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Exact and quasiclassical Green's functions of two-dimensional electron gas with Rashba-Dresselhaus spin-orbit interaction in parallel magnetic field

New exact and asymptotical results for the one particle Green's function of 2D electrons with combined Rashba - Dresselhaus spin - orbit interaction in the presence of in-plane uniform magnetic field are presented. A special case that allows an exact analytical solution is also highlighted. To demonstrate the advantages of our approach we apply the obtained Green's function to calculation of electron density and magnetization


INTRODUCTION
One particle Green's functions (GF) is widely used in quantum mechanics and solid state physics [1,2]. Although the simple analytical formulas for GFs of the free electron gas can be found in some textbooks (see, for example, [2]), in the presence of interactions and in external fields in most cases only complicated integral representations of GF are available. Normaly using the GF in the form of a multiple integral is extremely uncomfortable both for analytical analysis and for numerical computations. Therefore the search of exact, simplified and asymptotic results for GFs attracts the constant attention of theorists and mathematicians (see, for example, [3][4][5]).
In last decade the spintronics development has triggered off investigations of twodimensional (2D) electron systems with spin-orbit interaction (SOI) (for review see [6,7]).
Particularly the 2D systems with combined Rashba [8] and Dresselhaus [9] SOI possesses new perspective properties (for review see [10]). The GF for this case without magnetic field and its asymptotic behavior have been discussed in the papers [11,12]. Explicit GF for each Rashba and Dresselhaus spin-orbit Hamiltonians with uniform perpendicular magnetic field have been derived in Ref. [3].
In this paper we obtain some exact and asymptotic expressions at zero temperature for the time independent GF of 2D electron gas with combined Rashba -Dresselhaus SOI for arbitrary values of SOI constants and arbitrary uniform magnetic field strength and direction parallel to the plane of the conductor. The structure of this paper is as it follows. In Sec.
2 we discuss the Hamiltonian of the system, its eigenvalues and eigenfunctions. The GF in coordinate space is presented as the sum of two parts describing separated contributions of two spin -orbit splitted branches of electron energy spectrum. In Sec. 3 we reduce the general expression for the GF to single integral of well-known special functions. The obtained formula is valid at arbitrary values of parameters. In Sec. 4 we derive asymptotic formulas for GF for large value of coordinate variable. In Sec. 5 we find the exact GF for the special case of equal SOI constants and certain direction of magnetic field. In Sec.6 as a demonstration we apply our results for calculation of electron density of states and density of magnetization. We conclude the paper with the final remarks and summing up main results in Sec.7.

II. BASIC FORMULAS
Let us consider the Hamiltonian of a two-dimensional (2D) noninteracting electron gas with Rashba and Dresselhaus SOI in the magnetic field B = (B x , B y , 0). Using the Coulomb gauge A = (0, 0, B x y − B y x) , ∇ · A = 0, we write 2D Hamiltonian of the system as the sum of four termsĤ (1) σ 0 is the Hamiltonian of 2D free electron gas,Ĥ R = α(σ xKy −σ yKx ) and H D = β(σ xKx −σ yKy ) are Hamiltonians of Rashba and Dresselhaus SOI, respectively, is the Hamiltonian of interaction between electron spin and magnetic field,K x,y = −i∇ x,y is the wave vector operator, m is effective electron mass, σ x,y,z are Pauli matrices,σ 0 is unit matrix 2 × 2, α and β are Rashba (α) and Dresselhaus(β)constants of SOI, µ B is the Bohr magneton, g * is an effective g-factor of the 2D system. We rewrite the total Hamiltonian (1) in the following form where σ = (σ x , σ y , σ z ) is Pauli vector, The eigenvalues and the eigenfunctions of the Hamiltonian (2) are (see, for example, [13]) We introduce notations: The angles θ 1,2 define the average spin direction for two branches of energy spectrum (4) The electron GF corresponding Hamiltonian (2) for complex ǫ in coordinate representation can be written aŝ We used Eqs. (4), (6) and the identities ǫ 1,2 − ǫ 0 = ±R, In the Eq. (10) GF splits up into two independent parts describing separate contributions of every branch of energy spectrum (4).

III. EXACT RESULTS FOR GREEN'S FUNCTION
For α = β we introduce new variables of integrationk, f as follows where k 0 = (k x0 , k y0 ) is the point of branch touch (see, for example Ref. [14]). In coordinates (13) spin angles (6) depend only on the wave vector direction, the angle f, and SOI constants It can be shown that spin direction is symmetric with respect to the center θ 1,2 (f + π) = θ 1,2 (f ) + π, indicating a convenience of Eq. (13). The values k x0 , k y0 have been found out of the system of equations R x (k x0 , k y0 ) = 0, R y (k x0 , k y0 ) = 0. In a shifted polar coordinatesk, f (13) the energies ǫ 1,2 take the form Here The Eq.(16) make it possible to write the poles ǫ 1,2 k (1,2) ± , f = ǫ of the Green function (10) in a simple form By using the roots (19) in coordinates (13) one can write a simple expansion of the functions By means of identity (2121) the GF (10) is written as the double sum At r = 0 the integral overk in Eq. (22) can be expressed by means of trigonometric integral functions (see, for example [15]), where Si (z) and Ci (z) are sine and cosine integrals, The Eq.(23) is valid for any k 0 ∈ C excepting semiaxis Im k 0 = 0, 0 < Re k 0 < ∞, for which usually one introduces the retarded and advanced Green functions G R(A) (E, r) as the Using Eqs. (21), (23) one can rewrite the Green function (10) as where angle ϕ r defines a direction of coordinate r = r (cosϕ r , sinϕ r , 0). The formula (26) turns to the result of the Ref. [11] for the zero magnetic field. The obtained GF (26) is suitable for numerical calculation under arbitrary values of all parameters. It also gives analytical formulas in the quasiclassical case r → ∞.
For equal SOI constants, α = β, one cannot use coordinates (13) and we introduce usual polar coordinates After transformations similar to those performed above we find where k n are the roots of quartic polynomial equation without cubic term the function F (k, r) is given by Eq. (23), ϕ h defines the magnetic field direction, h = h (cos ϕ h , sin ϕ h , 0). Though the Eq. (29) has exact analytical solutions (see, for example, [16]) they are very complicated and not suitable for analytical calculation. Nevertheless the Eq. (28) may be pracically convinient in numerical analysis. As for the particular case of magnetic field direction along the symmetry axis k x = −k y we consider it in Sec.5. and the GF has been expressed by means of Bessel functions.

