Derivation of the shift charge current forD Bi2X3(X= Te,Se) Topological Insulators, using the Floquet formalism combined with Keldysh’s Green’s function method
Tobias Bouma
Bachelor’s project ( EC) A thesis presented for the degree of
Bachelor of Physics
Vladimir Gritsev | Erik van Heumen FNWI | ITFA
UvA | VU The Netherlands
Contents
Popular summary
Introduction
. From Hall to topology . . . . Edge states and Rice-Mele . . . . Characteristics and applications . . . . Light and topological insulators . . .
Deriving the expression for the shift charge current
. Describing current . . . .. Linear response theory . . . .. Derivation steps . . . . Deriving shift charge current using Floquet and Green’s functions . . . .. Description of the system . . . .. Derivation of Floquet Hamiltonian . . . .. The shape of the generalized conductivity . . . .. Other contributions to current . . . .. Simplifying the generalized conductivity . . . .. Term-wise calculation of the generalized conductivity . . . .. Final expression for the shift charge current . . .
Discussion and conclusion
. Validity of used approximations . . . . Comparison to experiment and other literature . . . . Link to the topology of the system . . . . Variations of the outlined approach . . . . Suggestions for further research . . . . Overview of the main derivation & Outlook . . .
Bibliography
A Supplementary material for the introduction
A. Derivation of the Berry phase . . . A. Chern number is always an integer . . . A. Spin and momentum are locked . . .
B Supplementary material for the main derivation
B. Floquet Hamiltonian . . . B.. Hamiltonian in terms of Floquet modes . . . B.. Short-hand for Floquet Hamiltonian . . . B.. The Green’s function . . . B.. Lesser Green’s function . . . B.. Expectation value of current operator . . . B. Implications of symmetry . . . B.. Parity . . . B.. Time-reversal . . . B. Generalized conductivity . . . B.. Initial simplification . . . B.. Proving the delta function equality . . . B. Spherical basis . . . B.. Calculating energy - a consistency check . . . B.. First orders of derivatives . . .
B.. Jacobian . . . B.. Eigenvectors . . . B.. Off-diagonals of Pauli matrices . . . B.. Derivatives in spherical basis . . . B.. Final calculations of conductivity . . . B.. Obtaining current expression . . .
C Supplementary material for the discussion
C. The high field expression . . . C. Second Harmonic Generation . . . C. Third-order non-linear response . . . C. Spin shift currents . . .
D Projections to recover constants in the main body
E Additional mathematical identities
Popular summary
Informatieopslag is een steeds groter probleem aan het worden. Een chip bestaat uit zeer veel kleine on-derdelen die informatie opslaan, transistoren, en volgens de wet van Moore verdubbelt dat aantal elke twee jaar. Op gegeven moment kan het niet kleiner: dan zijn de onderdelen waar een chip uit bestaat zo klein geworden dat ze zo groot zijn als een atoom. Voor verdere technologische ontwikkeling, moet dus een oplossing bedacht worden om meer informatie op te slaan op hetzelfde oppervlak.
E´en manier om dit te doen is om het aantal vrijheidsgraden te vergroten. Op een gewone harde schijf heb je bijvoorbeeld ´e´en vrijheidsgraad: we kijken naar een bit, en deze kan de waarde aannemen voor “aan”, en voor “uit”. Als we verder willen gaan, moeten we verder gaan dan alleen aan en uit, en een voorstel kan zijn om de spin van een electron mee te nemen als extra vrijheidsgraad. Een electron spin is een soort draaimoment wat opgesloten is in het deeltje, en kan wat betreft ori¨entatie spin omhoog en spin omlaag hebben.
Maar de vraag is: hoe doen we dat? Een voorbeeld van een materiaal wat dit gedrag zou kunnen beschri-jven is de topologische isolator: voor dit materiaal is het bekend welke kant de spin op staat langs de rand van het materiaal. De topologische isolator is een materiaal wat alleen geleidend is op het oppervlak, maar juist isolerend in het inwendig: de isolerende “bulk”. Let wel dat de topologie ook verstoord kan worden door speciale processen zoals een magnetisch veld, waarbij het systeem niet langer symmetrisch is in de tijd: als we de tijd achteruit laten lopen, dan veranderen we het systeem. De zogenaamde topologie van de isolator zorgt ervoor dat deze eigenschappen aanwezig zijn zolang we het niet verstoren. Typisch voor een Topologische Isolator is dat de deeltjes een bepaalde energie versus impuls relatie hebben, die uitgezet kan worden in eenD-grafiek als een kegel: de Dirac kegel.
Een topologische isolator kan in nabije materialen ook weer deze “spin” aantasten, en hiermee is de applicatie van informatieopslag geboren: we kunnen nu zelf bepalen wat de spin van de elektronen in een materiaal is, aangenomen dat het materiaal een be¨ınvloedbare spin heeft, natuurlijk. Een nieuwe vorm van elektronica is geboren: de zogenaamde spintronica.
Naast informatieopslag zijn er nog andere eigenschappen, zoals bijvoorbeeld het feit dat het materiaal niet kan opwarmen omdat de elektronen middels de topologie niet kunnen botsen op oneffenheden in het materiaal. Dit is een consequentie van het mechanisme wat samenhangt met de definitie van de topol-ogische isolator: geleidend op het oppervlak, maar isolerend in het inwendige. Dit maakt Topoltopol-ogische Isolatoren bijzonder geschikt om stroom door te geleiden: we verliezen weinig tot geen warmte tijdens stroomtransport, afhankelijk van de kwaliteit van de Topologische Isolator.
Een Topologische Isolator heeft echter alleen in een perfecte wereld al deze eigenschappen, zonder dat we ze moeten veranderen. Dit heeft te maken met dat we een energie van het systeem willen hebben, wat precies in het midden van de Dirac kegel zit, dus z = 0 and de Dirac kegel beschreven wordt door z = r in -ruimte. Om beter te begrijpen hoe we het energieniveau kunnen aanpassen, moeten we ook beter weten hoe een Topologische Isolator reageert op licht: op deze manier weten we bijvoorbeeld of we te hoog of te laag zitten, en kunnen we het materiaal aanpassen zodat we dichter bij dit kritieke punt zijn.
Wat we op de kleine schaal doen, als we licht schijnen op een materiaal, is dat we een stroom induceren. Hoe dit stroompje gaat lopen, kan ons dan meer vertellen over het materiaal. In mijn onderzoek heb ik geprobeerd theoretisch te beschrijven hoe deze stroom gaat lopen als het materiaal maar heel lichtjes ge¨exciteerd wordt door elektronen. Dit heb ik gedaan door het systeem uit te drukken in een speciale inverse impulsruimte, en door gebruik te maken van de functies van Green: een speciale vergelijking om de differentiaalvergelijkingen van de quantummechanica op te lossen.
