1. INTRODUCTION

Nonlinear phenomena have been widely studied since the invention of the laser by Townes et al. [1]. Subsequently, second harmonic generation (SHG) [2], third harmonic generation (THG) [3–5], higher harmonic generation (HHG) [6–9] and mixing frequency techniques [10–13] have been intensively studied theoretically and experimentally.

Another nonlinear optical phenomenon, studied soon afterwards, is electric-field-induced second harmonic (EFISH) generation, first observed experimentally by Terhune and coworkers using calcite crystals [5]. Later, the dependence of SHG with respect to a DC electric field for silver and silicon surfaces was reported by Lee and coworkers [14], where they already observed that for a $p$-polarized light the EFISH effect is large; however, for $s$-polarized light, it is almost negligible.

Experimentally it is worth mentioning that even in inversion symmetric materials, where dipolar SHG is forbidden (${\chi }_{ijk}^{\text{dipole}}=0$), bulk SHG can still occur due to symmetry breaking of the applied DC field. Therefore, the argument that only surfaces—and not the bulk—of Si contribute to dipolar SHG in inversion symmetric materials is then only correct if no static electric field is present in the sample. EFISH describes the interaction of matter with two fundamental electromagnetic waves with the addition of a DC electric field; thus, these three electric fields generate together a new signal with the doubled frequency of the impinging beams. SHG, generated through EFISH, is therefore a third-order nonlinear phenomenon. Third-order coefficients for EFISH have been measured experimentally for gases, liquids [15] and glass [16]. Also, EFISH was deeply and intensively studied experimentally, as well as theoretically, by Aktsipetrov and coworkers for metal-oxide-semiconductors structures and $\mathrm{Si}\text{-}{\mathrm{SiO}}_2$ interfaces [17], where in some cases it has been possible to separate the second harmonic (SH) signal originating in the bulk from the surface contribution by varying the applied DC field [18]. A very interesting quantum mechanical ab initio calculation of EFISH has been reported by Kikuchi and Tada [19]. Finally, it is worth mentioning that in the important contribution of Ohlhoff et al. [20] on EFISH at metal-oxide-semiconductor and metal-semiconductor structures, the difference between the $s$- and $p$-polarized EFISH responses is measured, but not explained. In that paper they compare experimental results with a model based on Green’s function due to Sipe [21] and also characterize the second harmonic intensity signal fitting Fourier coefficients for a series expansion.

In this work we argue based on group theoretical arguments why EFISH for light at normal incidence is not observed in all low index facets. However, for light with oblique incidence, a rather strong $p$-polarized EFISH and a rather weak $s$-polarized EFISH occur. A nonlinear bond model can be very illustrative [22,23] for explaining SHG. In a previous work, we sketched the main ideas for understanding the origin of EFISH using effective third-rank tensors and discussed the resulting symmetry after contracting it with the electric field in the $z$ direction, establishing an analogy with the surface break of the symmetry [24]. Also there, we extend the so-called simplified bond hyperpolarizability model (SBHM) [25,26] for EFISH. In the present paper, we analyze the full fourth-rank tensor description with the symmetry breaking the perturbing electric field, presenting explicitly the 81 elements (many of them zero, as is well known) in terms of the contracted (engineering or Voigt) notation for silicon and zincblende having (001), (011), and (111) facets, for both the $s$- and $p$-polarization.

It is worth mentioning that in this model, we compute the pure EFISH bulk effect of perfect crystalline Si, which will be in any SHG measurement superimposed with the coherent contribution of surfaces, crystal defects, and grain boundaries in the case of polycrystalline Si, etc. To describe effects at defect or impurity sites breaking the $O_h$ symmetry, group theory can be used in the context of crystal field theory (see, e.g., Ref. [27]), but this is beyond our aim of describing crystalline and defect free semiconductors.

The paper is organized as follows. Section 2 is a brief discussion of the “well-known” fourth-rank tensors or third-order susceptibility predicted by group theory for Si applied to the (001), (011), and (111) facets. The consequence of “symmetrizing” the fourth-rank tensor due to the presence of two identical electromagnetic fields is explained, reducing the number of independent elements to two. The resulting nonlinear polarizabilities (for incident $p$- and $s$-polarization) are calculated for these “symmetrized” tensors for each facet, and the resulting SH signal is provided. In Section 3, a classical oscillator model is used for a pictorial view of the interaction between the electric fields and the electrons in the crystal, pointing out the importance of the symmetry breaking through the direction of the oscillating electric field. Finally, conclusions are given in Section 4.

