach that has already been tested in studies of the magnetic-field-induced SHG in GaAs [21,22]. We choose the experimental geometry kω [001] for which the crystallographic SHG is symmetry forbidden. As one can see in Fig. 2, the SHG signals vanish in the exciton spectral range around the GaAs band gap. When the electric field is applied, a narrow resonant SHG peak at 2ω = 1.516 eV appears close to the 1s exciton energy at E1s = 1.5152 eV [43]. The integral peak intensity increases approximately quadratically with the electric field strength; see the inset in Fig. 2. The azimuthal dependencies of the induced nonlinear signals are important properties allowing one to distinguish different nonlinear contributions. Figure 3 shows the rotational anisotropies for SHG. As for (001)-oriented GaAs, no SHG signals are expected and anisotropies without an external field can only be observed due to a small sample misalignment. The SHG anisotropy is twofold for the geometry E2ω Eω and deformed sixfold for the geometry E2ω ⊥Eω . With an applied

FIG. 2. (Color online) SHG spectra of GaAs measured in different external electric fields in the kω [001] geometry. ϕ = 0◦ . Spectra are shifted vertically for clarity. Inset: Integrated SHG intensity as a function of applied electric field E.

FIG. 3. (Color online) Rotational anisotropies of the SHG intensity measured for GaAs at 1.516 eV in the geometries E2ω Eω (I2ω ) and E2ω ⊥Eω (I⊥2ω ). Points give experimental data; lines are fits by I 2ω ∝ |P2ω |2 with the nonlinear polarizations described by Eqs. (8)–(11).

electric field E, the SHG anisotropies are drastically changed and the SHG intensity is strongly increased in the direction parallel to E (compare upper panels in Fig. 3 at ϕ = 0◦ and the inset in Fig. 2). The SHG anisotropies are deformed fourfold for both geometries. The induced SHG is strongest in the parallel geometry E2ω Eω . B. Electric-field-induced THG

Results on the electric-field-induced THG are collected in Fig. 4. Here we also use the experimental geometry with kω [001], the same as in the SHG experiments, but contrary to the SHG signals, THG is allowed in this geometry in the ED approximation. This is evident from observation of a strong resonant signal in the absence of the electric field. Surprisingly, the THG signals decrease systematically down to about 50% of the initial signal level with the electric field increased up to E = ±2.9 kV/cm. The integrated THG peak intensity as a function of electric field is shown in the inset of Fig. 4. Figure 5 shows the rotational anisotropies for the THG. The anisotropy with and without external electric field have identical shapes, but are appreciably different for the E3ω Eω and E3ω ⊥ Eω geometries. For the E3ω Eω geometry, the THG rotational anisotropy has a squarelike shape, while for E3ω ⊥ Eω geometry, it shows a distorted fourfold pattern. The experimental results presented in this section show that the applied electric field has strong effects on both the SHG and THG signals related to the 1s exciton resonance. The induced intensity variations in SHG and THG demonstrate opposite trends. Worthwhile to note is that in the whole range of applied electric fields, we did not find a notable shift or broadening of the exciton resonance, which may be caused by the Stark effect. Obviously, the electric fields are too weak for that.

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PHYSICAL REVIEW B 92, 085202 (2015) V. THEORY

FIG. 4. (Color online) THG spectra of GaAs measured in different external electric fields in the kω [001] geometry. ϕ = 0◦ . Spectra are shifted vertically for clarity. Inset: The integrated THG intensity a function of applied electric field E.

Also we did not find any signals at the energies of 2s and 2p exciton states, which can be explained by their weak oscillator strength compared to the 1s exciton state. Note that in the spectral range above the 1s exciton resonance corresponding to the band-to-band optical transitions, no valuable changes of SHG or THG signals have been found in electric fields. This observation suggests the important role of the exciton resonance in the reported effects.

