# Chirality, charge and spin-density wave instabilities of a two-dimensional electron gas in the presence of Rashba spin-orbit coupling

###### Abstract

We show that a result equivalent to Overhauser’s famous Hartree-Fock instability theorem can be established for the case of a two-dimensional electron gas in the presence of Rashba spin-obit coupling. In this case it is the spatially homogeneous paramagnetic chiral ground state that is shown to be differentially unstable with respect to a certain class of distortions of the spin-density-wave and charge-density-wave type. The result holds for all densities. Basic properties of these inhomogeneous states are analyzed.

###### pacs:

71.10.Ay, 71.10.Ca, 72.10.-d, 71.55.-i^{†}

^{†}preprint: APS/123-QED

## I Introduction

Recent interest in the properties of the quasi-two dimensional electron and hole devices in the presence of structural (Rashba-Bychkov)Rashba (1959, 1960) or intrinsic (Dresselhaus)Dresselhaus (1955) spin orbit has brought to the fore the problem of the interacting chiral electron liquid. It is therefore important to revisit several of the fundamental notions of many-body theory for this intriguing system. The purpose of present paper is to begin a theoretical exploration of the relevance and special properties of a class of spatially non homogeneous spontaneously broken symmetry states of the electron liquid in the presence of Rashba spin-orbit coupling in two dimensions. Specifically we will focus our attention on spin density and charge density wave type states henceforth referred to for simplicity as SDW and CDW. SDW and CDW states, originally conceived by A. W. Overhauser,Overhauser (1960, 1962) are generally stabilized by the electron-electron interaction and are characterized by spatial oscillations of the spin density, the charge density, or both. In the absence of spin-orbit coupling, one can begin to describe SDW and CDW states by simply considering the electron number density for both spin projections:

(1) |

(2) |

In these expressions the wave vector spans the Fermi surface, i.e. does satisfy the condition . A CDW corresponds to , while a SDW state obtains for . Mixed state are also possible: one such state is beautifully realized in chromium.Overhauser and Arrott (1960); Corliss et al. (1959); Shull and Wilkinson (1953) As we will show, in the presence of linear Rashba spin orbit the corresponding distorted states are characterized by a more complex spatial dependence of the number density, the spin density and, where appropriate, a chiral density. As a first step towards establishing the fundamental properties of SDW- and CDW-like states in the presence of spin-orbit interaction we present here a generalization of the famous Overhauser’s Hartree-Fock (HF) instability theorem. The latter represents an important exact result in many-body theory for it establishes that, within HF, the homogeneous paramagnetic plane wave state does not represents a minimum of the energy for an otherwise uniform electron gas for it can be always variationally bettered by a suitably constructed distorted chiral SDW or CDW.com (a)

The paper is structured as follows: In Section II we discuss the relevant aspects of the theory of a two dimensional non-interacting electron gas in the presence of Rashba spin-obit coupling. Section III briefly discusses useful notions of the electron-electron interaction within Hartree-Fock approximation. Section IV is dedicated to the actual proof of the theorem and contains the main results. Finally the last Section contains the conclusions while a number of useful mathematical relations are derived in the two Appendices.

## Ii Two dimensional electron gas in the presence of Rashba spin-orbit

In the presence of linear Rashba spin-orbit coupling, the one-particle hamiltonian can be written as follows:

(3) |

where is the unit direction along the z-axis, the motion taking place in the plane.

The non interacting problem can be readily diagonalized to obtain the energy spectrum and the eigenfunctions:

(4) |

and

(5) |

(6) |

where labels a state’s chirality and is the angle spanned by the -axis and the two dimensional wave vector . A schematic of the lower energy sector of the spectrum is plotted in Fig. 1.

## Iii Hartree-Fock theory of a two-dimensional electron liquid in the presence of Rashba spin-orbit coupling

An accurate description of a realistic electronic system requires that electron-electron interaction be taken into account. A first step towards developing such a many-body theory is to investigate the results of a mean-field approach. The main idea behind the mean field procedure is to find an effective Hamiltonian which is quadratic in the electron creation and annihilation operators and can therefore be easily diagonalized. Within the HF theory, the ground state is approximated by a single Slater determinant made out of single particle wavefunctions, which in turn, are determined by imposing the requirement that the expectation value of the Hamiltonian over the Slater determinant be a minimumGiuliani and Vignale (2005). Using these wavefunctions as our basis set, a standard Wick decoupling procedureGiuliani and Vignale (2005) allows us to determine the effective HF potential. It can be easily proved that the non-interacting chiral states are indeed among the solutions of the corresponding HF equations. In this case, the HF potential is diagonal in wave vectors and chiral indices:

(9) |

The corresponding HF eigenvalues are given by:

(10) |

An evaluation of the Fermi energy of the two sub-bands leads to an interesting problem. Since one band will, in general, acquire more exchange energy than the other, this may result (in a first iteration) in two different Fermi levels. In order to equalize them (for elementary stability reasons), electrons from one subband will have to be moved to the other. This is the phenomenon of repopulation.