Introduction
Abstract
From the Hall effect at low temperatures emerges a quantized version, the quantum Hall effect. This eventually lead to the Quantum Spin Hall Effect and the concept of a Topological Insulator. Defining a Topological Insulator as conducting at the edges but insulating in the bulk, the smoothly modulated, time-dependent Rice-Mele model links certain quantities from the phase of the Hamiltonian, to the topol-ogy of the system. Finally some applications and previous studies are mentioned, motivating this study.
. From Hall to topology
The quantum Hall effect (QHE) is a quantum version of the well-known Hall effect, where the Hall con-ductance σµνnow only takes the values
σµν=
n
2π. ()
Here, subscripts µ, ν correspond to the dimension of j, E such that jµ= σµνEν. Various manifestations of this
effect exist, where subclassifications are made according to how n is quantized, and whether the electrons obey the known Landau levels. Examples of the former are, for instance, the integer QHE or the fractional QHE, while the latter is often called the Quantum Anomalous Hall Effect (QAHE).
In, Kane and Mele [] described another phase called the Quantum Spin Hall (QSH) Insulator. This new phase should average a non-zero spin-Hall conductance but zero charge-Hall conductance. The system in question comprises two separate QAH effects[] in opposite directions, such that the system as a whole is invariant under Time Reversal Symmetry (TRS), but not the individual spin channels. Sufficiently strong Spin-Orbit Coupling (SOC)[] then gives rise to the required (intrinsic) magnetic field, such that the QHE appears.
In the picture of the QSHI and QSHE, one could show that a topologically non-trivial phase would emerge in the presence of band inversion[]. This band inversion was a direct consequence of the SOC associated with the QSHE, and as such, it plays an important role in the emergence of a topological phase[, ].
Associated with this topological phase is the state of matter called the Topological Insulator (TI), quali-tatively defined as:
Matter that has conducting edge states but insulating bulk states.
The field of TIs can be split up further according to symmetries, often summarized in the so called “pe-riodic table of topological insulators”[]. Relevant are time reversal (Θ), particle-hole (Ξ) and chiral (Π) symmetry. Particular combinations of these symmetries can be related to a number, either an integer (n ∈ Z) or an integer modulo (mathematically denoted n ∈ Z2). Subclassifications according to these symmetries, are shown in table.
Using this table goes as follows: take for example the quantum Hall state. It has no symmetries, thus, in two-dimensions, the quantum Hall state is described by a Z topological invariant. Other TIs such as the QSHI can be classified similarly, and we would like to refer to Hasan and Kane [] for a more elaborate discussion on the relation between the topological invariants and system’s respective symmetries.
In this introduction, we will focus ourselves on a system described by the Chern invariant (a Z invari-ant), typically called the Chern Insulator. The Chern invariant can be expressed in terms of the so called Berry phase, where the Berry phase is a physical, gauge invariant quantity.
With n some integer and using natural units. Multiply by e2/~ for the SI representation. It is essential to have sufficiently low
temperatures and external magnetic field for the QHE to be observable, just like the Hall Effect.
In two-dimensions.
Pankratov et al. [] also described a variant much resembling the Topological Insulator in quantum wells, made by inducing
potential wells from sandwiching slices of e.g. HgTe between CdTe. See [] for experimental findings. See [, ] for D QSH effects and [] for a variants of the system with an intrinsic magnetic field.
Table: Periodic table of topological insulators according to their symmetries, limited to 1 ≤ d ≤ 3 systems. The first row obeys the naming scheme for symmetry classes from Altland and Zirnbauer []. Table adapted from Hasan and Kane [].
Symmetries d A&T Θ Ξ Π A Z AIII Z Z AI BDI Z D Z2 Z DIII - Z2 Z2 AII - Z2 Z2 CII - - Z Z2 C - Z CI - Z
Deriving the form of this Berry connection assumes adiabaticity and, in the easiest case, a non-degenerate Hamiltonian. Further, since the Schr¨odinger equation is satisfied for a wave function including a phase fac-tor, |Ψ (~R(t))i = e±iθ|n(~R(t))i, a gauge-invariant part of the phase can be derived to have the form[, ],
γn(t) = ∓i I C hn|∇ni d~r () = ∓i " C ∇ × hn|∇ni d ~S () = ∓i " C ∇ ×Ad ~~ S. ()
The related Chern invariant then obeys the following shape: C(1)= 1
2π Z
∇ ×Ad ~~ S. ()
Here, ~A is called the Berry connection. By a conceptual argument, it can be shown that the Chern number is always an integer, please refer to appendix A.. A similar result holds if the wave function is a superposition of the previously assumed form, i.e. the wave function obeys |Ψ (~R(t))i =P
ne
±iθn|
n(~R(t))i.
At this point, it is not directly evident what link exists between these quantities and the previously proposed “definition of the Topological Insulator”. In the section that follows, we attempt to illuminate this fact with the smoothly modulated, time-dependent Rice-Mele model. Note that this model is only pre-sented as an example of how edge states give physical meaning to the topological quantities of the system, and thus the topology of the system itself. For more elaborate models and a more detailed discussion on the link between the TIs and their topology, see for example [–].
Here, ∓ corresponds to the same minus or plus chosen in the exponent of the phase factor. For a full derivation, please see
appendix A..
Later on, more general forms of the geometrical phase were obtained for degenerate Hamiltonians[] and non-adiabatic
revolu-tions in parameter space[].
. Edge states and Rice-Mele
We will give physical meaning by considering a model of a chain described by a Hamiltonian. We will assume that we want the chain to be conducting at the edge and insulating at the bulk. Particles can be pumped around this chain by appropriate choice of parameters, and we will see that this choice of parameters naturally leads us to topological quantities: the Berry phase and the Chern number, respectively representing the phase change of the the Hamiltonian and the winding number. We will also see that these quantities lead us to the idea of a so-called topological invariant, a number protected by the topology of the system.
The model we will consider here, describes a chain with edge states. The edge states lie within the gap of the band structure, and the bulk states that form the conduction and valence bands. The chosen model describes these properties, and considers the pumping of particles over a chain with states at the edges of the chain. The chain consists of-dimers and assumes only short-range hoppings. Making things more concrete, the smoothly modulated Rice-Mele Hamiltonian is given by
H = u(t) N X m=1 |m, Bi hm, A| + h.c. + v(t) N −1 X m=1 |m + 1, Ai hm, A| + h.c. + w(t) N X m=1 |m, Bi hm, B| − |m, Ai hm, A| , ()
where u(t), v(t) and w(t) are the time-dependent onsite, intra and intercell potentials. Each site is denoted by m, and each subsite within the dimer is denoted A or B. The aforementioned potentials can be chosen, from which a pumping mechanism will emerge. Each particle can be tracked by using a position operator that links location to an eigenstate of the Hamiltonian. This position operator is given by
ˆ x = N X m=1 m |m, Ai hm, A| + |m, Bi hm, B| . ()
Suppose, now, that we have an open chain of N = 10 sites with the following potentials,
u(t) = sin(Ωt) ()
v(t) = v + cos(Ωt) ()
w(t) = 1, ()
with v, the average value of the intracell potential, chosen to be v = 1. The bulk momentum-space Hamil-tonian corresponding to the system reads
H(k, t) = ~d · ~σ = (v(t) + w(t) cos k) ~σx+ w(t) sin k ~σy+ u(t)~σz ()
with k the momentum.
