Interband Theory of Kerr Rotation in Unconventional Superconductors

TL;DR

This paper presents a theory explaining Kerr rotation in unconventional superconductors through interband effects and Coulomb interactions, suggesting detection of a superconducting order parameter in a band far from the Fermi energy.

Contribution

It introduces a strong-coupling interband theory that accounts for Kerr rotation and predicts a superconducting order parameter in unoccupied bands.

Findings

Coulomb interactions can induce order parameters in unoccupied bands.

The theory constrains the form of the order parameter across different bands.

Experimental Kerr rotation may indicate superconductivity in bands far from the Fermi level.

Abstract

Recent experiments have shown rotation of the plane of polarization of light reflected from the surface of some superconductors. This indicates that time reversal and certain mirror symmetries are broken in the ordered phase. The photon energy exceeds the electronic bandwidth, so that completely filled or completely empty bands must play a role. We show that in strong-coupling theory a Coulomb interaction may produce an order parameter in the unoccupied band that explains the observations. The theory puts tight constraints on the form of the order parameter in different bands. We propose that the experiments have detected, for the first time, the existence of a superconducting order parameter in a band far from the Fermi energy.

Equations

$$H-\mu N$$

$$\Delta \left(n,\vec{p}\right)=-\sum _{n^{\prime }{\vec{p}}^{\prime }}F\left(n^{\prime },{\vec{p}}^{\prime }\right)V\left(n,\vec{p};n^{\prime },{\vec{p}}^{\prime }\right)\Delta \left(n^{\prime },{\vec{p}}^{\prime }\right),$$

$$F$$

$${\Delta }_g^{\left(0\right)}\left[1-g_g{F}_g({\Delta }_g^{\left(0\right)})\right]$$

$${\Delta }_e^{\left(0\right)}$$

$$\mathrm{Re}{\epsilon }_{xy}\left(\omega \right)=\frac{64\pi }{{\omega }^2}\sum _{\vec{p}}{J}_x^{\left(mn\right)}\left(\vec{p}\right)J_y^{\left(mn\right)}\left(-\vec{p}\right)\times$$

$$\frac{\mathrm{Im}[\Delta \left(m,\vec{p}\right){\Delta }^{\ast }\left(n,\vec{p}\right)]}{E_m\left(\vec{p}\right)E_n\left(\vec{p}\right)}\delta \left[\omega -E_n\left(\vec{p}\right)-E_m\left(\vec{p}\right)\right],$$

$$\mathrm{Im}{\epsilon }_{xx}\left(\omega \right)$$

$$\frac{\mathrm{Re}{\epsilon }_{xy}}{\mathrm{Im}{\epsilon }_{xx}}$$

Full text

The subject of unconventional superconductivity is now over 30 years old, and the prime driving force in the field has been the determination of the form of the order parameter.  This issue has become more urgent in the era of topological superconductivity.  The determination has generally proved to be surprisingly difficult: unambiguous identifications remain remarkably few in number.  For this reason, it is useful to have experiments that test symmetry breaking directly, rather than after a train of reasoning and assumptions.  For time-reversal symmetry breaking, conceptually the simplest such experiment is Kerr rotation Kapitulnik et al. (2009), the rotation of the plane of polarization of reflected light through an angle $\theta_{K}$.  This effect in superconductors was already active in experiments Spielman et al. (1990) and theory Li and Joynt (1991); Yip and Sauls (1992) in the 1990s.   However, positive experimental results are relatively recent.  A nonzero signal that begins at the onset of superconductivity and grows as temperature decreases has been observed in Sr2RuO4 Xia et al. (2006), UPt3 Schemm et al. (2014), and URu2Si2 Schemm et al. (2015). The simplest theories of the pure system do not give a result large enough to explain the observed magnitude of $\theta_{K}\approx 10^{-6}$.  In Sr2RuO4 this has been interpreted in different ways.  One is to invoke impurity scattering Goryo (2008); Lutchyn et al. (2009).  Another is to attribute the effect to interband transitions between bands that cross the Fermi energy Wysokiński et al. (2012); Taylor and Kallin (2012). This paper focuses on clean UPt${}_{3},$ though we comment on other compounds below.  Since in UPt3 the bandwidth is less than the energy (0.8 eV) of the light T. Oguchi and A. J. Freeman ; Wang et al. (1987), the transitions induced by the light can only be of an interband nature.  UPt3 is the most clear-cut example of a material in which purely intraband effects cannot account for the observations.  We propose that the Coulomb interaction induces a nonzero order parameter in the completely unoccupied and completely occupied bands that is responsible for the Kerr effect.  This implies that the Kerr effect experiments probe, for the first time, a superconducting order parameter in a completely unoccupied or completely occupied band distant from the Fermi energy.  A somewhat similar effect was proposed for semiconductors with small gaps P. Nozieres and F. Pistolesi and may have been seen in a completely occupied band relatively near the Fermi energy in LiFeAs Hu et al. (2015).

