Floquet spin states in graphene under ac driven spin-orbit interaction
We study the role of periodically driven time-dependent Rashba spin-orbit coupling (RSOC) on a monolayer graphene sample. After recasting the originally system of dynamical equations as two time-reversal related two-level problems, the quasi-energy spectrum and the related dynamics are investigated via various techniques and approximations. In the static case the system is a gapped at the Dirac point. The rotating wave approximation (RWA) applied to the driven system unphysically preserves this feature, while the Magnus-Floquet approach as well as a numerically exact evaluation of the Floquet equation show that this gap is dynamically closed. In addition, a sizable oscillating pattern of the out-of-plane spin polarization is found in the driven case for states which completely unpolarized in the static limit. Evaluation of the autocorrelation function shows that the original uniform interference pattern corresponding to time-independent RSOC gets distorted. The resulting structure can be qualitatively explained as a consequence of the transitions induced by the ac driving among the static eigenstates, i.e., these transitions modulate the relative phases that add up to give the quantum revivals of the autocorrelation function. Contrary to the static case, in the driven scenario, quantum revivals (suppresions) are correlated to spin up (down) phases.
pacs:81.05.ue, 71.70.Ej, 72.25.Pn
One of the key features of relativistic (massless) free particle states is that they evolve, at least in effectively one-dimensional situations, in time without spreading. This in turn relies on the linear nature of the relativistic dispersion relation which for photons reads , with c the speed of light. The condensed matter relativistic-particle analog is found in the low energy approximation (long wavelength) of single layer graphene where the chiral massless particles move with a speed novoselov1 (). This linear spectrum provides graphene with remarkable transport properties such as high mobilitygeim (), Klein tunnelingguinearmp () and unconventional spin Hall effectsarmarmp (); kane (). The later stems from the interplay between intrinsic spin-orbit interaction and the coupling extrinsically induced by external gate voltages or an appropriate substrate. This extrinsic couplingmacdonald (); guinea (); stauber (), so called Rashba spin-orbit (RSOC) has been also found to give rise to spin polarization rashbapol () and relaxation fabianrelax (); guinearelax () effects.
Although the role of static RSOC on graphene has been extensively discussed in the literature, to the best of our knowledge, the role of periodically driven time dependent RSOC on graphene samples has not been analysed so far. Yet, recent works have focused on the dynamical features of charge currents induced by means of time dependent extrinsic spin-orbit interaction on mesoscopic semiconductor quantum rings where Rabi oscillations are shown to appear as well as collapse and revival phenomenapeeters1 (). The main motivation of our work is twofold: First at all, we are interested in determining the feasibility of ac driven fields to generate and modulate a finite spin polarization of carriers in graphene for states that under static conditions remain unpolarized. In addition, we are also interested in the dynamical modulation of the effective Rashba coupling strength which would allow to explore regimes beyond the static limit domain.
Taking advantage of the periodicity of the problem, the evolution equations can be solved via Floquet theory. A standard approach here consists of expressing the Hamiltonian in a Fourier mode expansion leading to an infinite-dimensional eigenvalue problem for the so-called quasi-energies milena (); chu (). This quasi-energy spectrum carries nontrivial information on the topological nature of the system under studytopological1 (), and for semiconductor quantum wells with a zincblende structure it has been recently shown that ac driving can induce a topological phase transitionfti ().
Practically, in order to treat the infinite eigenvalue problem one has to truncate at an order of the harmonic expansion chosen appropriately to yield well-converged results. An alternative approach to the Floquet problem which does not rely on Fourier expansions has been devised by Magnusmagnus1 (). This method appears to be somewhat less popular and amounts in formulating the time evolution operator as the exponential of a series of nested commutators. It has the virtue of both preserving unitarity at any order in the series expansion (in contrast to truncation of the Dyson series within a perturbative approach) and avoids the infinite-dimensional eigenvalue problems. Followingmagnus2 () we will make use of the Magnus expansion approach combined with Floquet theory in order to generate semi-analytical solutions of the dynamics induced by periodic RSOC.