The simulation corresponding to the chosen parameters can be seen in fig. and tells us several things. Firstly, we find that the bulk-momentum space vector associated with the Hamiltonian, ~d, describes the surface of a torus, see (b). Since each energy eigenstate can be tracked using the formula for ˆx as previously assumed, we can interpret (c). In (c), we can see that during each cycle:
• All particles at the bottom are pumped towards the right edge of the chain. • One particle at the right edge is pumped to the top part of the chain. • All particles at the top are pumped to the left.
• . . . And finally the leftmost particle at the top part of the chain is pumped to the bottom part.
An alternative to the smoothly modulated case is one made up of parts that are defined discretely. This gives simpler calculations
but does not allow to derive the results that follow later on.
This is specifically the position operator corresponding to Wannier states in the thermodynamic limit. Since the argument made
here is primarily conceptual, the specifics of this operator will be omitted here, and can be found in e.g. []. Wannier states are also discussed in much more detail in this paper.
Figure : Time-dependent Rice-Mele model for N = 10 and parameters conform eq. (). (Top left) Time-dependence of the three parameters (Top right) trace by individual components of ~d as seen in the bulk momentum-space Hamiltonian (Bottom left) Time-periodic eigenstates of the bulk momentum-space Hamiltonian, along with the edge state eigenstates - left is blue, right is red (Bottom right) Amplitude of the wave functions corresponding to either an edge state with an exponential tendency towards localization on a particular site within the cell (d = right, tendency on site B, e = left, tendency towards site A), or a bulk state (f). Adapted from Asb ´oth et al. []
The cross-shaped band structure of conducting edge states turns out to be similar in the instantaneous k (rather than t)-dependent eigenstate E(k): it is a typical characteristic of the Topological Insulator and translates to three dimensions in a natural way, and as such it is called the Dirac cone. Lastly, at the bottom right, wave amplitudes corresponding to specific parts of the (E, t) graph are highlighted.
To see which states are edge states, we can apply the constraint that the wave function |Ψ i is larger in amplitude about some cell than some arbitrary critical amplitude. For example, for the left edge we can set
N X m=1 |hΨ |m, Ai|2+ |hΨ |m, B||2 ≥Acrit, ()
with Acrit= 0.6. (The right edge can be determined similarly by changing the summation appropriately.) Say, now, that we take some energy somewhere in a small region of the graph (e.g. between t0< t < t1), and we take ≈ 0. Every (single instance of an) edge state branch, and thus particle, will cross this region either from E < to E > , or from E > to E < . The number of particles that corresponds to this crossing from E < to E > or E > to E < can be denoted as respectively N+and N−. The difference Q between these two numbers of particles will be the net number that is pumped about the edges, during a cycle. Note that the choice of has no effect on the number Q, as long as it satisfies aforementioned conditions.
Note also that Q is uniquely defined for the system, since, if we could find another number Q0 , Q at 0 > , this would imply an edge state branch would enter the region described by the set of energies < E < 0, but never exit it. Since we are away from the bulk gap, this would lead to an unphysical system. Ergo, Q must be a conserved quantity.
The Dirac cone actually gives us another piece of information. Namely, it has a point where the two edge states cross, and thus
we have E(k) = E(−k). This point is known as the Dirac point, and it obeys Kramer’s theorem, which requires a twofold degeneracy, a Kramer’s pair. What Kramer’s theorem tells us, is that the Hamiltonian associated with this Dirac point commutes with the time reversal operator Θ, i.e. [H, Θ] = 0. Hence, the edge states are protected by Time Reversal symmetry.
Figure: Adiabatic deformations of the dispersion relation, showing which crossings are allowed. Essen-tially, any crossings that keep Q conserved and E(kx) single-valued are allowed. Adapted from Asb ´oth et al.
[].
Next, we argue that Q is a topological invariant. Suppose we have a system as in fig.. The band struc-ture as in (a) could describe an edge state to the right side of the chain of dimers, with three bands emerging from the valence band and submerging in the conduction band. Adiabatic and continuous deformation will not change the system. In the figure, it is clear that although the number of crossings through E = 0 can change, the net number of particles cannot, unless the bulk gap is somehow closed. This is not possible, by the earlier argument that Q must be conserved in this system. Furthermore, the criterium that E(kx)
is uniquely defined requires single-valuedness of the energy at all times, ruling out options (e) and (g) as possible smooth deformations of the Hamiltonian. This makes a strong argument that Q is conserved and topologically protected. Note that the consequences of this last fact are quite immense: since the properties are linked to the topology of the system, it follows that the materials are insensitive to impurities of the material.
Moving forward, to track where each particle is, we have to calculate the expectation value of the oper-ator defined in eq. (). To do so, we can use the expression for the so called “Wannier states” |w(j)i, which are the inverse Fourier transforms of the Bloch plane wave states |Ψ i,
|Ψ i= |ki ⊗ |u(k)i ()
with |ki defined in the usual way, |ki =√1
N
PN
m=1eimk|mi. The Wannier state |w(j)i is then defined as
|w(j)i =√1 N N δk X m=δk e−ijk|Ψ(k)i , ()
where we omitted a potential phase factor for brevity.
Using these expressions, we can let ˆx act on the integral representation of the Wannier state center as ˆ x |w(0)i = 1 2π Zπ −π X m eikm|mi ⊗ |u(k)i dk () = i 2π Z X m eikm|mi ⊗ |∂ku(k)i dk, ()
where we have used partial integration to go to the second line. The expectation value is then hw(0)| ˆx |w(0)i = i
2π Zπ
−π
hu(k)|∂ku(k)i dk + j, ()
Obviously at the loss of generality, however note that for the end result, the phase factor does not contribute.
which tells us is that the displacement of the centers, and therefore the location of each particle, is changing evenly during each adiabatic cycle through parameter space. It also follows that the location between particles is evenly spaced, which is trivial given our system is a lattice in the shape of a chain.
