## Periodic Reporting for period 1 - PhotoWann (Bulk Photovoltaic effect via Wannier functions)

Reporting period: 2019-10-01 to 2021-09-30

Quick and efficient conversion of light into electricity

is key for future clean-energy technologies.

Recently a focus of renewed attention, the bulk photovoltaic effect (BPVE)

is a nonlinear absorption process

that converts light into electrical current intrinsically,

i.e. without the need of any pn-junction for driving the photoexcited electrons.

The BPVE can furthermore

surpass the Shockley-Queisser limit for the solar-cell efficiency,

thus opening the way for devices largely exceeding

current capabilities.

Alongside, the photovoltage attained in the BPVE

is not limited by the band gap of the material, giving rise to huge measured

values.

In the last years, the study of the BPVE, and in particular the shift-current contribution,

has witnessed a huge progress.

Groundbreaking discoveries in the field include the measurement of

large nonlinear photovoltages in Weyl semimetals as well as

in nanotubes made of tungsten disulfide.

A magnetically switchable nonlinear light-matter interaction

has been recently proposed, adding potential functionalities to the BPVE.

An extremely promising variant effect was also proposed recently,

the so-called flexo-photovoltaic effect

that turns widely used semiconductors like silicon into

nonlinear photoelectrics. The study of many of these effects is still in its infancy,

and its development is important for society

in order to substantially push the field of renewable energy resources.

As of today, many of the theoretical studies of the aforementioned effects are performed using

phenomenological models that contain parameters that are unknown in principle.

Furthermore, there is growing evidence that the model approach may not be suitable for studying

nonlinear responses, as the subtle quantities involved in the light-induced transition matrix elements

seem to present problems in this context.

The aim of the present project is to develop alternative and more reliable ab-initio computational tools that will improve

our ability to study the aforementioned effects in a variety of materials. We plan to do so by including the essential quantum-mechanical ingredients

involved in the process using the first principle method based on density functional theory.

is key for future clean-energy technologies.

Recently a focus of renewed attention, the bulk photovoltaic effect (BPVE)

is a nonlinear absorption process

that converts light into electrical current intrinsically,

i.e. without the need of any pn-junction for driving the photoexcited electrons.

The BPVE can furthermore

surpass the Shockley-Queisser limit for the solar-cell efficiency,

thus opening the way for devices largely exceeding

current capabilities.

Alongside, the photovoltage attained in the BPVE

is not limited by the band gap of the material, giving rise to huge measured

values.

In the last years, the study of the BPVE, and in particular the shift-current contribution,

has witnessed a huge progress.

Groundbreaking discoveries in the field include the measurement of

large nonlinear photovoltages in Weyl semimetals as well as

in nanotubes made of tungsten disulfide.

A magnetically switchable nonlinear light-matter interaction

has been recently proposed, adding potential functionalities to the BPVE.

An extremely promising variant effect was also proposed recently,

the so-called flexo-photovoltaic effect

that turns widely used semiconductors like silicon into

nonlinear photoelectrics. The study of many of these effects is still in its infancy,

and its development is important for society

in order to substantially push the field of renewable energy resources.

As of today, many of the theoretical studies of the aforementioned effects are performed using

phenomenological models that contain parameters that are unknown in principle.

Furthermore, there is growing evidence that the model approach may not be suitable for studying

nonlinear responses, as the subtle quantities involved in the light-induced transition matrix elements

seem to present problems in this context.

The aim of the present project is to develop alternative and more reliable ab-initio computational tools that will improve

our ability to study the aforementioned effects in a variety of materials. We plan to do so by including the essential quantum-mechanical ingredients

involved in the process using the first principle method based on density functional theory.

In semiconductors that break inversion symmetry,

a bulk photocurrent can be generated by applying light with frequency larger than the gap.

During the project, we have shown that for a gap centered at a mirror-invariant k-point,

two classes of band-edge responses are possible depending

on the relative parity of conduction and valence bands.

When the relative parity is even, as for example in GeS,

the current flows parallel to the electric field polarization.

With odd relative parities, however, the current flows instead

purely perpendicular to the electric field and its

strength is proportional to the Berry curvature dipole.

This remarkable quantum-mechanical effect had been overlooked in the literature up to now.

As a proof of principle,

we have validated our prediction via ab initio calculations on graphitic BC2N,

which exhibits a extraordinarily large transverse shift current

that ranks amongst the strongest nonlinear responses reported.

As a complementary outcome of the project, we have proposed

the measurement of this sizable effect as a benchmark test to clarify

the longstanding debate concerning the atomic structure of this promising material.

In order to understand the nonlinear shift-current response in depth, we have

performed a study of the position operator

in describing optical properties of acentric materials.

This is important, as tight-binding models often employed for studies of nonlinear responses

implicitly assume a rather drastic approximation

concerning the position operator, which

amounts to discarding all of its off-diagonal matrix elements.

In turn, our ab initio analysis is based on a Wannier-interpolation scheme that naturally

incorporates such matrix elements into the formalism, hence it is particularly well suited for the task.

Our results show that while the linear dielectric

function is mildly affected by the approximation,

it can induce serious numerical errors in the case of the shift current.