In order to reveal the microscopic mechanisms responsible for the electric-field-induced SHG and THG, the effect of the field on the exciton states has to be analyzed. For this consideration, the spin properties of the excitons as well as their translational motion can be ignored. The excitons formed from the conduction band of 6 symmetry and the degenerate valence band of 8 symmetry are eightfold degenerate for exciton wave vector kexc = 0. The relative motion of the electron and the hole can be described similar to the motion of a hole from the 8 valence band bound to an acceptor [44,45]. The resulting symmetry of the exciton states is not the same as of pure s and p states as in the hydrogenlike model. In fact, the lowest even s state (orbital momentum l = 0) has admixtures of d symmetry wave functions (orbital momentum l = 2) and the lowest odd p state (orbital momentum l = 1) has admixtures of the f symmetry functions (orbital momentum l = 3). These admixtures are controlled by the ratio β of the reduced masses of the light-hole exciton to the heavy-hole exciton: the admixture is large for small values of this ratio and can be neglected for β close to unity. In GaAs, the value of β is close to unity because of the small effective electron mass compared to the hole masses. Accounting for these small admixtures would only change the values of the matrix elements describing the electric-field-induced mixing of even and odd states, but does not lead to qualitative changes of the results. In the following consideration, we therefore neglect this admixture and calculate the electric-field-induced matrix elements using the hydrogenlike radial wave functions R10 (r), R20 (r), and R21 (r) of the 1s, 2s, and 2p exciton states [35], respectively (r is the coordinate of the relative electron-hole motion). In a crystal of cubic symmetry at zero electric field, the i 2p exciton state with energy E2p is degenerate with respect to the three x, y, and z directions of the envelope function polarization. The electric field inside the crystal given by E = E(ex ,ey ,ez ) mixes the states of s symmetry with the 2p states, which have nonzero projection on the electric field direction. In the hydrogenlike model, the Hamiltonian including the electric field in the basis of the envelope functions 1s , 2s , and 2p = ex 2p,x + ey 2p,y + ez 2p,z is given by ⎛

i E1s ˆ ⎝ 0 H (E) = 3γ eaB E

FIG. 5. (Color online) Rotational anisotropies of the THG intensities measured for GaAs at 1.516 eV in the geometries E3ω Eω and E3ω ⊥Eω . Points give the experimental data; lines are fits by I 3ω ∝ |P3ω |2 with the nonlinear polarizations described by Eqs. (12) and (13).

0 i E2s 3eaB E

⎞ 3γ eaB E 3eaB E ⎠, i E2p

(14)

where aB = 11.5 nm is the exciton Bohr radius in GaAs and e is the elementary charge. The main diagonal elements are the i i i energies E1s = 1.5152, E2s = 1.5183, and E2p = 1.5189 eV [46,47] of the unperturbed exciton states. The off-diagonal elements describe the mixingof the exciton states ∞due to the Stark ∞ effect. The factor γ = 0 R10 R21 r 3 dr/ 0 R20 R21 r 3 dr = 8 − √12 236 ≈ −0.25 takes into account the different electric-fieldcoupling strength between the 1s/2p and the 2s/2p excitons. Diagonalization of the Hamiltonian (14) gives the energies of the new eigenstates E f (f = 1s,2s,2p) of the exciton states modified by the electric field E. For simplicity, we notate final mixed states by zero-field states from which they originate.

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The resulting wave functions contain all three unperturbed components due to mixing by the electric field, f

f

f

f (E f ) = C1s (E)1s + C2s (E)2s + C2p (E)2p , (15)

with the unperturbed wave functions j with j = 1s,2s,2p f and the field-dependent admixture coefficients Cj (E). In analogy with Eqs. (A18)–(A20) from Ref. [26], we write these coefficients as

i γ 3eEaB E f − E2s =

, i 2 i 2 i 2 i 2 E f − E1s E f − E2s + (3eEy aB )2 E f − E1s + γ 2 E f − E2s i 3eEaB E f − E1s f C2s (E) =

, i 2 i 2 i 2 i 2 E f − E1s E f − E2s + (3eEy aB )2 E f − E1s + γ 2 E f − E2s f f i i E − E2s E − E1s f C2p (E) =

. i 2 i 2 i 2 i 2 E f − E1s E f − E2s + (3eEy aB )2 E f − E1s + γ 2 E f − E2s f C1s (E)

The calculated energies of the mixed exciton eigenstates which emerge from the 1s, 2s, and 2p excitons at zero field as a function of the electric field E are shown in Fig. 6(a) by lines. The 1s energy is nearly constant up to E = 3 kV/cm. The 2s/2p excitons undergo a repulsion caused by their strong mixing with each other. Figure 6(b) shows the behavior of the admixture coefficients for the 1s exciton state in electric field.