2. GROUP THEORETICAL DESCRIPTION OF EFISH FOR ZINCBLENDE AND DIAMOND (001), (111), AND (011) FACETS

The symmetries associated with a crystal can be described using matrices and, when connected to symmetries of space and time, by tensors. All the matrices describing the operations in a crystal form a group or, more specifically, a point group. The surfaces and interfaces are a natural break of the symmetry in a crystal, but this breaking can also be generated by other external factors. For example, in the nonlinear phenomenon known as EFISH generation, the break of symmetry is due to the application of a static electric field (DC electric field). Mathematically, for these two frequencies, $\omega$ and zero, the linear and nonlinear response of the system is modeled through the resulting polarization:

In particular, the second term in Eq. (1) for $\overrightarrow{P}(2\omega )$ precisely describes the correspondent third-order phenomena, which can be rewritten in terms of its components for the particular EFISH case by

As mentioned before, ${\chi }_{ijkl}^{(3)}$ is a fourth-rank tensor; the “$i$” index is the resulting component of the polarization. The indices “$j$” and “$k$” are contracted with the electric field, and these indices can be permuted, whereas the index “$l$” is contracted with the DC field; hence, it cannot be permuted with the other indices. A general fourth-rank tensor can be represented as a $3\times 3$ matrix, which also has $3\times 3$ matrices as elements, such as [28]

There are 81 elements, which are labeled as follows. The first index “$i$” in ${\chi }_{ijkl}^{(3)}$ corresponds to the rows and the second index “$j$” to the columns in the main matrix (the external one). In the same way, the indices “$k$” and “$l$” will correspond to the usual way of labeling a $3\times 3$ matrix, namely, the rows and columns in the inner $3\times 3$ matrix, respectively. For example, the element ${\chi }_{3213}$ corresponds to the element in the external matrix within position third row and second column but now this element is also a matrix and inside that matrix we should look for the element in the first row and third column.

On the other hand, there is a more compact way to label the elements in the tensor, known as the engineering or Voigt matrix representation. Using this representation, the tensor in Eq. (3) can be written in the form of a $6\times 6$ matrix, given by

However, the transformation rule for tensors can only be applied under the form given by Eq. (3), which is the explicit fourth-rank tensor with four subscripts. The general transformation is then

A. EFISH of Zincblende (001) Facet

It is well known that silicon (zincblende) crystals have $O_h(T_d)$ symmetry. However, when we study their surfaces, the symmetry is reduced, depending on the direction in which the crystal has been cut. Also, for the case of a static electric field applied in a particular direction, the symmetry is broken due to the perturbation, and the resulting group has less symmetry elements.

On the other hand, the response of the conventional cell can be very well simulated through SBHM only by using one tetrahedral element. A tetrahedron has $T_d$ symmetry, but when the symmetry is broken in one direction, as mentioned before, the symmetry is lowered. When a DC electric field is applied in the $z$ direction in the coordinate system shown in Fig. 1, the electrons are shifted from their equilibrium position inside the bulk (in the middle between the atoms), and the bonds are no longer equivalent or equal. Thus, the symmetry is now reduced to $C_{2v}$ point group, with a two-fold axis of rotation around $z$ [20].

 

Fig. 1. Conventional silicon cell.

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Mathematically, the tensor for an $O_h$ as well as a $T_d$ point group can be stated in contracted notation, without any symmetry breaking through an external DC field [23]:

 

Fig. 2. (a) SHG, inside the high index nonlinear material, has a very small angle with the normal. There is a coherent contribution just for a certain depth $d$. (b) High index prims keeps a large angle with the normal inside the nonlinear material.