Suppose now, we call the above expression x(ti), and we split it up into infinitesimal pieces. First
calculate the difference across such an infinitesimal piece while omitting the k argument for brevity
x(ti+ ∆t) − x(ti) = i 2π Zπ −π hu|∂kui ti+δt− hu|∂kuiti dk () = 1 2π I ~ A · d ~R, ()
where we enclose a small rectangle with element d ~R with edges (and thus integration interval) ∂~R, and integrate over Berry connection ~A. The resulting expression can then be recast in a locally smooth gauge as
= 1 2π
Z
Adkdt, ()
where A is the Berry curvature. Since we have split up the cycle into parts, a complete cycle will now give us
∆x(t) = 1 2π
I
Adkdt, ()
that is, the number of particles displaced during each cycle is the Chern number. Upon further investi-gation, we can calculate that in the smoothly modulated Rice-Mele model, this displacement per cycle is .
Concludingly, the assumption about edge states, bulk states and the chosen Hamiltonian gave meaning to the Berry phase and the Chern number. Specifically, we started from the assumption of conducting edge states and insulating bulk states. The Hamiltonian we chose to consider in this system, could be described by a vector ~d dot multiplied by the Pauli matrices ~σ . This vector ~d traced out a torus, thus a geometrical shape with genus. The above assumptions led us to the Berry curvature, and to the Chern number. The Berry curvature can be shown to correspond to the current, while the number of particles pumped during each adiabatic cycle is the Chern number.
In closing, we wish to iterate that a material that is conducting at the edge but insulating in the bulk, is the definition of the Topological Insulator. This prerequisite is not exclusively satisfied by the symmetry class of Chern Insulators alone.
. Characteristics and applications
Topological Insulators host a variety of different physical properties related to their topology that make them useful.
Firstly, since the electric field is perpendicular to the surface, spin is locked to momentum. An argument for this claim presents itself readily, from the spin-Zeeman term from atomic physics,
HSOC= −µ · ~B. ()
The above term can be rewritten by writing ~B in terms of ~E ∝ ˆr, and then in terms of the gradient of the vector potential. Since ~k of the edge states is parallel to the surface, spin must also be locked to the surface. A more complete derivation can be found in the appendix. Closesly related to this is Rashba splitting[].
The Berry curvature can be interpreted as the phase change as H(~k) cycles through the parameter space of k.
The Chern number is a special number associated with the Berry phase, in terms of a winding number of the Brillouin zone
according to the topology of the band structure. Different physical meaning can be given by using mathematical equivalence. For example, [] describes the Chern number as being equivalent to the number of magnetic monopoles in a system, while the Berry curvature and connection respectively correspond to the magnetic field and vector potential.
Not proven here. Please refer to [] for more detail.
Figure: A typical shape of the Dirac cone in energy-momentum space. Adapted from [].
Writing the electric field as a potential gradient, and combining it with the assumption of the Quantum Spin Hall insulator leads to the splitting of spin channels.
It can be shown that even crude models ofD topological insulators have a conical dispersion relation at the surface (edge states): a Dirac cone. This Dirac cone is typical for the Topological Insulator, and is represented graphically in fig.. An example of a model where a modified Dirac equation and a Topological Insulator state gives rise to a Dirac cone can be found in e.g. [].
Further, there is a phenomenon called the Kramers theorem, which considers systems of particles with half-integer spin. This theorem states that each energy eigenvalue of the Hamiltonian is at least doubly de-generate. For the Dirac point, this means that both spin up and spin down would have identical energies, since each edge has spin locked to momentum. Chemically inducing the energy of a Topological Insulator, for it to lie at the Dirac point would then obviously lead to a spin-coherent phase, since both spin channels would have identical energies. It has been shown that this spin-coherent phase could then also be used to adjust spins of nearby ferromagnetic materials, opening the possibilities to a host of applications[].
Another consequence of the topological nature of the conducting edge states is that electrons are pro-tected from back-scattering. We propose to illuminate why this is in a crude, semi-classical way[]. Sup-pose we have an electron encountering an impurity on the lattice. As a consequence, it moves around the impurity and reverses direction in a clockwise or counterclockwise direction. As is well-known, the elec-tron picks up a minus sign under a full (2π) rotation, however under the assumption we have Kramer’s pair, implying we also have TRS, we know that the other particle at −~k additionally picks up the minus sign. Therefore, the contributions of the impurity cancel out, and we effectively have no backscattering. Please see [] for a more detailed discussion on this effect and its origins.
Topological Insulators in general have several applications. Firstly, the mechanism of Rashba splitting provides the necessary coherence of the spin channel. This in turn allows the use of spin as an extra de-gree of freedom and could prove useful in the synthesis of spintronic devices as an evoluting of electronic devices. Another application stems from the fact that topological insulators have surface states that are protected from backscattering. Therefore the heat dissipation from magnetic impurities is nullified, de-creasing the overal energy loss from resistance R.
Finally, [] showed that Topological Insulators are among the best thermoelectric devices because the thermoelectric size dependency was almost zero, making it a prime candidate for future studies on ther-moelectrics and thermoelectric applications.
. Light and topological insulators
An obvious reason why the photoresponse of any material would be interesting is to understand the band structure of a material, which gives many clues about more general characteristics of the material, e.g. the conductivity.
The optical response of Topological Insulators has special interest, however, since the action of the
I.e. at k = 0.
And thus the equations of motion.
material was directly related to the topology. This meant that knowing how local E and B fields affected photons passing through the material was of direct importance. With this reason, it was that so many experiments focused on measuring the Kerr and Faraday effects for TIs[–].
Another field to study is the photocurrent response of Bi2X3 (X = Te, Se) TIs, i.e. the study of inducing currents originating from the material by exciting electrons in it, and thus creating electron hole pairs. Photocurrent response is useful specifically since it is the primary driving force for the application of optical spintronic devices.
Firstly, we consider the dispersion relation of the material. Experimentally, it is approximated by fig., i.e. a single Dirac cone. Additionally, there is no spin-degeneracy at the edge states, but there is at the bulk (this is due to the spin-momentum locking of the edge states). Since it was shown that the spins parallel to the wave vector of the incident light are preferentially excited[], the necessary symmetry breaking can be made to induce such a photocurrent. If the excited electron stays inside the material, we specifically speak of the photovoltaic effect.
Previous attempts to describe this photocurrent response focused on linear approximations of an effect known to be non-linear[]. The resulting expressions only confirmed experiment within their approxi-mation, but not in the more general non-approximated numerical simulations[]. Existing experiments of the shift photocurrent response have shown that a warping current Jwarpin a Dirac surface state, should be of order J ∼ 10 pA · m−1[]. This study focuses on the shift charge current of Bi2
X3(X = Te, Se) in the low field E, assuming the inversion symmetry is broken and thus a photocurrent can exist. The study uses a Hamiltonian (taken from []). It works towards the expected current response to second order in the electric field as in [], and it hopes to confirm the experimental result from e.g. [].
The topology-dependent term is called axionic term of the action and can be extracted from field-tensor expression for action S.
For more information please refer to [, ].