Since RSOC couples the spin and pseudospin degrees of freedom the problem is, for a given wave vector, generically four-dimensional. However, by an appropriate unitary transformation, the evolution equations can be recast as a set of two equivalent two-level Schrödinger equations related by time reversal. In this way we can explicitly analyse which static states get dynamically coupled.
Our main results are the following: The ac driven RSOC induces a quasi-energy spectrum where the original gap due to static spin-orbit coupling is dynamically closed. In particular, at the Dirac point () the dynamics is exactly solvable with zero quasi-energy Floquet states. This quasi-energy spectrum is two fold-degenerate as a consequence of time reversal invariance of spin-orbit interaction, and the closing of the original gap is due to the destructive interference induced among the initially uncoupled positive and negative energy RSOC eigenstates. Then we show that sizeable alternating out of plane spin polarization ensues on states that under static conditions remain unpolarized. We also find that the uniform interference pattern shown by the autocorrelation function for static RSOC gets distorted due to the interlevel mixing of the static eigenstates which dynamically modulates the relative phases that add up in the quantum revivals of the autocorrelation function. In the driven case quantum revivals (suppressions) are directly correlated to spin up (down) phases of the out of plane spin polarization. Since the autocorrelation function is related to the Fourier transform of the local density of stateskramer (), and because spin probes can be more demanding in practical implementations than charge detection, its evaluation yields useful indirect information on the spin degree of polarization. We believe these findings have the potential to provide interesting new strategies to dynamically control spin properties of charge carriers in graphene for future spintronics applications.
The paper is organized as follows. In section II we describe the spectum for static RSOC and introduce the model hamiltonian for periodically driven RSOC. Here we also present the exact solution to the dynamical equations corresponding to the Dirac point . The main results of the Floquet-Magnus approach for the semi-analytical solution of the evolution operator at finite momentum are presented in section III. Next, in section IV we compare the quasi-energy spectrum obtained through Magnus approach to that given by making a rotating wave approximation. We also evaluate and discuss the out of plane spin polarization as well as the onset of quantum revivals for the autocorrelation function. Finally, in section V we give some concluding remarks and discuss an experimental scenario where our results could be tested.
We consider a graphene monolayer sample subject to periodic time dependent spin-orbit interation of the Rashba type. In graphene RSOC interaction emerges as a consequence of and orbital mixingyao () and stems from the induced electric field due to the substrate over which the graphene sample lies or by applied gate voltages. Then, a periodic modulation can in principle be implemented by means of time dependent gate voltages or by the induced time varying electric field within a parallel plates capacitor coupled to a LC circuit. Under these circumstances the induced RSOC perturbation could be given a periodic time dependence , where the driving amplitude will be assumed to be periodic with , the coupling strenght in the static case.
Concerning energy scales the value of the intrinsic and extrinsic spin-orbit coupling parameters and in graphene have been obtained by tight binding guinea (); zarea () and band structure calculations macdonald (); yao (). They gave estimates in the range , much smaller than any other energy scale in the problem (kinetic, interaction and disorder). However, the RSOC strength has recently been reportedrashbaexp () to be of order , where is the value of the first-neighbor hopping parameter for graphene within a tight binding approach.
The formulation of the problem is as follows. In momentum space and and taking into account the energy scales of spin-orbit coupling we can work within the so called long wavelength approximation, where the total hamiltonian for monolayer graphene in presence of time dependent RSOC can be described by the hamiltoniankane ()
where is the Fermi velocity in graphene, is a vector of Pauli matrices, with describing states on the sublattice and so called pseudo spin degree of freedom, whereas describes the so called Dirac points and , respectively. In addition, is the momentum measured from the point and (i=0,x,y,z) represents the real spin degree of freedom, with the identity matrix. In addition, gives the time dependence of the RSOC and we have neglected the intrinsic spin-orbit contributions. Now since RSOC does not mix the valleys, we can focus on any of the two Dirac points, say and then the results for the point are found by the substitution . Yet, we will formulate the problem in an isotropic way such that the results for the Dirac point will inmediately follow. Before dealing with the time dependent problem we summarize the main results for static RSOC.