(16a)

(16b)

(16c)

This state obtains an increasing admixture of the 2p state with increasing electric field. As a result, at E = 3 kV/cm, the 1s 1s C1s coefficient decreases from 1.0 down to 0.83, while the C2p coefficient increases from 0 to 0.55. The 1s and 2s states are not mixed directly with each other by the electric field, but they become mixed with the 2p state, which results in a weak increase 1s of the C2s coefficient from 0 up to 0.03 at the largest field. Let us turn now to calculation of the nonlinear susceptibilities. In this analysis, we consider only the geometry kω z and apply the electric field perpendicular to the k vector: E = E(ex ,ey ,0). Therefore, only the induced susceptibility components with l = x,y in Eq. (3) for SHG and the crystallographic susceptibility in Eq. (5) for THG have to be considered. For calculating the induced SHG susceptibilities, we follow the general scheme in the framework of second-order perturbation theory (see Ref. [26]). For the SHG susceptibilities in electric field E in Eq. (4), one obtains ind ind χyyyy (E f ,E) = 2χyxyx (E f ,E)

1 f f f ∝ C2p (E) C1s (E) + √ C2s (E) , (17) 8 ind χyxxy (E f ,E) = 0.

√ The factor |R20 (0)/R10 (0)| = 1/ 8 takes into account the reduced oscillator strength of the 2s exciton. Keeping in mind 1s that the C2s coefficient is negligibly small [see Fig. 6(b)], one obtains, for the 1s state, ind 1s 1s χyyyy (E 1s ,E) ∝ C2p (E)C1s (E).

FIG. 6. (Color online) (a) Calculated energy levels of the 1s, 2s, and 2p exciton states at zero field vs applied electric field. Symbols show experimental peak energies for the SHG and THG lines. (b) Calculated electric field dependence of the admixture coefficients Cj1s (E) of the 1s exciton wave function. (c) Electric field dependence of the normalized SHG intensity: symbols are experimental data from the inset of Fig. 2 and the line is the calculation after Eq. (18). (d) Electric field dependence of the normalized THG intensity: symbols are experimental data from the inset of Fig. 4 and the line is the calculation with Eq. (22). For the electric field in all panels, the values outside the crystal are given, i.e., they correspond to E.

(18)

One can see from Eq. (17) that for the considered exciton mixing mechanism, the electric-field-induced SHG is described by only one independent component of the susceptibility tensor. Using the relations for the induced susceptibilities, the rotational anisotropy expressions (10) and (11) can be simplified to P2ω ∝ (5 − 2 cos2 ϕ) cos ϕ,

(19)

for the parallel geometry E2ω Eω , and P⊥2ω ∝ (3 − 2 cos2 ϕ) sin ϕ, for the crossed geometry E2ω ⊥ Eω .

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(20)



As discussed above, there is no electric-field-induced contribution to the THG in this geometry. However, the crystallographic contribution can be modified by the electric field. Indeed, for all THG crystallographic susceptibilities that are nonzero for the s symmetry exciton, one gets in electric field at the exciton state with energy E f :

2 1 f f χijcrkl (E f ,E) ∝ C1s (E) + √ C2s (E) . (21) 8 For the particular case of the 1s state, one can see from Fig. 6(b) 1s that C2s is small, and therefore

1s 2

1s 2 χijcrkl (E 1s ,E) ∝ C1s (E) ≈ 1 − C2p (E) . (22) Here, i,j,k,l = x,y. The reduction of the THG signal by the electric field at the particular energy is due to the reduction of the s symmetry state contribution to the wave function of this state. VI. DISCUSSION