This Bi-based compound specifically has rhombohedral symmetry and is thus noncentrosymmetric[].
Deriving the expression for the shift charge current
Abstract
Describing shift charge current in general requires a model of non-linear optical response. The Floquet two-band model is used as an approximation. In the low field limit, a Floquet Hamiltonian describing an anti-crossing between the valence band + photon and the conduction band is used to describe the exciton formation. Using Green’s functions, the current expectation value of the two anti-crossings is calculated using time reversal and inversion symmetry arguments. A final expression is calculated within the spherical basis, using a Bi2X3(X = Te, Se) Hamiltonian, taken from [].
In the discussion that follows, e = ~ = vF = 1. Appropriate projections to recover these constants can be found in
the appendix. Furthermore, many intermediate steps are not shown in the body text, and can also be found in the supplementary material.
. Describing current
.. Linear response theory
Suppose we look to describe the current ~J of an arbitrary system – of which a photocurrent would be a particular case. This current can be split up into any number of different directions, and can be proportional to any order of the electric field. In mathematical terms this general form can be denoted as
Jj= X i χjiEi+ X il χiljEiEl+ X ilm χilmj EiElEm+ . . . , ()
where subscripts below the summations imply summation over all (physical) dimensions that the parame-ter can take, χ refers to the generalized conductivity and Ei is the magnetic field in the i’th direction.
An example application of eq. () would be Ohm’s law, which describes the resistivity of a material, often used in the context of an insulator. Ohm’s law is sometimes formulated as
~J = σ ~E, ()
with σ ≡ χ1 a conductivity in the case of a linear proportionality to the electric field ~E, which is often a good approximation.
Oftentimes the leading order in eq. () is kept to make an approximation of the current, which for small fields is accurate, since higher orders of a small number are zero in the limit of Ei→0, i.e., the
low-field approximation. As was mentioned in the introduction, for the shift charge photocurrent, we expect a non-linear shift charge response
~J ∼ ~ E 2
. This proportionality suggests that linear approximations will not yield the results obtained from experiment.
Going towards the proposed proportionality of order is done here by the use of Floquet theory. The theory treats the Hamiltonian H(~k) as time-periodic, and expands H(~k) in k-space by the regular Fourier formulae. Then using the Green’s function approach to calculate the expectation value of the current, yields the desired proportionality. Generally, the Green’s function will not have a shape that is easy to calculate, however, in this case, in Fourier space, the Green’s function takes a form that is significantly easier to handle and to perform calculations with. The aforementioned framework will utilize this fact.
For a more extensive discussion on the methods applied here, please refer to [–]. .. Derivation steps
We will derive the expression in several steps.
. We will first describe the system which we look to describe with Floquet theory. . We will then derive the Floquet Hamiltonian.
k
E
Figure: From bottom to top, a valence band in blue, a conduction band in green, and a valence band with the additional energy of a photon again in blue. The circled area shows the anti-crossing. The non-equilibrium steady state describes the non-linear optical response, and is described by the Floquet Hamil-tonian HF. Adapted from [].
. Afterwards we will determine a formula χil
j from eq. () and the Green’s function approach.
. Finally, we will calculate χil
j for our system.
. Deriving shift charge current using Floquet and Green’s functions
.. Description of the system
The derivation here is similar to [] combined with []. In the low field limit, the system can be described by a Floquet two-band model. The proposed two-band model describes a steady-state d = 3 system, i.e. a D system with all parameters held constant, with a transition between the two bands at the (Dirac) d = 2 surface state of a topological insulator similar to Bi2Te3.
We look to describe a valence band with the addition of a photon from monochromatic light as donor, and a conduction band as acceptor for the electron-hole pair mechanism. Monochromatic light only has a single harmonic and will simplify the overall calculational complexity considerably. The non-linear optical response is described by the anti-crossings between the valence band plus photon and the conduction band. A schematic overview can be seen in fig..
Note that throughout the calculations done in the body text, only a single anti-crossing at +~k is sidered, while in reality, from the calculation of expectation value onwards, both anti-crossings at ±~k con-tribute. The omission of the second calculation is done because most of the math is identical in −~k. Any point where the contributions of current at +~k and −~k do not conspire, it will be mentioned specifically. .. Derivation of Floquet Hamiltonian
Suppose we have a system with a gauge potential and some momentum-dependent Hamiltonian in k-space, i.e. H(~k). This Hamiltonian can be expanded to first order in the time-dependent gauge potential A(t),
I.e., a second generation TI. In accordance with [, , ].
where A(t) ≡ E(t)/Ω (E intrinsic), by the use of a Taylor approximation about A(t) = 0 H1(t) = H0[k + A(t)] () = ∞ X n=0 A(t)d dk !n H0(k) n! () ≈H0(k) + A(t)dH0(k) dk ()
The obtained Hamiltonian H1(t) can now be expressed in terms of the so-called Floquet modes m, n by a Fourier expansion. The Fourier expansion is done over a characteristic timescale, such that Ω =2πT and
Hmn= 1 T ZT 0 exp [i(m − n)Ωt] H1(k)dt, ()
with which we can determine the Floquet Hamiltonian with the expression
HF= Hmn−nΩδmn. ()
This essentially takes the equilibrium Hamiltonian in Fourier modes, Hmn, and subtracts a quantum of
energy (from the photon) nΩ, according to the chosen Floquet modes. In this way we can add one photon to the conduction band n = 1, but none to the valence band m = n−1 = 0. See e.g. []. Note that the Floquet modes obey translational invariance, and as such, m and n can be chosen to correspond to any respective row and column index, as long as the relation m = n − 1 is obeyed.