The spectrum of the noninteracting Hamiltonian
is given by the linear dispersion relation
whereas its eigenbasis is spanned by the spinors
with , and () describes electron (hole) states.
When a static RSOC interaction term is present the hamiltonian near the point reads
and the energy spectrum changes to with
Since RSOC mixes the and atomic orbitals it induces a gap at the Dirac point . The static Rashba hamiltonian in Eq.(5) is diagonalized by the unitary transformation given explicitly as
where . In this basis the static RSOC hamiltonian reads
This particular choice of basis will simplify the calculations that follow.
We are interested in analyzing the emergent dynamics of Dirac fermions in monolayer graphene when the amplitude of RSOC is a periodically varying function of time , with and the amplitude and frequency of the driving term. Then we would have to deal with the following evolution equations
with both sub-blocks periodic functions of time i.e. . Therefore the unitary transformation represented by (7) simplifies considerably the mathematical resolution of the evolution equations by recasting the problem as two time reversal pairs of coupled two level problems. In addition, it has the physical appealing feature of clearly giving the subset of states which are dynamically coupled through the time dependent interaction.
Let us then focus on the upper block which reads
whereas the lower block is obtained by changing the sign of the amplitude . Because of this symmetry relation among the two subspaces their quasi-energy spectra are identical. This is to be expected since RSOC is time reversal invariant.
First of all, we notice that in the static limit the reduced hamiltonian (10) is block diagonal
We also note that at the Dirac point , one gets
In this case the resulting dynamics
is exactly solved by the eigenspinors
As will be discussed below, these solutions correspond to zero quasi-energy Floquet states. The corresponding evolution operator is diagonal and given as
Iii Magnus-Floquet approach
Although we have shown that at the dynamics is exactly solvable, this is no longer true for finite . Then, we need to resort to approximate solutions. As we discuss below, a semi-analytical approach known as Magnus-Floquet expansion will be suitable for dealing with the dynamical equations of periodically driven systems. Since the Magnus-Floquet approach is not so popular in the literature we now briefly summarize its main results (see reference magnus2 () for more detailed derivations).
In the language of differential equations, the matrix solution of a -dimensional system of dynamical evolution equations (here we omit the orbital degrees of freedom for ease of notation),
i.e. a matrix that satisfies
is called a fundamental matrix solution if all its columns are linearly independent. If in addition, there is a time such that is the identity matrix, then is called a principal fundamental matrix solution. To solve Eq.(19) Magnusmagnus1 () proposed to find exponential solutions to the evolution operator in the form
and then wrote as an infinite series
where each term is given as a combination of nested commutators, with the first terms reading as
where and are matrices, such that is periodic and is time independent. Floquet’s theorem is the time dependent analog of Bloch’s theorem in solid state physics for spatially periodic structures and it provides a time dependent transformation such that the so called Floquet states evolve according to the time independent matrix . This time dependent transformation is implemented by .
One important remark is in order since although the interaction is periodic, the corresponding evolution matrix is not, i.e. . In fact carries indeed non trivial information on the dynamics of the periodic system. The eigenvalues of are called Floquet exponents . These Floquet exponents can be found by diagonalizing . Yet, they are not uniquely defined since leaves invariant.
In order to determine those exponents one standard approach consists of performing an expansion in the (infinite) eigenbasis of time periodic functions (Fourier modes). In that periodic basis the evolution operator is diagonalized and the Floquet exponents are the logarithms of the eigenvalues of the evolution operator evaluated at , i.e. . Then, in order to deal with the infinite eigenvalue problem one resorts to a truncation procedure in order to determine the Floquet exponents.
The Magnus approach avoids the need to solve the infinite dimensional eigenvalue problem and has the physical virtue of preserving unitarity of the evolved state to any order in the expansion. The connection between Magnus expansion and Floquet’s theorem is found in reference magnus2 () where the authors present a solution of the evolution equations that consists of writing the periodic part as an exponential
and then they proceed to expand both operators and in power series
Now, since is by construction a principal fundamental matrix solution is the identity matrix and one gets for all values of
Introducing the Bernoulli numbers (, , ) such that for , the exponent operator term contributions satisfy a recurrence relation in terms of two auxiliary time dependent operators and , according to the relations
In turn, the and are given through the iterative relations
In practical calculations, the relations in the last two lines serve to initialize the iterative procedure.