Reflection of light of frequency $\omega$ incident on a sample along the $z$-axis is controlled by the components $\varepsilon_{xx}\left(\omega\right)$, $\varepsilon_{yy}\left(\omega\right)$, $\varepsilon_{xy}\left(\omega\right)$, and $\varepsilon_{yx}\left(\omega\right)$ of the dielectric tensor.  UPt3 is hexagonal and we can assume that tetragonal distortions that give rise to the inequality of $\varepsilon_{xx}\left(\omega\right)$ and $\varepsilon_{yy}\left(\omega\right)$ (linear birefringence) are absent so $\varepsilon_{xx}\left(\omega\right)=\varepsilon_{yy}\left(\omega\right)=\varepsilon$.  In any case, the experiments are designed not to be sensitive to linear birefringence Kapitulnik et al. (2009).  In a field $\vec{u}$ that breaks time reversal symmetry, including an effective field from an order parameter, the Onsager relation is $\varepsilon_{xy}\left(\omega,\vec{u}\right)$ $=\varepsilon_{yx}\left(\omega,-\vec{u}\right)$ L.D. Landau and E.M. Lifshitz and since the assumption of a linear relation between $\varepsilon$ and $\vec{u}$ is well-founded in the present case, we find $\varepsilon_{xy}\left(\omega\right)$ $=-\varepsilon_{yx}\left(\omega\right)$.  Then the normal modes in the metal for a propagation direction $\vec{k}=k\hat{z}$ are circularly polarized.  For each frequency there are two wavevectors $k_{+}$ and $k_{-}$ that correspond to the two helicities: $k_{\pm}^{2}=\left(\omega^{2}/c^{2}\right)\varepsilon_{\pm}$ and $\varepsilon_{\pm}\left(\omega\right)=\varepsilon_{xx}\left(\omega\right)\pm i\left|\varepsilon_{xy}\left(\omega\right)\right|$. The different dispersion and absorption for $k_{+}$ and $k_{-}$ gives the Kerr rotation $\theta_{K}=\mathrm{Re}\left(\varepsilon_{xy}/\varepsilon_{xx}^{3/2}\right)$ in a metal (for which $\left|\varepsilon_{xx}\right|\gg 1)$. To have $\varepsilon_{xy}\neq 0$ or $\varepsilon_{yx}\neq 0$, we need breaking of both mirror symmetries $x\rightarrow-x$ and $y\rightarrow-y$ and time-reversal Wang et al. (2014).   For a spatially uniform conventional singlet superconductor with order parameter $\Delta$, time-reversal means $\Delta\rightarrow\Delta^{\ast}$ and a change of gauge is $\Delta\rightarrow\Delta e^{i\phi},$ so any time-reversal transformation is equivalent to a gauge transformation and as a result there is no non-trivial notion of time-reversal.  For an unconventional superconductor with a momentum-dependent order parameter, this is not the case: we might have $\Delta(\mathbf{p})=c\Delta_{0}p_{z}\left(p_{x}+ip_{y}\right)$.  Then $\Delta^{\ast}(\mathbf{p})=c\Delta_{0}p_{z}\left(p_{x}-ip_{y}\right)$ and no uniform phase factor relates $\Delta$ and $\Delta^{\ast}$.