In calculating Hmnwe here assume the gauge potential has the standard form, A(t) = iAe−iΩt−iA∗e−iΩt
(such that E = A/Ω). The drift velocity vi is defined by dH0(~k)
dki = v i
mn, where m, n again denote the Floquet
modes. Under these assumptions, the following expression holds:
Hmn= 1 T Z T 0 exp [i(m − n)Ωt] H0(k)dt + 1 T Z T 0
exp [i(m − n)Ωt] A(t)dH0(k) dki
dt ()
= H0(k)δmn+ ivimnAδm,n+1−ivmni A
∗
δm,n−1. ()
The now derived equation puts us in a position to determine the Floquet Hamiltonian formula, eq. (), and gives us HF= 1 − Ω −iA∗ v12i iAvi21 2 ! , ()
where vi is the velocity in the i’th direction and indices 1, 2 corresponding to the valence and conduction
band respectively. For future calculations it is convenient to recast the expression in terms of the short-hands ~ and ~d, where ~ is the energy of the original Hamiltonian, and ~d represents the deviation from this equilibrium,
= ~ + ~d · ~σ ()
Some math gives us the following set of equations (superscripts suppressed for brevity): =1 2(1− Ω+ 2) dx= 1 2(iAv21−iA ∗ v12) dy= − 1 2(Av21+ A ∗ v12) dz= 1 2(1− Ω −2) ()
Calculating the derivative of this will also give us the appropriate velocities to calculate the current densities later on. In particular, this will be of use when we wish to calculate J ≡ h ˜vi with h·i the expectation value and ˜v the (DC) current operator. We get
˜ vjl= ∂HFl ∂kj = v11j −iA∗ ∂vl 12 ∂kj iA ∂vl 21 ∂kj vj22 ()
which can be identically recast in the short-hand, where b0 is simply an off-set and bi is the DC current
operator split up into its individual spatial components,
= ~b0+~b · ~σ ()
with (indices suppressed)
b0= 1 2(v11+ v22) bx=1 2 iA ∂v21 dk −iA ∗∂v12 ∂k ! by= −1 2 A ∂v21 dk + A ∗∂v12 ∂k ! bz= 1 2(v11−v22) . ()
.. The shape of the generalized conductivity
The idea is now to calculate χilj from the expectation value of the current, by noting that it is the constant of proportionality from the E2term in eq. (). The expression for the expectation value can be calculated by noting that the so called “lesser Green’s function” can be viewed as a density matrix, such that
J = h ˜vi = Z
TrhvG˜ <idω, ()
where the integration over ω lets us go back to “real” (i.e. non-Fourier) space. The lesser Green’s function is given by G<= GRΣ<GA. More on the lesser self-energy Σ<later.
To first obtain the Green’s functions, we can make use of the Dyson equation, which is a master equation that stems from the assumptions of the Green’s function method. The Dyson equation relates the Green’s function to the frequency of the system, the Floquet Hamiltonian and the self-energy. The Dyson equation tells us GR GK 0 GA ! = (ω − HF+ Σmn) −1 . ()
Here, GR/K/Aare respectively the retarded, Keldysh and advanced Green’s function and ω the frequency in the system. To avoid ambiguity, we note here that the RHS has a dependency on the Floquet modes mn, such that we find a 2 × 2 matrix of individual 2 × 2 matrices.
This dependency can be understood from the shape of the Floquet mode dependent self-energy Σmn.
This energy has its origins in Quantum Field Theory (QFT) and will not be derived here, but some physical interpretation will be given to it. In brief, it measures the strength of the coupling of the system to a surrounding “heat bath”, and is approximatedin terms of Floquet modes with the formula[]:
Σmn= iΓ δmn 1 2 −1 + f (ω + mΩ) 0 −1 2 ! , ()
The self-energy will only have this form in the wide band limit. A discussion on derivation of self-energies in the context of this
paper, can be found in [].
with Γ the field strength of this QFT effect.
Effectively, only the diagonal elements of the Dyson expression need to be considered for this part of the calculation. This stems from the fact that the lesser Green’s function only considers the advanced and retarded Green’s functions, for which only the diagonal elements of the self-energy contribute. In our approximation, we will assume that the total self-energy is approximately given by the self-energy dependent on Floquet modes, as in [].
Inserting eqs. () and () into eq. (), we get
(ω − HF+ Σmn) = ω − ~ + ~d · ~σ±iΓ 2δmn () (ω − HF+ Σ)mm= ω − ± iΓ 2 ! · I −dxσ~x−dyσ~y−dzσ~z () = ω − ±iΓ2−dz −dx+ idy −dx−idy ω − ±iΓ 2 + dz ()
and from this we can readily determine the appropriate forms of the retarded and advanced Green’s func-tions, which were previously determined to suffice for our calculation:
(ω − HF+ Σ) −1 mm= ω − ±iΓ2−dz −dx+ idy −dx−idy ω − ±iΓ 2 + dz −1 () = ω − ±iΓ2· I+ ~d · ~σ ω − ±iΓ22−d2 . ()
By using the now determined shape of the Green’s functions,
GR GK 0 GA ! = (ω−+iΓ2)·I+ ~d·~σ (ω−+iΓ2)2−d2 GK 0 (ω−−iΓ2)·I+ ~d·~σ (ω−−iΓ2)2−d2 , ()
we can determine the general shape of G<:
G<= GRΣ<GA () = h ω − +iΓ2· I+ ~d · ~σiΣ<hω − −iΓ 2 · I+ ~d · ~σi ω − +iΓ22−d2 ω − −iΓ 2 2 −d2 . ()
with Σ<is the lesser self-energy. It can be shown that the lesser self-energy can be written as[] Σ<= iΓ1 + σz 2 = iΓ 1 0 0 0 ! , ()
again derived from QFT.
Using this shape of the lesser self-energy, we are finally in a position to calculate the lesser Green’s function and thus also J. To do so, we can take the trace and then perform the residue method for com-plex integrals over variable ω, where only poles of ω are considered that obey Im(ω) > 0. The resulting expression can be split up into partial contributions to the total current,
J = Z
(j1+ j2+ j3) d~k ()
Note that we here relate the Green’s functions according to the Floquet modes of the self-energy. Afterwards, the expression is no
longer Floquet mode dependent since the δ function was already applied. Therefore, the equality of Floquet modes is taken outside the braces to denote the complete expression is valid on the diagonals in the Dyson equation.
where the individual components can be grouped according to the first order terms in the z direction and the field strength Γ ,
j1= Γ 2 −dxby+ dybx d2+Γ2 4 , () j2= dz dxbx+ dyby d2+Γ2 4 , () j3= bz dz2+Γ 2 4 d2+Γ2 4 + b0. ()
By inspection, the first term can be rewritten using equalities from our original matrices as
−dxby+ dybx= Imhdx+ idy bx−ibyi () = Im " (iAv21) −iA ∗ ∂v ∂k ! 12 !# () = |A|2Im " v21 ∂v ∂k ! 12 # , ()
from which follows (with indices):
jj1= |A|2 Γ 2Im v21i ∂vl ∂kj 12 d2+Γ2 4 () = E 2 Ω2 Γ 2Im v21i ∂vl ∂kj 12 d2+Γ2 4 , ()
giving us the expected proportionality of the current ~J ∼ E~ 2
. From the last expression we read off the first contribution to the generalized conductivity as
χjil1 = 1 Ω2 Z 1 (2π)d Γ 2Im v21i ∂vl ∂kj 12 d2+Γ2 4 d~k, ()
where the superscript in χ denotes the contribution of the i’th partial current ji to the total generalized
conductivity χilj.
.. Other contributions to current
The second term j2can identically be evaluated by rewriting the terms as follows, dxbx+ dyby= Re h dx+ idy bx−iby i () = Re " iAv21 −iA ∗ ∂v ∂k ! 12 !# () = |A|2Re " v21 ∂v ∂k ! 12 # . ()
However, under time reversal symmetry, we have Θ : v217→ −v21, Θ : (vk)12 7→(vk)12 and Θ : dz 7→ −dz.