The theoretical approach developed in the previous section was used for calculating electric field dependencies of SHG and THG intensities. To that end, the respective nonlinear polarizations, which are proportional to the nonlinear susceptibilities, have to be squared. The results of these calculations for the 1s exciton state are shown by the green lines in Figs. 6(c) and 6(d). They are in good agreement with the experimental data shown by the symbols. It is remarkable that two different trends, namely the increase of the SHG intensity and the decrease of the THG intensity, are well reproduced. This confirms the validity of the chosen theoretical approach and of the suggested microscopic mechanism for electric-field-induced SHG and THG on the GaAs excitons. In (001)-oriented GaAs, the electric field induces SHG signals and reduces the crystallographic THG intensity. In the case of SHG, the field-induced effect is caused by the nonlinear polarization χijindkl (E) [see the second term in Eq. (3)], which induces new contributions. In the case of THG, the field-induced action emerges from the crystallographic susceptibility χijcrkl (E); see the first term in Eq. (5). As discussed in Sec. II for precisely (001)-oriented GaAs in the kω [001] z geometry, no SHG signals are expected. However, small SHG signals without an external field are observed due to a small sample misalignment. The fitted curves for the SHG intensity I 2ω ∝ |P2ω |2 using Eqs. (8)–(11) are shown in Figs. 3(a) and 3(b) by the lines. One finds good agreement with the experimental data shown by data points in these figures. The small background due to the crystallographic SHG contribution leads to a Fano-like effect close to the exciton resonance, which gives rise to an asymmetric shape of the SHG resonance; see Fig. 2. This is explained by an optical phase shift

[1] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984). [2] R. W. Boyd, Nonlinear Optics (Academic/Elsevier, Burlington, 2008).

of the electric-field-induced SHG signal across the peak energy, while the crystallographic background SHG contribution has a constant phase. The resulting constructive and destructive interference below and above the peak maximum position leads to the asymmetric shape of the exciton resonance. The SHG intensity dependence on the electric field in the inset of Fig. 2 is not perfectly symmetric with respect to inversion of the field polarity. This is also due to constructive and destructive interference of the crystallographic and induced nonlinear contributions to the SHG intensity. The THG dependencies are almost symmetric with respect to the field polarity inversion. Calculations of the THG rotational anisotropies according to Eqs. (12) and (13) are shown in Figs. 5(a) and 5(b) by the lines. At zero electric field [Fig. 5(a)], the anisotropy is described solely by χijcrkl [see the first term in Eq. (5)], which describes well the experimental data. In the electric field [Fig. 5(b)], the shape of the anisotropy is the same, only its intensity decreases. It is also well fitted by accounting only for the field effect on the crystallographic THG susceptibility χijcrkl (E). This confirms that the induced THG susceptibility is small compared with the crystallographic one. One can see in Fig. 6(a) that the experimental data points for the SHG and THG line maximum are shifted by about 1 meV to higher energy relative to the exciton resonance measured in linear spectroscopy. The latter corresponds with the calculated green line and has been checked for the studied sample by measuring a reflectivity spectrum (not shown here). We suggest that this shift may be caused by the exciton-polariton dispersion in GaAs for which the wave-vector conservation condition is better fulfilled at the shifted energy. In conclusion, we have observed strong electric-fieldinduced phenomena related to SHG and THG in bulk semiconductor GaAs. Narrow SHG and THG lines are observed around the 1s exciton. The applied electric field acts oppositely on the SHG and THG intensities; it induces the SHG signal for (001)-oriented GaAs but reduces the crystallographic THG signal. Model considerations based on the mixing of exciton states of different parities due to the Stark effect show that both the SHG and THG intensity variations are mainly due to the mixing of the 1s and 2p exciton states. The effect should be observable also on excitons in other semiconductors, where it is expected to scale accordingly with the Coulomb interaction. ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft via Sonderforschungsbereich TRR142, the Russian Government Project No. 14.B25.31.0025, and the Russian Foundation for Basic Research (Projects No. 13-02-00754 and No. 15-52-12015). The research stay of V.V.P. at TU Dortmund University was supported by the Alexander-von-Humboldt Foundation.