An appropriate model Hamiltonian for the multiband superconductor UPt3 is:

[TABLE]

where $\xi\left(n,\vec{p}\right)$ are the single-particle energies measured relative to the chemical potential, $\sigma$ is the pseudospin, $n$ and $n^{\prime}$ are band indices, $a_{\sigma}^{{\dagger}}\left(n,\vec{p}\right)$ creates an electron in the state $n\vec{p}\sigma,$ and $V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$ is a singlet pairing interaction.  The sum runs over all bands within 0.8 eV of the Fermi energy.  Since the bandwidths are of order $B\approx$ 0.2 to 0.3 eV, this includes completely full and completely empty bands as well as the usual partially occupied bands.  Because of the narrow bandwidth, there is no pair of partially occupied bands that have energies as much as 0.8 eV apart.

We have restricted our model to give singlet pairing only for ease of presentation.  The conclusions are essentially the same for triplet pairing. $\ H$ is treated in the mean field approximation in a straightforward generalization of the usual BCS-Gor’kov procedure Suhl et al. (1959).  However, we include strong coupling in that we make no assumptions about frequency cutoffs for the function $V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$.  This leads to a set of coupled gap equations

[TABLE]

where $F(n,\vec{p})=\tanh[\beta E(n,\vec{p})/2]/\left[2E\left(n,\vec{p}\right)\right]$, $E(n,\vec{p})=[\xi^{2}(n,\vec{p})+|\Delta(n,\vec{p})|^{2}]^{1/2}$, and $\beta$ is the inverse temperature.  Since the experiments are done near the critical temperature, we linearize these equations with respect to $\Delta$ and $F\left(n,\vec{p}\right)=\tanh\left[\beta\xi\left(n,\vec{p}\right)/2\right]/\left[2\xi\left(n,\vec{p}\right)\right]$.  In this case $F\left(n,\vec{p}\right)$ has the full symmetry of the lattice.

The point group of the system is $D_{6h}$ for UPt3. The case of interest is that of unconventional superconductivity.  Let $R$ be a group operation not the identity.  We have that $\Delta\left(n,R\vec{p}\right)\neq\Delta\left(n,\vec{p}\right)$ for all $n$.  It is also true that $V\left(n,R\vec{p};n^{\prime},R\vec{p}^{\prime}\right)=V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$ so $V$ can be decomposed into channels corresponding to the irreducible representations of $G$.  Regarded as a function of $\vec{p},$ we seek the highest eigenvalue of $V,$ which then determines the representation actually realized. Calculations for UPt3 using experimental data to estimate $V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$ were done years ago, but were not conclusive Putikka and Joynt (1988); Norman (1990) and first principles calculations using the functional renormalization group have been done for other systems W.S. Wang, M. Gao, Y. Yang, Y.Y. Yang, and Q.H. Wang , but not for UPt3.

The split transition in UPt3 R. Joynt ; Fisher et al. (1989) implies that this representation is multi-dimensional, which for singlet superconductivity means $E_{1g}$ or $E_{2g}$.  We choose the former for definiteness, but our conclusions apply equally to these two representations.  It is important to note that in the linear regime, Eq. (2) determines the representation, but not which combination of basis functions is chosen by the system.  This degeneracy is broken at higher order and there must be complex coefficients for a Kerr rotation to occur.  Thus we have that $\Delta\left(n,\vec{p}\right)=\Delta_{0}\left(n,\vec{p}\right)~{}p_{z}(p_{x}\pm ip_{y}),$ where $\Delta_{0}\left(n,R\vec{p}\right)=\Delta_{0}\left(n,\vec{p}\right)$ for all $R$.

We may separate the bands into partially filled bands, of which there are 5 in UPt3 indexed by $1\leq n\leq 5$ and completely filled or empty bands, indexed by $n>5$.  There are 6 separate Fermi surfaces in UPt3.  For $n>5,$ $F\left(n,\vec{p}\right)$ is of order $1/B$ or less.  We expect $\left|\Delta\left(n^{\prime},\vec{p}\right)\right|$ for $n^{\prime}>5$ to be induced by a coupling $V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$ that is off-diagonal in the band indices, connecting partially filled to completely filled or completely empty bands.

Then there are two questions that are crucial for the calculation of $\theta_{K}$.  1. How are the Ising-like variables $\pm$ in the equation $\Delta\left(n,\vec{p}\right)=\Delta_{0}\left(n,\vec{p}\right)~{}p_{z}(p_{x}\pm ip_{y})$ determined as $n$ varies?  2. What is the order of magnitude of $\left|\Delta\left(n,\vec{p}\right)\right|$ for $n>5$?