This is confirmed in two separate ways; i) by explicitly assessing the impact of

off-diagonal position matrix elements in ab-initio calculations and,

ii) by means of a two-band k.p model that assumes implicitly the diagonal

tight-binding approximation. For the latter, we have developed a

numerical scheme for extracting k.p coefficients using quasi-degenerate (Lowding) perturbation

theory, a tool that we have implemented into the free-software package Wannier90 and we expect it will be useful for

researchers in the field.

In summary, this part of the project has highlighted the strong sensitivity of the shift-current mechanism to

wavefunction localization.

Finally, the project has also contributed in a nearby field where the PI has previous experience, namely studying the response of magnetic

materials and, more precisely, magnetic nanostructures. In this field, we have studied the impact of

quantum spin-fluctuations on the magnetic exchange interactions, i.e. the fundamental physical quantities that

define how magnetic atoms interact with each other. By developing an appropriate theoretical scheme and performing

ab-initio calculations, we have shown that quantum fluctuations are needed in order to explain previous

experimental results based on scanning tunneling microscope (STM).

a bulk photocurrent can be generated by applying light with frequency larger than the gap.

During the project, we have shown that for a gap centered at a mirror-invariant k-point,

two classes of band-edge responses are possible depending

on the relative parity of conduction and valence bands.

When the relative parity is even, as for example in GeS,

the current flows parallel to the electric field polarization.

With odd relative parities, however, the current flows instead

purely perpendicular to the electric field and its

strength is proportional to the Berry curvature dipole.

This remarkable quantum-mechanical effect had been overlooked in the literature up to now.

As a proof of principle,

we have validated our prediction via ab initio calculations on graphitic BC2N,

which exhibits a extraordinarily large transverse shift current

that ranks amongst the strongest nonlinear responses reported.

As a complementary outcome of the project, we have proposed

the measurement of this sizable effect as a benchmark test to clarify

the longstanding debate concerning the atomic structure of this promising material.

In order to understand the nonlinear shift-current response in depth, we have

performed a study of the position operator

in describing optical properties of acentric materials.

This is important, as tight-binding models often employed for studies of nonlinear responses

implicitly assume a rather drastic approximation

concerning the position operator, which

amounts to discarding all of its off-diagonal matrix elements.

In turn, our ab initio analysis is based on a Wannier-interpolation scheme that naturally

incorporates such matrix elements into the formalism, hence it is particularly well suited for the task.

Our results show that while the linear dielectric

function is mildly affected by the approximation,

it can induce serious numerical errors in the case of the shift current.

This is confirmed in two separate ways; i) by explicitly assessing the impact of

off-diagonal position matrix elements in ab-initio calculations and,

ii) by means of a two-band k.p model that assumes implicitly the diagonal

tight-binding approximation. For the latter, we have developed a

numerical scheme for extracting k.p coefficients using quasi-degenerate (Lowding) perturbation

theory, a tool that we have implemented into the free-software package Wannier90 and we expect it will be useful for

researchers in the field.

In summary, this part of the project has highlighted the strong sensitivity of the shift-current mechanism to

wavefunction localization.

Finally, the project has also contributed in a nearby field where the PI has previous experience, namely studying the response of magnetic

materials and, more precisely, magnetic nanostructures. In this field, we have studied the impact of

quantum spin-fluctuations on the magnetic exchange interactions, i.e. the fundamental physical quantities that

define how magnetic atoms interact with each other. By developing an appropriate theoretical scheme and performing

ab-initio calculations, we have shown that quantum fluctuations are needed in order to explain previous

experimental results based on scanning tunneling microscope (STM).

The impact of quantum dipole selection rules on the directionality of the shift current response had been overlooked

in the literature up to now. In the project, we have worked out in detail the way these fundamental selection rules affect the

shift current; in particular, we have shown how could this effect be used to obtain a current response that is purely

perpendicular to the polarization of the incident light, and we have furthermore proposed an explicit material for observing this

effect based on ab-initio calculations.

We have also made progress that goes beyond the state of the art concerning the understanding of the role of the position

operator on optical responses, in particular the nonlinear shift current. We have found that including the full matrix structure of the

position operator is essential for giving an accurate account of the shift current. Given the growing interest of nonlinear responses, and the fact

that most theoretical studies up to now use rather uncontrolled approximations concerning the position matrix elements,

our work will improve the accuracy of current calculations.

Finally, concerning the work on quantum spin-fluctuations, their impact on the magnetic exchange interactions

had never been analyzed up to now within a first-principles framework, hence this achievement

represents an important leap forward in the state-of-the-art of magnetic nanostructures.

in the literature up to now. In the project, we have worked out in detail the way these fundamental selection rules affect the

shift current; in particular, we have shown how could this effect be used to obtain a current response that is purely

perpendicular to the polarization of the incident light, and we have furthermore proposed an explicit material for observing this

effect based on ab-initio calculations.

We have also made progress that goes beyond the state of the art concerning the understanding of the role of the position

operator on optical responses, in particular the nonlinear shift current. We have found that including the full matrix structure of the

position operator is essential for giving an accurate account of the shift current. Given the growing interest of nonlinear responses, and the fact

that most theoretical studies up to now use rather uncontrolled approximations concerning the position matrix elements,

our work will improve the accuracy of current calculations.

Finally, concerning the work on quantum spin-fluctuations, their impact on the magnetic exchange interactions

had never been analyzed up to now within a first-principles framework, hence this achievement

represents an important leap forward in the state-of-the-art of magnetic nanostructures.