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This radiation, however, has only a $z$-component; that is, it is equivalent that the polarization direction for the frequency doubled light of the dipoles is along $\widehat{z}$. The resulting radiation can therefore not be detected in normal incidence transmission or reflection SHG, because in the ambient medium (air or vacuum), the polarization of the displacement field has to be normal to the propagation direction. Under oblique incidence, however, the $s$-polarized EFISH signal should become weakly measurable in transmission (because refraction is towards the surface normal) and to an even weaker extent in reflection.

The notation for the angle of incidence and refraction and the second harmonic angles is depicted in Fig. 2(a). Due to Snell’s law, the refracted beam always propagates rather close to the surface normal in silicon; so, the radiated intensity proportional to ${|\overrightarrow{P}|}^2$ will be rather weak and will only be detected for a $p$-polarization, which will be discussed now.

For a $p$-polarization, the electric field will only have components in the $x$ and $z$ directions ${\overrightarrow{E}}_p=E_p(-\cos {\theta }_2(\omega ),0,\sin {\theta }_2(\omega ))$, and the DC field is again along $z$. The result is

It is worth noticing that, in general, $n_i(\omega )\ne n_i(2\omega )$ and ${\theta }_i(\omega )\ne {\theta }_i(2\omega )$ ($i=1$, 2). Additionally, for optically dense materials, the propagation direction approaches the normal, and there is a very small contribution for the $x$-component in the $p$-polarization when the incident angle is small. Therefore, for normal incidence, there is no measurable EFISH response for the $s$-polarization, and it will only show up for the $p$-polarization, whereas Eq. (10) reduces to Eq. (9) as should be under normal incidence.

B. EFISH of Zincblende (111) Facet

As the orientation is different, we need to rotate the crystal to a system of reference with the $z$-axis perpendicular to the plane (111). To do this, two rotations must be performed; the first one is around the $z$-axis by an angle of $\pi /4$; then in the new system of reference, the second rotation will be around the $y$-axis by $\beta /2$, where $\beta$ is the angle between the bonds, which it is equal to 109.47° for silicon. Therefore, after applying Eq. (5) to the tensor given in Eq. (7) and taking account of the “symmetrization” in indices “$j$” and “$k$,” the resulting tensor is

This tensor still has $O_h$ symmetry, but in the new system of reference it looks rather different. Two independent elements remain, but now there are five different values in the tensor. Although, when this fourth-rank tensor Eq. (13) is contracted with the DC field, the resulting symmetry group for the third-rank tensor is $C_{3v}$ (the three-fold axis of rotation is the $z$ direction), as it should be. This can be checked directly doing the contraction, as mentioned before, or for physical arguments when the symmetry operations involving the $z$ coordinate are eliminated.

As before, we can contract the tensor in Eq. (13) with all the electromagnetic fields defined in the same way and get the final expressions for the incident polarizations. For the case of $s$-polarization,

C. EFISH of Zincblende (110) Facet

For the same reasons exposed in the previous section, here we are going to start with the tensor given in Eq. (7) after the symmetrization in the indices “$j$” and “$k$.” However, the rotations will be first $\pi /4$ around the $z$-axis and then $\pi /2$ around the $y$-axis. These two combined rotations result in the following tensor:

Please note that even when the number of independent coefficients in the tensor is the same, there are four different values in it. The contraction of this tensor Eq. (17) with the DC field and the electromagnetic fields results in

In analogy with Eq. (15), this expression for ${\overrightarrow{P}}_p(2\omega )$ can be used as a test for relative magnitudes via Brewster’s angle. It is interesting that in this case, if ${\theta }_1(\omega )=0$, a different result is obtained than the one for $s$-polarization, whereas in the previous directions Si(001) and Si(111), they reduce to the same, just changing $E_y\to E_p$. This means that for the Si(110) facet, the directions $x$ and $y$ are not fully equivalent, a fact that is already known from linear optics. In linear optics, the polarization sensitive technique reflection difference/anisotropy spectroscopy can distinguish crystal orientations for the case of Si(110) facets [32].