The spatially homogeneous chiral states are just one of the possible Hartree-Fock solutions. A detailed analysis of the possible solutions corresponding to symmetric occupations in momentum space can be done by systematically minimizing the total energy as a function of spin orientation and generalized chirality of the systemChesi and Giuliani (2007a). More general solutions correspond to non-symmetric occupations of the single particle chiral states. The problem has been studied and the corresponding very interesting phase diagram has been exploredChesi and Giuliani (2007a, b). As we will presently discuss there also exists an interesting class of spatially non-homogenous solutions to the problem.

## Iv Proof of the instability theorem

We will proceed by showing that it is always possible to lower the energy of the homogeneous paramagnetic chiral ground state by introducing a suitable real space distortion which is periodic with wave vector . The general approach follows that of Fedders and Martin Fedders and Martin (1966) and is based on an Ansatz which represents a generalization of that given by Giuliani and Vignale for the case of the three dimensional electron gas.Giuliani and Vignale (2005)

Let us consider first the putative HF ground state of our many-body system . A complete, and, as we shall see convenient, description of this determinantal state can be achieved in terms of the corresponding single-particle density matrix elements here given by:

(11) |

where here and label the one-particle states which are used to build the Slater determinant.

Now, within the space of Slater determinants, any slightly modification of the state can be described in terms of a corresponding infinitesimal change of the single-particle density matrix elements. Let us indicate such a change by . At this point the next task consists in trying to evaluate the change of the total HF energy in terms of these quantities.

Since is a solution of the HF equations,com (b) the first order variation in the energy must vanish so that the problem at hand is reduced to determining the sign of the energy change to second order in the ’s. The relevant expression is therefore given by:Giuliani and Vignale (2005)

(12) |

where the notation means that only states situated on opposite sides of the Fermi sea are considered in the summation. In this formula, we indicate the Hartree-Fock eigenvalues as and the corresponding occupation numbers as .

The next step in our procedure consists in constructing a Slater determinant for which the HF energy is lower than . In order to do, we follow Overhauser’s idea and choose the new one-particle states to be suitable linear combinations of chiral plane waves states situated near opposite points on the Fermi surface, the distortion being limited to a very narrow strip. The width of this strip will play the role of a variational parameter. We then carefully devise an expression for the wave vector dependent amplitude of the coupling between the plane waves and construct the corresponding Slater determinant. In the last step, we calculate the change of the Hartree-Fock energy due to this perturbation to leading order in the distortion amplitude from Eq. (12).

### iv.1 Instability for the case of chirality equal one

Because it presents a formally simpler problem, the first case to be treated is that in which only the lower subband is occupied while the upper one is just about to be filled.com (c) This situation is depicted in Fig. 2.

Let us build the new trial wavefunctions as mixtures of the wavefunctions corresponding to wave vector with those corresponding to wave vector , i.e.

(13) |

Here, as anticipated, as shown in Fig. 2.

In evaluating the HF energy change, only the states situated on opposite sides of the Fermi sea are relevant. We will consider and . Here, the occupation numbers are , while the amplitude satisfies the condition: . The only non-zero matrix elements of have the form:

(14) |

These wavefunctions indeed describe SDW/CDW-like states. A simple calculation shows that retaining only the linear order in the amplitude of the distortion, the spin and the charge densities exhibit spatial oscillations with wave vector . Specifically:

(15) |

The change in the HF energy is obtained by substituting the expression of the non-zero density matrix elements from Eq. (14) into Eq. (12). The resulting expression can be expressed as:

(16) |

where we have defined

(17) | |||||

(18) | |||||

(19) | |||||

The Hartree and exchange terms in Eqs. (18)-(19) contain combinations of the matrix elements of the electron-electron interaction. By employing Eq. (II), after simple algebraic manipulations, we obtain:

(20) |

and

(21) |

In order to explicitly evaluate the change in the Hartree-Fock energy, we need to assume a specific expression for the distortion amplitudes. As a first condition, we will perturb only a narrow region near the Fermi surface. Following the same pattern of the proof of reference Giuliani and Vignale (2005), we propose for the present problem the following educated variational guess:

(22) |

where is our small parameter and the second arbitrary . This expression is intentionally chosen to display singularities for and . These singularities are crucial to the present proof.