IV. QUASICLASSICAL GREEN'S FUNCTION
Quasiclassical approximation can be applied in physical investigations, if characteristic length scales of the problem are much larger than Fermi wavelength λ F which has of the order of inverse wave vector k −1 at Fermi level. Since the GF oscillates as a function of the coordinate r on a scale r ∼ k −1 in framework of quasiclassical approximation in most cases the asymptotic formulas for large kr the GF could be used. Below we find the asymptotic expressions for GF (26) at r → ∞. For real ǫ = E the equality gives two branches of the electron energy spectrum. The positive roots of Eq.(32) describe the isoenergetic contours k = k (j) ± (E, f ) corresponding physical electron states in k -space for given energy E. If E > E 0 , the roots k For E < E 0 reals roots of equation (32) ± take positive values for the angles f in which λ j > 0. Below we will not consider values of energies E for which the GF exponentially decreases with coordinate rassuming k (j) ± , E ∈ R. At first we substitute asymptotic expansions for Si (z) and Ci (z) at the large z (see, for example [15]) to the function F (k 0 , r) (23). At large r → ∞ and real k 0 the main term of expansion reads as As the second step one derive the asymptotic formula for (23) by the stationary phase method [17]. Stationary phase points f = f (j) st must be found from equation which leads to the condition r n v , where n v is the unit vector along the electron velocity v (j) = ∇ k ǫ j / (see also [18,19]) As the result of standard calculations we find the asymptotic of the GF (26) We assume S j (f ) ∈ R, r > 0, S j f  st (s) for given direction of vector r(see Ref. [12]). It is possible in the cases when isoenergetic is an absolute value of electron velocity. The Eq. (36) coincides with the results of Ref. [12] in the case of B = 0. We do not adduce the GF for single inflection points, which can exist on the isoenergetic contour ǫ 2 (f ) = E for certain values of SOI constants. It can be simply derived in the same way.

V. EXACT RESULTS FOR SPECIAL CASE OF EQUAL SOI CONSTANTS
Let us consider the special case: α = β and the magnetic field is directed along the 1, 1, 0). Under these conditions the Eq. (10) can be presented in the formĜ where we introduce new functions of energy dimension The spin angles (6), that correspond with parts of the sum with ǫ ± , keep the constant directions θ + = 3π 4 and θ − = − π 4 . The Eq. (41) can be rewritten aŝ where for the function G ± (ǫ) one obtains We point out that G 2D (ǫ, r) is well-known GF of free 2D electrons. Particularly the retarded GF reads as where H (n) 0 (x) is the Hankel function and K 0 (x) is the McDonald function.

VI. DENSITIES OF ELECTRON STATES AND MAGNETIZATION
As an example of our results applications we calculate the electron density of states ρ (E) and the density of vector magnetization m (E) at α = β, which are important characteristics of 2D conducting system (compare with results of Ref. [13]).
Density of states can be found from the relation Substituting the retarded GF (25) at r = 0from Eqs. (22), (25) one can derive electron density of states, where λ (j) and ξ (j) are defined by Eqs. (17) and (20). Note the importance of the relation between electron energy E and the energy E 0 (18) of branch touch point. The Eq.(47) shows that the density of states is the same as for free 2D electron gas for energies E ≥ E 0 . In opposite case E < E 0 the density of states ρ (E) depends on the magnetic field and constants of SOI. The features of ρ (E) related to minima (steps) and saddle points (peaks) on the energy surfaces (32). These points k ν , f ν should be found out of the system of equations from whichk It is clear that the Eq.(49) can be satisfied, if ξ (j) = 0 (see Eq.(20)). Therefore The energy minima ǫ 1,2 k ν , f ν = E min 1,2 correspond to negative second derivativeλ (1,2) (f ν ) < 0 and saddle points ǫ 1,2 k ν , f ν = E sad 1,2 meet the caseλ (1,2) (f ν ) > 0. The electron density should be found by integration over all available energies below We find density of magnetization using its relations with retarded GF Substituting the GF (22) into Eq. (54) after calculations similar to carried out above one that follows from symmetry relations At E < E 0 the density of magnetization becomes If Fermi energy E F ≥ E 0 the magnetic moment corresponds to Pauli's paramagnetism of the free electron gas without SOI and it does not depend on SOI constants. Here m x,y;j (E) are the two items in the sum over j in Eq. (57).

VII. SUMMARY
To some up, the exact and asymptotical expressions for the Green's function (GF) of 2D noninteracting electron gas with combined Rashba -Dresselhaus spin-orbit interaction in parallel magnetic field at zero temperature are derived. We split the GF into two parts For the equal SOI constants and magnetic field direction along one of the symmetry axis we express the GF by means of Bessel functions, Eq. (43). Although this exact result describes the special case, it may be used for qualitative analysis of different problems for