Iv Results and discussion
In order to apply the previous results to our problem one just has to make the following identifications , . Then, the characteristic exponents are proportional to the quasi-energies .
For our calculation we choose a time dependence of the form , where is the frequency of the driving and the RSOC strength and proceeded to evaluate by the iteration procedure described in the previous section.
We have found a characteristic behavior of the (adimensional) quasi-energies , which qualitatively do not change when one goes beyond third order in the Magnus-Floquet expansion (in the appendix we briefly describe the calculations up to fifth order). For finite they explicitly read as
where we have defined the adimensional quantities and .
In FIG.1 we show the typical behavior of the quasi-energies (blue and red, thick continous lines) given in equations (34) as function of the adimensional quasiparticles non interacting energy . We also show (gray, thin, dashed lines) the corresponding static eigenvalues of the RSOC hamiltonian as described by , as well as the eigenenergies of the non interacting hamiltonian (thin, dashed, black lines). We have also included (red and blue, thick, dashed lines) the result from a -mode Fourier expansion of the quasi-energies. We have expressed all quantities in units of the first neighbor hopping parameter and for finite have set and change the effective coupling . This is true for all the figures within the dynamical case.
Since mixes the static eigenstates of the relative phases among them are dynamically modulated giving rise to interference phenomena and we find that this leads to a dynamical closing of the original gap . Therefore, the exact solutions at correspond to vanishing quasi-energies (modulo ). We further find (see arrows in FIG.1) that in the limit , corresponding to a highly oscillating field, because then the influence of the driving quickly tends to vanish on average and intuitively one expects to recover the non interacting linear spectrum. The result from a mode Fourier expansion shows qualitative agreement to the Magnus-Floquet approach, however a small discrepancy is found and the Magnus result converges faster in the limit of highly oscillating fields , as depicted by thick arrows in FIG.1.
Within the Magnus-Floquet approach the evolution operator corresponding to the hamiltonian is found to be given as
where is a vector of Pauli matrices and the non vanishing components of the vector are given as
whereas those of are
At the Dirac point we find that all vanish, whereas , and thus we recover the exact solution (17).
Before we proceed to evaluate other physical quantities of interest we make a brief disgression on the importance of taking into account the full time dependence of the RSOC. In particular, due to smallness of the RSOC strength one can try a rotating wave approximation (RWA) where for a given finite value of only near to resonance () terms are kept in the interacting hamiltonian. Now we show that for the present model this approach gives unphysical results, for instance, it predicts a gap opening at . This in turn would imply finite quasi-energy mode contributions at the Dirac point which contradicts the previously described exact result.
Since the results are easily found within the -dimensional formulation we briefly return to the original basis and make a full dimensional discussion. Within a RWA approach, the hamiltonian reads (here we omit the spatial degrees of freedom for ease of notation
which for a given value of describes near resonance spin and pseudo spin flipping processes and neglects the so called secular or counter-rotating terms that oscillate rapidly. In this case the solution is exact and the adimensional quasi-energy spectrum reads
where describes the resonance and .
When we evaluate at and finite one would get the gaps for the originally gapped states and a dynamically opened gap on the static degenerate states . These results are in disagreement with the exact solution to the full equations (15) where we had found zero energy solutions at so this approach is not suitable to describe the dynamical features of the periodic driving. In FIG.2 we depict the quasi-energy spectrum as a function of non- interacting energies for this RWA solution.
After this brief discussion we turn back our attention to the Magnus-Floquet solution in order to get additional information on the dynamical behavior induced on the system. This can be found by evaluating some other physical quantities of interest from an experimental point of view. First we analyse the out of plane spin polarization , where and the identity matrix in pseudo spin space. Using again the transformation (7) the out of plane spin polarization reads and is found to be isotropic and block anti-diagonal
with given explicitly as
with a vector of Pauli matrices and the two dimensional identity matrix. The anti-diagonal structure of in this basis reflects the fact that spin polarization is not conserved in presence of RSOC.