1. The first question is fairly easy to answer in our model. $V\left(n,\vec{p};n^{\prime},\vec{p}^{\prime}\right)$ couples only bands with a $(p_{x}+ip_{y})$ with other bands with a $(p_{x}+ip_{y})$ gap and couples only bands with a $(p_{x}-ip_{y})$ with other bands with a $(p_{x}-ip_{y})$ gap, i.e., it is diagonal in the $\pm$ degree of freedom.  However, this coupling can be of either sign.  Of course there are no symmetries in the band index, so the couplings have no particular relation to each other.  In a Ginzburg-Landau approach, we may define $\Delta\left(n,\vec{p}\right)=\Delta_{0}\left(n,\vec{p}\right)~{}p_{z}(\eta_{x}p_{x}+\eta_{y}p_{y})$ where the “internal” order parameter $\vec{\eta}=\left(\eta_{x},\eta_{y}\right)$ depends on the band index.  The free energy in $E_{1g}$ is then

[TABLE]

with a summation convention over the band indices $m$ and $n$ in effect.  To break time-reversal symmetry we need the $\gamma_{m}$ to be positive, and we need some of the $J_{mn}$ to be positive for some pair $\left(m,n\right)$ of bands that differ in energy by 0.8 eV.  Then we have a problem of determining the ground state of an Ising magnet with more-or-less random couplings.  We may expect both $(p_{x}+ip_{y})~{}$and $(p_{x}-ip_{y})$ to occur in the absence of physical considerations to the contrary.

2. The second question is more complicated.  The size of $\left|\Delta\left(n,\vec{p}\right)\right|$ for the partially occupied bands ($n\leq 5)$ is at least partially constrained by experiment.  We expect at least one and perhaps more of the gaps to be of order $2k_{B}T,$ i.e., about 10*-4*eV.  The superconductivity for $n>5$ is induced from the partially occupied bands.  For estimation purposes, we choose 2 bands from Eq. (2), denoting them by $g$ for partially occupied and $e$ for empty.  We consider the separable forms: $V\left(g,\vec{p};g,\vec{p}^{\prime}\right)=-g_{g}f(\vec{p})f^{\ast}(\vec{p}^{\prime})$ and $V\left(g,\vec{p};e,\vec{p}^{\prime}\right)=-g_{e}f(\vec{p})f^{\ast}(\vec{p}^{\prime})$ with $f(\vec{p})=p_{z}\left(p_{x}+ip_{y}\right)$, so that $\Delta\left(g,\vec{p}\right)=\Delta_{g}^{\left(0\right)}f(\vec{p})$ and $\Delta\left(e,\vec{p}\right)=\Delta_{e}^{\left(0\right)}f(\vec{p})$. The assumption that $\Delta_{e}^{\left(0\right)}$ is induced means that the corresponding component of $V\left(e,\vec{p};e,\vec{p}^{\prime}\right)$ is small and we set it to zero.  Then Eq. (2) yields

[TABLE]

where $F_{g}(\Delta_{g}^{\left(0\right)})=\sum_{\vec{p}}F(g,\vec{p})|f(\vec{p})|^{2}$ and $F_{e}(\Delta_{e}^{\left(0\right)})=\sum_{\vec{p}}F(e,\vec{p})|f(\vec{p})|^{2}$. At $T=0$ we have that $F_{g}(\Delta_{g}^{\left(0\right)})\approx N_{g}\left(0\right)\ln\left(\omega_{c}/\Delta_{g}^{\left(0\right)}\right)$ while $F_{e}(\Delta_{e}^{\left(0\right)})$ is of order $1/\omega_{c}$. The latter estimate also requires that the cutoff $\omega_{c}$ is not too much less than the bandwidth, justified if the interaction comes from the Coulomb interaction.  We find that $\left|\Delta_{e}^{\left(0\right)}/\Delta_{g}^{\left(0\right)}\right|\sim\left|g_{e}/g_{g}\right|$.  Since all the bands are $f$-like in UPt${}_{3},$ the Coulomb matrix elements at short distances are expected to be comparable, and this gives reason to suppose that $\left|\Delta_{e}^{\left(0\right)}/\Delta_{g}^{\left(0\right)}\right|$ is of order unity.