[3] E. Garmire, Resource letter NO-1: Nonlinear optics, Am. J. Phys. 79, 245 (2011). [4] A. H. Reshak, in Second harmonic generation signal from biological materials using multi-functional two-photon laser

085202-7

D. BRUNNE et al.

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14] [15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]


scanning microscopy, Laser Scanning, Theory and Applications, edited by Ch.-Ch. Wang (InTech, 2011), p. 233. M. Fiebig, V. V. Pavlov, and R. V. Pisarev, Second-harmonic generation as a tool for studying electronic and magnetic structures of crystals: Review, J. Opt. Soc. Am. 22, 96 (2005). S. A. Denev, T. T. A. Lummen, E. Barnes, A. Kumar, and V. Gopalan, Probing ferroelectrics using optical second harmonic generation, J. Am. Ceram. Soc. 94, 2699 (2011). J. Wang, B.-F. Zhu, and R.-B. Liu, Second-order nonlinear optical effects of spin currents, Phys. Rev. Lett. 104, 256601 (2010). H. Yokota, J. Kaneshiro, and Y. Uesu, Optical second harmonic generation microscopy as a tool of material diagnosis, Phys. Res. Intl. 2012, 704634 (2012). C. Y. Fong and Y. R. Shen, Theoretical studies on the dispersion of the nonlinear optical susceptibilities in GaAs, InAs, and InSb, Phys. Rev. B 12, 2325 (1975). M.-Z. Huang and W. Y. Ching, Calculation of optical excitations in cubic semiconductors. II. Second-harmonic generation, Phys. Rev. B 47, 9464 (1993). W. Y. Ching and M.-Z. Huang, Calculation of optical excitations in cubic semiconductors. III. Third-harmonic generation, Phys. Rev. B 47, 9479 (1993). J. L. P. Hughes and J. E. Sipe, Calculation of second-order optical response in semiconductors, Phys. Rev. B 53, 10751 (1996). J. L. P. Hughes and J. E. Sipe, Comparison of calculated optical response in cubic and hexagonal II-VI semiconductors, Phys. Rev. B 58, 7761 (1998). J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000). S. N. Rashkeev, W. R. L. Lambrecht, and B. Segall, Efficient ab initio method for the calculation of frequency-dependent second-order optical response in semiconductors, Phys. Rev. B 57, 3905 (1998). S. N. Rashkeev, W. R. L. Lambrecht, and B. Segall, Secondharmonic generation in SiC polytypes, Phys. Rev. B 57, 9705 (1998). J. L. Cabellos, B. S. Mendoza, M. A. Escobar, F. Nastos, and J. E. Sipe, Effects of nonlocality on second-harmonic generation in bulk semiconductors, Phys. Rev. B 80, 155205 (2009). S. Bergfeld and W. Daum, Second-harmonic generation in (2) , Phys. GaAs: Experiment versus theoretical predictions of χxyz Rev. Lett. 90, 036801 (2003). H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS, Phys. Rev. B 58, 10494 (1998). R. V. Pisarev, B. Kaminski, M. Lafrentz, V. V. Pavlov, D. R. Yakovlev, and M. Bayer, Novel mechanisms of optical harmonics generation in semiconductors, Phys. Status Solidi B 247, 1498 (2010). V. V. Pavlov, A. V. Kalashnikova, R. V. Pisarev, I. Sänger, D. R. Yakovlev, and M. Bayer, Magnetic-field-induced secondharmonic generation in semiconductor GaAs, Phys. Rev. Lett. 94, 157404 (2005). I. Sänger, D. R. Yakovlev, R. V. Pisarev, V. V. Pavlov, M. Bayer, G. Karczewski, T. Wojtowicz, and J. Kossut, Spin and orbital quantization of electronic states as origins of second

[23]

[24]

[25]

[26]

[27]

[28] [29] [30]

[31]

[32]

[33] [34] [35] [36]

[37]

[38]

[39]