Then we have to evaluate
for any initially prepared state . Next we separate explicitly the four spinor in upper and lower components as
where are normalized two dimensional spinors. After some algebra one gets for the spin polarization in terms of adimensional parameters
For a finite value of we now choose the initial spinor configuration , in such way that the out of plane polarization vanishes for a static RSOC.
In figure FIG.3 we show a density plot of the resulting out of plane spin polarization for the exact solution . In this case, the only relevant parameter is the adimensional amplitude of the driving field . As expected, for the system remains unpolarized for all values of the adimensional time (given in units of ). For finite values of the effective coupling an alternating pattern of spin phases (denoted as and representing up and down, respectively) are seen to apper as time evolves. They are symmetrically distributed among the values where and thus the relative phases among the static RSOC eigenstates vanish.
However, as shown in figure FIG.4, once is finite this panorama qualitatively changes. In this case, the additional interference due to level mixing induces a pattern of alternating maxima and minima for , . The reason for such a behavior is that for a given the evolution operator is given by and thus the polarization maxima and minima depend essentially on the quasienergy spectrum properties. Then, increasing makes these alternating maxima and minima to get closer. When we move to the next period (corresponding to in figure FIG.4) the number of alternating maxima and minima is doubled, and so on. Therefore, dynamical coupling produces a non vanishing out of plane spin polarization with an oscillating time pattern resulting from the mixing of the static eigenstates such that the changing relative phases, modulated by the time dependent interaction prevent total destructive interference to happen. Therefore ac driven RSOC provides a suitable means to dynamically control the degree of spin polarization and could in principle serve to generate non vanishing and non trivial spin polarized phases in otherwise unpolarized states in monolayer graphene.
In order to complement the just described physical picture of the spin polarization scenario we evaluated the autocorrelation function . This is given by the projection of the evolved state along a given (in principle arbitrary) initial spinor configuration , i.e.
The absolute value of the autocorrelation function provides information on the so called recurrences or quantum revivals of the dynamics, i.e. those values of the time parameter for which such overlapping is a maximum. In addition, its Fourier transform is proportional to the local density of stateskramer ().
In figure FIG.5 (FIG.6) we plot the absolute value of obtained by means of the exact (semi-analytical) evolution operators. For the exact solution shown in figure FIG.5 only large values of induce an appreciable phase change and the system remains mostly correlated to the initial state. As for the case of the out of plane spin polarization, for the autocorrelation gives maxima corresponding to red (dark gray) zones in the figure. These signal the return of the system to initial vanishing spin polarization. Maxima and minima of spin polarization correspond to partial quantum revivals. Given the definition of the autocorrelation as the probabibility that the system returns to its initial state we find that for giving the loci of correspond to vanishing of spin polarization and are thus equally spaced as . On the contrary, the maxima and minima of spin polarization correspond in this case to quantum suppressions and are shown as blue (dark) zones in the figure. As soon as we move towards finite values of (FIG.6) interference phenomena come again into play and we find recurrences represented as read (dark gray) zones and suppressions, described by purple (black) zones. These arise because of the constructive and destructive interference effects, described previously and are modulated by the ac driving.
To check that this physics is different from the static scenario, in Figures FIG.7 and FIG.8 we depict the corresponding countour plots for time independent RSOC interaction. For the case shown in FIG.7, the alternating pattern of quantum revivals and suppressions is separated by the inverse of the energy gap . This is due to the fact that the in this situation the relative phase among the non interacting eigenstates is set by the energy separation among them, i.e. , and according to Heisenberg’s time-energy uncertainty relation one should have inversely proportional to the energy separation. As soon as is finite, a quasi homogeneous pattern of recurrences is seen. Therefore although in the static case where the spin polarization vanishes for all values of RSOC strength, there appear recurrences indicating there is no longer correlation between and , which for the system under study are only correlated for time dependent driving.