In order to calculate $\theta_{K}$ we need the diagonal complex dielectric function $\varepsilon_{xx}(\omega=0.8~{}\text{eV})$.  This has been determined by reflectivity measurements and a Kramers-Kronig analysis J. Schoenes and J.J.M. Franse ; Sulewski et al. (1988).  In this frequency range it is necessary to include several Lorentz oscillators to fit the data, showing that there are interband transitions at $\omega=0.8$ eV.  This is in agreement with band calculations T. Oguchi and A. J. Freeman ; Wang et al. (1987).  We extract the approximate values $\mathrm{Re}\varepsilon_{xx}(\omega=0.8~{}\text{eV})\approx 3$ and $\mathrm{Im}\varepsilon_{xx}(\omega=0.8~{}\text{eV})\approx 25$ from these results.

The key quantity is of course the off-diagonal dielectric function $\varepsilon_{xy}\left(\omega=0.8~{}\text{eV}\right)$.  The result for a single pair of bands $(m,n)$ is

[TABLE]

obtained from the anomalous part of the lowest-order bubble diagram.  This is not the total dielectric function. To get that we must also sum over all pairs.   Here $J_{x,y}^{\left(mn\right)}\left(\vec{p}\right)$ are the interband matrix elements of the current operator between single-particle states in the partially occupied $m$-band and the empty $n$-band.  An analogous expression would hold for transitions from a completely full band to a partially occupied band.  This expression for $\varepsilon_{xy}$ is to be compared to that for the familiar normal-state dielectric function

[TABLE]

Although Eqs. (5) and (6) contain quantities which are poorly known, only the ratio is involved in the Kerr angle $\theta_{K}$.  It is only this that allows us to give an order of magnitude estimate for $\theta_{K}$.

To achieve this, we adopt a simple model of the bands in which the single-particle energies $\xi_{m}\left(\vec{p}\right)$, $\xi_{n}\left(\vec{p}\right)$ are random variables that are uniformly distributed over a bandwidth $B,$ and the center of the $m$ and $n$ bands are separated by an energy $B$.  In the model the averages over the current matrix elements are assumed to be the same for the two bands, and there are no correlations in momentum space between the gap functions $\Delta\left(m,\vec{p}\right)$, $\Delta^{\ast}\left(n,\vec{p}\right)$ and $J_{x}^{\left(mn\right)}\left(\vec{p}\right)$, $J_{y}^{\left(mn\right)}\left(\vec{p}\right)$.  The computation of the ratio then reduces to a determination of the ratio of the density of states parts of Eqs. (5) and (6).   The result is:

[TABLE]

Here $\Delta_{m}$, $\Delta_{n}$ are the average values of $\left|\Delta(m,\vec{p})\right|$, $\left|\Delta(n,\vec{p})\right|$, taken to be approximately equal to $2k_{B}T_{c},$ $\omega_{c}$ is the cutoff for the gap $\Delta\left(m,\vec{p}\right)$ and we have assumed that $\omega_{c}$ does not differ by orders of magnitude from $B$, which we take to be $B=0.2$ eV.  $I_{s}$ is the normalized angular integral over the anisotropic gap functions.  We discuss it further below.  Eq. (7) gives the contribution to the ratio $\mathrm{Re}\varepsilon_{xy}/\mathrm{Im}\varepsilon_{xx}$ from one pair of bands.  If we sum over all bands and the ratio does not vary much from pair to pair, then we may combine this value with the normal-state experimental value of $\varepsilon_{xx}\left(\omega\right)$ quoted above to find $\theta_{K}\sim 2\times 10^{-7}$ at zero temperature, which is about 20% or so of the value one would get if the experimental results measured near $T_{c}$ are extrapolated to $T=0$. Considering the approximations involved, and our general ignorance about the mechanism of superconductivity, this is about all that can be expected.  Intraband theories typically give $\theta_{K}\sim(\Delta/\omega)^{2}\sim 10^{-8},$ which is smaller.