Consequentially, when integrating over ~k, each value of the integrand at +~k has a partner at −~k, such that the integrals sums up to zero.
Similarly, the projection Θ : bz7→ −bzimplies j3has no contribution to the expectation value.
This can also be understood from mathematical perspective, where the angular integration from 0 to 2π has no square of the
(co)sine, such that the integral vanishes and there is no contribution. Please refer to the next sections for more elaboration.
.. Simplifying the generalized conductivity
The next step is simplifying the generalized conductivity χ as much as possible before calculating it explic-itly.
One way to do this is to apply the resonance constraint of the proposed system, which implies that the detuning dzis zero. We will do this by inserting a delta function,
χilj = 1 Ω2 Z 1 (2π)d Γ 2Im v21i ∂vl ∂kj 12 d2+Γ2 4 d~k () = π Ω2 Z 1 (2π)d Γ 2 q |A|2|v12|2+Γ2 4 δ(dz) Im " v21i ∂v l ∂kj ! 12 # d~k ()
We can now make our expression even more explicit, which we will do by applying an appropriate warping Hamiltonian and transforming the system into spherical coordinates. For this we use a trivial constraint hz= h cos θ (trivially true within the spherical basis). The Bi2X3warping Hamiltonian is (as in []) Hwarp= h −ky, kx, λk3 x−3kxky2 i ·σ = ~h · ~~ σ , ()
where the short hands relate to the spherical basis as
hx= h sin θ cos φ = −kx () hy= h sin θ sin φ = ky () hz= h cos θ = λ kx3−3kxk2y . ()
Next, between the second and the third line, an approximation assuming small λ is made and higher orders of λ are discarded. Fu [] estimates a warping of λ ≈ 250 eV ˚A3=∧
4.01 · 10−47 J · m3, small enough to justify this approximation. Making the substitution, we find
χilj = π Ω2 Z 1 (2π)2δ(1−2− Ω) Im " v21i ∂v l ∂kj ! 12 # dkxdkyδ(hz−h cos θ)dhz () = 1 4πΩ2 Z h2δ(2h − Ω) Im " vi21 ∂v l ∂kj ! 12 #
δ(λ(h3cos3θ sin3φ − 3h3cos θ sin2θ sin3φ) − h cos θ) sin θdhdθdφ () ≈ 1 4πΩ2 Z h2δ(2h − Ω) Im " v21i ∂v l ∂kj ! 12 # δ(h cos θ) sin θdhdθdφ () = 1 4πΩ2 Z hδ(2h − Ω) sin θδ(cos θ) Im " v21i ∂v l ∂kj ! 12 # dhdθdφ. ()
.. Term-wise calculation of the generalized conductivity
At this point it is required to calculate derivatives of the warping Hamiltonian to determine χ in a particular direction. This can be done by calculating e.g. (vx)cv ≡
dH
warp
dkx
cv for the drift velocity in the x direction
from a transition from the conduction to the valence band. The derivatives are taken from the warping
Postulated here as a mathematical trick, it circumvents the need of calculating an indefinite integral by noting that the r
(h)-dependence is entirely enclosed by delta functions.
Henceforth, subscript 1 becomes subscript c and 2 becomes v, to give more physical meaning to the Floquet modes (although they
are essentially the same past the moment we determined the Floquet Hamiltonian onwards.
Hamiltonian in eq. (). The off-diagonal elements of the Pauli matrices are given by
(σx)cv= − cos φ cos θ − i sin φ ()
σy
cv= i cos φ − sin φ cos θ ()
(σz)cv= sin θ ()
and allow us to derive:
(vx)cv= i cos φ − sin φ cos θ + 6λh2sin2θ cos θ sin2φ ()
vy
cv= i sin φ + cos φ cos θ + 3λh
2sin2φ sin θ cos 2θ ()
dvx dkx ! cv = −6λh sin2θ sin φ () dvx dky ! cv = −6λh sin2θ cos φ () dvy dky ! cv = 6λh sin2θ sin φ, () which gives us χyyy = 1 4πΩ2 Z hδ(2h − Ω) sin θδ(cos θ) Im " (vy)vc ∂vy ∂ky ! cv # dhdθdφ () = −3λ 16, () χxxy = 1 4πΩ2 Z hδ(2h − Ω) sin θδ(cos θ) Im " (vx)vc ∂vx ∂ky ! cv # dhdθdφ () =3λ 16 = χ xy x , () χyxx = 1 4πΩ2 Z hδ(2h − Ω) sin θδ(cos θ) Im " (vy)vc ∂vx ∂kx ! cv # dhdθdφ () =3λ 16. ()
The other terms of χ evaluate to zero, which can be understood by revisiting the warping Hamiltonian, eq. (), and from an inversion symmetry argument.
In the projection kx7→ −kx, we get
Hwarp x7→ − Hwarp
x. However, in a similar fashion, the projection
ky7→ −kydoes not imply
Hwarp y7→ − Hwarp
y, i.e. Hwarpis only symmetric about x. A consequence is that
instances of χ in which kyappears an even number of times will not contribute to the expectation value of
the current.
This can be understood from the fact that we have two anti-crossings, and one of the terms cancels the other (this was also mentioned in the model specifications), in other words, for each point +~k there is a partner at −~k that manifestly cancels the contribution to the expectation value J.
.. Final expression for the shift charge current
Since Jj is given by Jj=PijχiljEiEl, we can calculate for e.g. Jx:
Jx= X ij χilx () = χyxx EyEx+ χ xy x ExEy () =3λ 16 EyEx+ ExEy . ()
Using linearly polarized light, ~E(t) = E0[cosφ ˆx + sin φ ˆy] cos(Ωt), it follows that ExEy=1
2E 2
0sin 2φ cos2Ωt. ()
A similar calculation for Jygives a total shift charge current of
~J =3λE 2 0
16 (sin 2φ ˆx + cos 2φ ˆy) . ()
Discussion and conclusion
. Validity of used approximations
The used approximations in this paper were as follows.
. First we assumed the Floquet two-band model accurately described exciton formation.
. Second, we assumed a small electric field, such that the first order Taylor approximation was accurate. . Third, we used an approximated form for the self-energy, Σtotal≈ Σmn.
. We further approximated the lesser self-energy Σ<,
. and discarded higher orders of λ in the final stages of calculating χil j.
As is evident from the above list, we can see that the approximations made had decreasing impact on the final result as the derivation progressed.
Looking at our biggest simplification, we can improve the result by refining the used model. For exam-ple, since the proposed band model is gapped, we could additionally allow a scattering process from bulk to edge state (or reverse) to occur by use of additional resonance conditions for the detuning.