We now notice that in the expressions of the interaction matrix elements, there appear factors of the type:

(23) |

Since we are only interested in the leading order expansion with respect to , these cosines can be simply taken to be equal to unity, since in the region where the amplitude of the distortion is non-vanishing, both and are of order . Accordingly we will assume

(24) |

At this point, we recall that and must lie on opposite sides of the Fermi sea (i.e and ), which implies that .

We can now proceed to the evaluation of the three components of from Eqs. (18)-(19) to leading order in .

For the first step is to calculate 10), with energies expressed in Ry, and , we have: . Using (

The quadratures appearing in this expression can then be manipulated by making use of the results of Appendix A. The result is:

(26) |

where the logarithmic term accounts for the divergence of the derivative of the HF single particle energy near the Fermi level. Here, is a constant approximately equal to .

By substituting (26) and (22) in (17) we obtain:

(27) |

an expression that, to leading order in , reduces to:

(28) |

where is the number of particles.

The Hartree term (containing ) can in turn be evaluated as follows. By making use of the assumed amplitude in (22), we write:

(29) |

At this point, using the result (66), the leading order in of this quantity is given by:

(30) |

The last term of (16), the exchange energy contribution, is clearly negative and therefore will certainly lower the energy. Its evaluation is formidable, for it involves several complicated and seemingly difficult quadratures. Rather than attempting to actually calculate it, we will establish a lower limit for its magnitude.

We will restrict ourselves to the region in which both angles and are in the first quadrant. Since this excludes some contributions of the same sign, the exchange energy will be underestimated. We therefore have:

(31) |

It is simple to see that this expression will turn out to be proportional to . This is due to the presence of three singularities in the denominator of the integrand. Of these one stems from the divergence of the Coulomb potential, while the other two come from the upper limit of the angular integrations.

Another simplification is provided by the use of the inequality:

(32) |

Using the same changes of variable as in Appendix B, i.e. and , we can rewrite this integral as follows:

(33) |

It is clear now that the main contribution to the integral comes from the region around the upper limit of integration for both and . In order to retain the leading order term, a good approximation will be to replace both ’s with in the denominator of the first square root. In this way, we can separate the angular integrations in (31) to obtain:

(34) |

The last two integrals are evaluated using (66) and other changes of variables (, ) :

(35) |

The last quadrature is calculated in (69), leading us to a very simple inequality for the exchange contribution:

(36) |

This term contains a logarithmic factor , which allows the negative change in the exchange contribution to control all the remaining terms. This concludes the proof for this case.

The same chain of arguments does apply to the case in which the generalized chirality is greater than one. The coupling that produces this kind of instability is schematically shown in Fig. 3.

All the formulas we derived in the previous case do still apply. The only difference lies in the lower integration limits of Eq. (IV.1), but no relevant contribution ensues from this. The matrix elements related to the Hartree and exchange contributions are the same, and, as a consequence, the leading order approximation is identical.

### iv.2 Instability for the case of chirality less than one

The argument of the previous Section can be applied when the chirality is less than one, i.e. when both chiral subbands are occupied. We can try to break the symmetry by coupling states with the same chirality as well as states with different chiralities. When same chirality states are coupled, there is nothing new, as one simply just adds a chirality index to the various quantities. In this case, the wave vectors characterizing the oscillations are given by: and the trial wavefuntions can be written as:

(37) |

This type of coupling is depicted in Fig. 4.

The corresponding distortion of the components of the spin density and the number density can be again calculated up to the first order in the amplitudes:

(38) | |||||

(39) | |||||

(40) | |||||

(41) |

We proceed in this case by choosing an amplitude not unlike the one assumed above:

(42) |

The proof of the corresponding instability theorem proceeds then in exactly the same manner.

As anticipated, there is also not much difference when we try to couple states with opposite chirality (see Fig. 5). Although some of the expressions involved in the derivation do change, the main features of the argument remain unchanged. The coupling vector in this case is given by . Here, we try to find a lower energy state by coupling wavefunctions with wave vector with those with wave vector and opposite chirality. The trial wavefunctions then read:

(43) |

For the only non zero variations of the matrix elements of the single-particle density matrix operator acquire the following form:

(44) |

In this case, the new state is characterized by a similar spin and density modulation:

(45) | |||||

(46) | |||||

(47) | |||||