Two comments are in order here. The first one is concerning the validity of the Dirac fermion hamiltonian approximation in order to deal with spin-orbit related phenomena in graphene monolayer. As it is discussed in referenceyao () the large gap of the hamiltonian describing the orbitals implies that the effective hamiltonian including SOC is essentially given by the long wavelength or Dirac Hamiltonian considered in our model. In addition, M. Gmitra, et almitra () used first principles calculations to discuss the relevance of spin-orbit related physics in graphene near and beyond the Dirac points. The second point to remark is the role of localized impurities. As has been discussed incastro (), the presence of local impurities can enhance the value of the static intrinsic spin-orbit coupling strength because of the induced distortion which leads to a hybridization of the and orbitals and as shown in referencemitra () RSOC only respond to this hybridization. Therefore, we would expect our results to be robust or even enhanced if localized impurities were included in the model.
Now we would like to compare to other proposals of dynamical modulation of energy gaps under ac drivenin graphene. For instance, Oka and Aokioka () found that circularly polarized intense laser fields can induce a photovoltaic effect in graphene, that is, a Hall effect without magnetic fields. This in turn relies on the gap opening at the Dirac point . However, as shown recently by Zhou and Wuwu () in analyzing the optical response of graphene under intense THz fields, the contributions from large momenta make the effective gaps opened to get also dynamically closed. However, it is found that the quasi energy spectrum for the linear polarization leads to linear quasi energy spectrum (see also foa ()). Our model could in principle be mapped into their scenario of linearly polarized radiation field. Yet, in both works wu (); foa () the case of linear polarization leads to linear quasienergy spectrum. Therefore, from the semi parabolic quasi energy spectrum shown in figure (FIG.1) we can infer that the bending of the quasienergy spectrum makes the spin-orbit driven scenario qualitatively different from these other approaches to dynamical control of graphene electronic properties. Our argument relies on the fact that the topological properties of periodically driven systems are characterized by the quasienergy spectrumtopological1 () so we can conclude that the physics related to ac driven spin-orbit does reveal new physical interesting electronics properties which are absent in the static regime.
We have investigated the role of periodically driven RSOC in monolayer graphene and recasted the original -dimensional problem as an equivalent set of two two-level problems. Due to the induced modulation of the relative phases among the static eigenstates we found a closing of the static gap at the Dirac point . This result is in agreement with the available exact solution and differs from RWA where an unphysical gap is seen to appear. This physical picture is confirmed through a Fourier mode expansion and we found that Magnus Floquet approach indeed has the advantage of providing the quasienergy spectrum with less computational effort. We also found that the generation and manipulation of out of plane spin polarization for otherwise spin unpolarized states requires the time driving to be realizable within this set up. Due to the induced interlevel mixing among the static eigenstates we found a set of alternating positive and negative spin phases in clear distinction to the well separated spin phases at the Dirac point corresponding to the exact solution. The dynamical onset of quantum revivals described through the autocorrelation function is directly correlated to the appearance of maxima for either spin phases. However, in the static case, such a correlation does not ensues since the spin polarization vanishes identically whereas the quantum revivals are still present. Concerning the actual experimental realization of our proposal we believe it could be implemented by means of Magnetic Resonance Force Microscopy as reported inrugal (). Within this scheme, single spin polarization could be detected by means of the frequency shift induced on a cantilever that is used to scan the sample. The sign of the cantilever’s frequency shift can be associated to the spin polarization. This detection is achieved by means of low intensity magnetic fields under resonant conditions, thus no magnetic coupling terms need to be included in the description of the dynamics of the charge carriers in graphene. In this way, although in principle RSOC in graphene has a small static strength, the time dependent efective phenomena produce some interesting spin controlling strategies which we believe could provide a route to new implementations of graphene in spintronic devices with appropiate spin detection techniques.
Acknowledgments– Z.Z.S. thanks the Alexander von Humboldt Foundation (Germany) for a grant, and A.L. thanks DAAD-FUNDAYACUCHO for financial support. This work has been supported by Deutsche Forschungsgemeinschaft via GRK 1570.
Appendix A derivation of iterative terms and quasi energy spectrum
Using the simplifying notation for the time dependent interaction one gets to fourth order in the iteration procedure that
and performing the corresponding calculations one gets
whereas the even contributions all vanish. The quasienergies are obtained through diagonalization of the selfadjoint matrix and up to this order one gets
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