Since the derivation done here assumes low field, one could naively expect high field corrections to follow by approximating χilj = π Ω2 Z 1 (2π)d Γ 2 q |A|2|v12|2+Γ2 4 δ(dz) Im " v21i ∂v l ∂kj ! 12 # d~k, ()
in the limit |A|2|v12|2 Γ2
4. However, the resulting expression, χjil= 1 8πΩ Z Γ E × 1 v i 12 δ(dz) Im " v21i ∂v l ∂kj ! 12 # dkxdky, ()
proves unphysical, as some terms diverge. This suggests the need for an approach with either more terms in the Taylor polynomial in the gauge potential A(t). One thing the above expansion does tell us is that the high field shift current could be approximated with a linear proportionality to the field strength, J ∼ Γ E, rather than J ∼ E2. In this fashion, the previously proposed models using Fermi’s Golden Rule could yield accurate results[].
Why our model will not be accurate in the high field, could follow from a consideration relating to the Floquet two-band model. Namely, in the high field limit, we are describing electron saturation. This suggests that no more electron-hole pairs exist which satisfy the resonance condition dz= 0. Consequence:
our assumption that the Floquet Hamiltonian accurately describes the system, no longer holds.
. Comparison to experiment and other literature
The obtained dispersion relation, resonance condition and the individual coefficients for Jwarpare shown in fig.. Immediately striking is that, indeed, the Hamiltonian from [] does approximately obey the Dirac cone shape as claimed in the introduction.
The obtained expression can additionally be numerically evaluated by rewriting the electric field as EiEl = 2I0
0c
, and using the Fermi velocity estimated from Fu [], v
F = 2.55 eV · ˚A =
∧
3.87 · 105 m · s−1 (divide by ~ to go from natural units to SI units, dimensional analysis confirms this) and λ = 250 eV · ˚A3=∧
This is done by taking the time average of the Poynting vector, see e.g. [].
Figure: Dirac cone, resonance condition and warping in (a) with energies and momenta made unitless through multiplication of resp. √vF/λ and
q
vF3/λ, (b-d) show coupling strength to vy, (vy)ky and vy(vy)ky,
resp. Adapted from [].
4.005 · 10−47kg · m5·s−2. The obtained approximation for the amplitude in terms of the intensity of light I0 is Jj= 31.602 · 10−19A · s34.005 · 10−47kg · m5·s−2 161.055 · 10−34m2·kg · s−13(3.874 · 105m · s−1)2 2I0 kg · s−3 8.854 · 10−12s4·A2·m−3·kg−1 (2.998 · 108m · s−1) () ≈132 · 10−12I0 A 3·s3·kg · m5·s−2· kg · s−3 m6·kg3·s−3·m2·s−2·s4·A2·m−3·kg−1 ·m · s−1 = 132 · 10 −12 I0 A m, ()
in agreement with the CPGC (circular photogalvanic current) estimation in Hosur [] (with a deviation of ∼30%) and Lindner et al. [] (deviation ∼ 80%), but two orders of magnitude compared to experiments using oblique incident light as presented in the introduction by [] (cf. McIver et al. [], Olbrich et al. [], assuming a 1 mm2sample size).
As described in the previous subsection, in the high field a linear proportionality is estimated using the theoretical framework outlined in the main body. Estimations similar to the one made using FGR in Junck et al. [] may prove useful there.
. Link to the topology of the system
The correlation functions calculated above can be shown to have a direct relation to the geometrical phase. Specifically, Morimoto and Nagaosa [] that the following expression holds:
∂v ∂k ! 12 = v12 " ∂ϕ12 ∂k + ~A1−A~2 # = v12Rk () where ϕ21≡arctan d y dx
and the part between the brackets defined as Rk. This expression was called the
shift vector, contained Berry connections Ai for the conduction/valence bands and was gauge invariant
(necessarily so, since the exact calculations from the spherical basis onwards were similarly gauge invari-ant). Using this shift vector notation, Sipe and Shkrebtii [] obtained an identical expression for the shift charge current for semiconductors.
. Variations of the outlined approach
Besides the first harmonic generation, the second-order non-linear response and the shift charge current, a variety of variations of the derivation exist. See for example [] and [], in which second harmonic generations, third order non-linear responses and spin shift currents were additionally calculated. For details and steps of these calculations, please refer to appendices C. to C..
. Suggestions for further research
Firstly, the E-proportionality in the high field limit requires further experiment to be determined. A first attempt at estimating this proportionality theoretically, is made in eq. (). A further calculation in the appendix shows our expression in this limit diverges, suggesting a different approach is needed to derive the shift charge current in high E. A possible solution is to consider more terms of the gauge potential A(t) to calculate the Floquet Hamiltonian. Naturally this would lead to substantially more cumbersome calculations. An obvious recommendation is the use of numerical methods.
Secondly, a gross simplification is made in the shape of the self-energy. Both the Floquet self-energy as well as the lesser self-energy were approximated in the calculations made in the main body. Solidifying a particular shape for this self-energy with experiment, could prove or disprove the accuracy of the method determined here.
Finally, a recommendation is made for a generalization to higher order harmonics. The delta functions that originate from the SHG approximation show surprising regularity, as is expected with the straight-forward method of calculation for the shift current expression. Capturing a generalized method for the n-order harmonic and determining whether this limit converges would be a first step towards proposing the Floquet two-band model as a general method for describing photocurrent response in Topological In-sulators. An obvious caveat would be irregularities in the n > 2 order harmonics of the photocurrent.
. Overview of the main derivation & Outlook
Since the shift charge current is described by an ~ E 2
proportionality, a framework containing non-linear response was required. To describe the shift charge current of a Bi2X3(X = Te, Se) in low field, an expression for the shift current was derived using the Floquet formalism combined with the Green’s functions method. In the Floquet formalism, we used a Floquet two-band model with a valence band and a photon, and a conduction band. An anti-crossing between the two bands described the non-linear optical response of the system.
The resulting Floquet Hamiltonian and DC current operator allowed us to calculate the lesser Green’s function. Time reversal symmetry reduced the three terms jithat followed from this expression to just one,
j1. The then derived expression could be calculated in the spherical basis, by taking a Bi2X3 (X = Te, Se) warping Hamiltonian from Fu [], from which we calculated shift charge coefficients. Some coefficients could be discarded using inversion symmetry. The obtained current Jwarp = 0.13I0 nA · m
−1
agrees with experiment to two orders of magnitude at worst and% at best.
Further research on the accuracy of the self-energy expressions used throughout the derivation may improve the accuracy of estimations, or suggest a direction for a better model. Qualitative calculations of the derived expressions in higher E suggested the model was not accurate in the high field limit. This suggests a need for an alternative description of the anti-crossing in high field, which is possibly given by the higher order Taylor expansion of the time-dependent gauge potential A(t), or a more general form of the self-energy which can be used outside the wide-band limit.
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