085202-8

harmonic generation in semiconductors, Phys. Rev. Lett. 96, 117211 (2006). I. Sänger, D. R. Yakovlev, B. Kaminski, R. V. Pisarev, V. V. Pavlov, and M. Bayer, Orbital quantization of electronic states in a magnetic field as the origin of second-harmonic generation in diamagnetic semiconductors, Phys. Rev. B 74, 165208 (2006). B. Kaminski, M. Lafrentz, R. V. Pisarev, D. R. Yakovlev, V. V. Pavlov, V. A. Lukoshkin, A. B. Henriques, G. Springholz, G. Bauer, E. Abramof, P. H. O. Rappl, and M. Bayer, Spininduced optical second harmonic generation in the centrosymmetric magnetic semiconductors EuTe and EuSe, Phys. Rev. Lett. 103, 057203 (2009). M. Lafrentz, D. Brunne, B. Kaminski, V. V. Pavlov, A. V. Rodina, R. V. Pisarev, D. R. Yakovlev, A. Bakin, and M. Bayer, Magneto-Stark effect of excitons as the origin of second harmonic generation in ZnO, Phys. Rev. Lett. 110, 116402 (2013). M. Lafrentz, D. Brunne, A. V. Rodina, V. V. Pavlov, R. V. Pisarev, D. R. Yakovlev, A. Bakin, and M. Bayer, Second-harmonic generation spectroscopy of excitons in ZnO, Phys. Rev. B 88, 235207 (2013). J. R. Meyer, C. A. Hoffman, F. J. Bartoli, and L. R. RamMohan, Intersubband second-harmonic generation with voltagecontrolled phase matching, Appl. Phys. Lett. 67, 608 (1995). R. W. Terhune, P. D. Maker, and C. M. Savage, Optical harmonic generation in calcite, Phys. Rev. Lett. 8, 404 (1962). J. B. Khurgin, Current induced second harmonic generation in semiconductors, Appl. Phys. Lett. 67, 1113 (1995). O. A. Aktsipetrov, V. O. Bessonov, A. A. Fedyanin, and V. O. Val’dner, DC-induced generation of the reflected second harmonic in silicon, JETP Lett. 89, 58 (2009). B. A. Ruzicka, L. K. Werake, G. Xu, J. B. Khurgin, E. Ya. Sherman, J. Z. Wu, and H. Zhao, Second-harmonic generation induced by electric currents in GaAs, Phys. Rev. Lett. 108, 077403 (2012). E. F. Gross, B. P. Zakharchenya, and L. M. Kanskaya, Investigation of the Stark exciton effect in oriented monocrystals of copper oxide, Soviet Phys. Solid State 3, 706 (1961). J. D. Dow and D. Redfield, Electroabsorption in semiconductors: the excitonic absorption edge, Phys. Rev. B 1, 3358 (1970). E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures (Springer, Berlin, 2007). H. A. Bethe and E. E. Salpeter, Quantum Mechanics of Oneand Two-electron Atoms (Springer-Verlag, Berlin, 1957). V. T. Agekyan, B. S. Monozon, and I. P. Shiryapov, The fine structure of Wannier-Mott excitons in a cubic crystal and its behavior in an electric field, Phys. Status Solidi B 66, 359 (1974). T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation, Opt. Lett. 27, 628 (2002). R. Haidar, Ph. Kupecek, and E. Rosencher, Nonresonant quasiphase matching in GaAs plates by Fresnel birefringence, Appl. Phys. Lett. 83, 1506 (2003). A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Phase matching using an isotropic nonlinear optical material, Nature (London) 391, 463 (1998).



[40] R. R. Birss, Symmetry and magnetism (North-Holland, Amsterdam, 1966). [41] S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Institute of Physics Publishing, Bristol and Philadelphia, 1995). [42] C. J. Johnson, G. H. Sherman, and R. Weil, Far infrared measurement of the dielectric properties of GaAs and CdTe at 300 K and 8 K, Appl. Opt. 8, 1667 (1969). [43] Note that in the spectral range above the 1s exciton resonance corresponding to band-band optical transitions, no valuable changes of SHG or THG signals have been found in electric fields. This suggests the important role of the exciton resonance in the reported effects.

[44] B. L. Gel’mont and M. I. D’yakonov, Acceptor levels in diamond-type semiconductors, Fiz. Tekh. Poluprov. 5, 2191 (1971) [,Sov. Phys.–Semicond. 5, 1905 (1971)]. [45] A. Baldereschi and N. O. Lipari, Spherical model of shallow acceptor states in semiconductors, Phys. Rev. B 8, 2697 (1973). [46] J. S. Michaelis, K. Unterrainer, E. Gornik, and E. Bauser, Electric and magnetic dipole two-photon absorption in semiconductors, Phys. Rev. B 54, 7917 (1996). [47] S. I. Gubarev, T. Ruf, M. Cardona, and K. Ploog, Magnetoluminescence of GaAs in the quasiclassical limit, Phys. Rev. B 48, 1647 (1993).

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