These order-of-magnitude considerations all assume that $\theta_{K}$ does not vanish by symmetry, which of course can happen if the angular integral in Eq. (5) vanishes: $I_{s}=0$.  Resolving this question amounts to a symmetry analysis of the factor $J_{x}^{\left(mn\right)}\left(\vec{p}\right)J_{y}^{\left(mn\right)}\left(-\vec{p}\right)~{}\left[\Delta\left(m,\vec{p}\right)\Delta^{\ast}\left(n,\vec{p}\right)\right]$ for each pair of bands that are separated by the laser photon energy 0.8 eV.   Both the real and the imaginary part of $~{}\left[\Delta\left(m,\vec{p}\right)\Delta^{\ast}\left(n,\vec{p}\right)\right]$ contribute to $\theta_{K}$.  The product $J_{x}J_{y}$ transforms according to the $E_{2g}$ representation of $D_{6h}$.  The representation of $\left[\Delta\left(m,\vec{p}\right)\Delta^{\ast}\left(n,\vec{p}\right)\right]$ is in general reducible, and if we define it as $\Gamma_{\Delta},$ then the integral vanishes if and only if the $E_{2g}\times\Gamma_{\Delta}$ representation does not contain the $A_{1g}$ (identity) representation.  $\theta_{K}$ itself will vanish if and only if the integral in Eq. (5) vanishes for every pair of bands.  Again we will use the example that $\Delta\left(m,\vec{p}\right)$ and $\Delta\left(n,\vec{p}\right)$ both transform as the $E_{1g}$ representation, but the considerations apply to all representations.  For this case we have essentially two possibilities.   The first is $\Delta\left(m,\vec{p}\right)\Delta^{\ast}\left(n,\vec{p}\right)\sim p_{z}^{2}\left(p_{x}+ip_{y}\right)\left(p_{x}+ip_{y}\right)^{\ast}=p_{z}^{2}(p_{x}^{2}+p_{y}^{2}),$ and we find $\Gamma_{\Delta}=A_{1g}$ and since $E_{2g}\times A_{1g}=E_{2g},$ $I_{s}=0$ and there is no contribution to $\theta_{K}$ for a pair of bands both of which are of the $\left(p_{x}+ip_{y}\right)$ type.  The second case is $\Delta\left(m,\vec{p}\right)\Delta^{\ast}\left(n,\vec{p}\right)\sim p_{z}^{2}\left(p_{x}+ip_{y}\right)\left(p_{x}-ip_{y}\right)^{\ast}=p_{z}^{2}\left(p_{x}^{2}-p_{y}^{2}+2ip_{x}p_{y}\right),$ which gives $\Gamma_{\Delta}=E_{2g}$.  Since $E_{2g}\times E_{2g}=A_{1g}+A_{2g}+E_{2g}$, $I_{s}\neq 0$ a pair of this type can contribute to nonzero $\theta_{K}$.  Thus, continuing the analogy with the Ising model, each band being of the $\pm$ type, we see that a “ferromagnetic” state with all bands of the $+$ type or all bands being of the $-$ type does not lead to a Kerr rotation.  We must have a mixture of $"+"$ and $"-"$ gaps on different bands.  Qualitatively, we may perhaps think of the Kerr rotation as the creation of a broken pair that must be able to absorb both the energy and the angular momentum of the photon - this can only occur if the chirality of the two gaps involved is different.

The observation of a nonzero Kerr rotation in UPt3 is seen to have a multitude of consequences, going well beyond the fact that it demands that there must be time-reversal symmetry breaking.  There must also exist a pairing field in a completely filled or a completely empty band, since the photon energy exceeds the bandwidth.  The observation of this pairing field is novel: in fact we believe it is the first time that it has been observed in experiments in a band that is very distant in energy from the Fermi energy.  The existence of this pairing field is also not enough to explain the phenomenon: there must be a subtle pattern of relative symmetry between the bands involved.  Finally, we note that if there is superconductivity with opposite chiralities $p_{z}\left(p_{x}+ip_{y}\right)$ and $p_{z}\left(p_{x}-ip_{y}\right)$ coexisting in the same sample, it raises interesting questions concerning topology issues.  How does the coexistence modify the theory of Majorana fermions in the vortex cores, and protected surface states?

We thank Po-Ho Yang and A. V. Chubukov for useful discussions. Financial support from MOST of Taiwan (Grant No. 105-2112-M-003-005) is acknowledged.