As a summary of this section, we would like to point out that for Si(001) and Si(110) no EFISH signal should be detected for normal incidence, neither in transmission nor in reflection, because the second harmonic electric field only has a component for the $z$ direction. In a Si(111) facet, a strong EFISH signal for both the $s$- and $p$-polarizations should be detected due to the breaking of symmetry in the bulk for this bond configuration. In addition, we point out that the electric field generated through the EFISH effect, which is propagating along the $z$ direction, could interfere destructively at the detector if the region where the static electric field is applied is longer than the coherence length between the fundamental and harmonic. However, for usual doping concentrations and semiconductor depletion regions, this will hardly occur in transmission, but it could be significant in reflection.

In the next section, we will present a phenomenological explanation of EFISH using a bond model, where the physics of this phenomenon is very clear by considering the movement of electrons along the bonds due to the action of the electromagnetic fields.

3. CLASSICAL MODEL FOR EFISH

There is no dipole-related SHG signal in silicon and, in general, in any centrosymmetric crystal due to bulk contribution, at least attributed to a second-order nonlinear phenomenon. Still, there could be contributions from several other mechanisms, for example, built-in strain [22], but we are only going to discuss EFISH here, where the breaking of the symmetry in the electronic distribution originates from a static electric field applied in the $z$ direction.

Now, we would like to pictorially explain EFISH mechanism, arguing about the break of the symmetry in a conventional silicon cell and in the physics of the modifications suffered by the electric charge density along the bonds when the DC electric field is applied.

As shown mathematically before, there is no EFISH signal for normal incidence ($s$-polarization) for facets Si(001) and Si(110). This is because even when breaking the symmetry (in the $z$ direction because of the DC field), there are always bonds opposing each other [see Figs. 3(a) and 3(b)], and canceling in this way, the signal contribution is generated by the incoming radiation, with only one component in its electromagnetic field. So, even though the symmetry is broken by the static field along $z$, for normal incidence, as seen in Figs. 3(a) and 3(b), the middle blue silicon atom forms a two-dimensional in plane ($x–y$) inversion center.

 

Fig. 3. Equilibrium position of the electric charge is shifted due to the DC field $E_Z$. Facets (a) Si(001), (b) Si(110), and (c) Si(111).

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However, under oblique incidence this statement is not fully true anymore: $p$-polarization has two components in the electromagnetic field, and therefore, due to the interaction of the two orthogonal components driving the electrons, the opposing bonds no longer equilibrate each other, resulting in an EFISH signal.

In the case of the facet Si(111), the DC field also breaks the symmetry in the direction $z$, but here due to the orientation, there is no opposing bond that represents the response of the conventional cell [see Fig. 3(c), where down bonds are below the $xy$ plane and one is oriented directly below the negative $y$-axis]. In fact, the bond parallel to the $z$-axis provides no contribution at all for EFISH under normal incidence, because for normal incidence this wave function has still a 2D ($x–y$) inversion symmetry, whereas the other three bonds pointing down have not (a 2D $x–y$ inversion symmetry) and contribute to EFISH.

For the three bonds, which are tilted with respect to the static electric field, we now define their equation of motion by taking into account the static field potential: classically in terms of the atomic potential only containing even powers [33]. Thus, the potential for each tilted bond material is

The interesting result from Eq. (22) is that the strength parameter $b$ for THG will be related to the second harmonic signal originated through EFISH mechanism.

Remarkably, Kikuchi and Tada [19] have already shown, using quantum mechanical calculations, that EFISH can be modeled with only one independent parameter in the fourth-rank tensor if adequate conditions exist. Therefore, this result is very reasonable and agrees with experimental results for SHG [34,35] and EFISH [20].

4. CONCLUSIONS

The EFISH generation is clearly a bulk phenomenon. Whereas group theory requires only two independent elements in the fourth-rank susceptibility tensor (after symmetrization), the classical oscillator bond model fits the experimental data reasonably well with only one free parameter. Moreover, both approaches predict an EFISH signal for facet Si(111) but none for facets Si(110) and Si(001) for normal incidence. We also propose experimental setups, which can test the validity of SBHM for EFISH generation. In general, these results are valid for zincblende crystals, but it could be difficult to separate the dipole allowed contribution from the EFISH effect. However, we think that an appropriated combination of the angle and polarization dependence should enable one to distinguish between these two phenomena.

The existence of a relationship between the parameter that characterizes THG and the second harmonic signal originated through EFISH is a very interesting result, and the validity should be checked by experiments or ab initio theories.