Friedel oscillations in 2D electron gas from spin-orbit interaction in a parallel magnetic field I. V. Kozlov and Yu. A. Kolesnichenkoa)
B.I. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, pr. Nauki, 47, Khar’kov, 61103, Ukraine
Effects associated with the interference of electron waves around a magnetic point defect in two-dimensional electron gas with combined Rashba-Dresselhaus spin-orbit interaction in the presence of a parallel magnetic field are theoretically investigated. The effect of a magnetic field on the anisotropic spatial distribution of the local density of states and the local density of magneti- zation is analyzed. The existence of oscillations of the density of magnetization with scattering by a non-magnetic defect and the contribution of magnetic scattering (accompanied by spin-flip) in the local density of electron states are predicted.
1. Introduction absence of a center of inversion of a bulk crystal, result in Interest in research on quasi-two-dimensional [hereinafter spin-orbit interaction (SOI) (see the monograph,12 which sig- for brevity, two-dimensional (2D)] conductive systems results nificantly affects the thermodynamic and kinetic characteris- from new capabilities for studying various quantum effects tics of such 2D systems.13 The FO near an individual point that are absent or are very small in mass conductors. Among defect in 2D electron gas with Rashba SOI18,19 were experi- such effects are spatial oscillations of the density of electrons mentally observed in Refs. 14–17, and in this case a rather 1 20–27 n(ϵF, r) theoretically predicted by Friedel, depending on the large number of theoretical works were devoted to the distance r from a point defect or conductor edge (ϵF is Fermi analysis of the oscillatory dependence of the LDS. The LDM energy). Friedel oscillations (FO) are associated with interfer- was theoretically studied in Ref. 28 for electron scattering of ence of incident and reflected electron waves and in an isotro- surface states of Au(111) on an adsorbed Co atom, taking pic conductor have a period Δr = πħ/pF (where pF is the into account the Rashba SOI and the Kondo effect. Fermi impulse). Direct observation of FO became possible In a number of 2D systems, combined Rashba19 and with the development of scanning tunnel microscopy (STM).2 Dresselhaus31 SOI (R-D SOI) is produced, based on semicon- Thus, when studying in Refs. 3 and 4 the surface (111) of ductors such as zincblende (III – V zincblende and wurtzite) noble metals by means of STM, FO of the local density of and SiGe (see Refs. 29 and 30). With R-D SOI, the scattering states (LDS) ρ(ϵ,r)=dn/dϵ|ϵ = ϵF were detected, arising as law of charge carriers is anisotropic, and this results in signifi- the result of the scattering of 2D surface states on individual cant modification of the FO. Thus, FO beats were predicted in adsorbed atoms. It was further shown that the study of the Ref. 32, and it is shown in Ref. 33 that the FO are also essen- periods and contours of the constant phase of FO makes it tially anisotropic, and contain more than two harmonics for a possible to obtain new information on the local characteris- certain relationship of the constants of SOI. tics of the spectrum of charge carriers and on the process of As a consequence of the dependence on the wave vector their scattering by an individual point defect of known nature of the direction of electron spin in 2D systems with SOI, the (see reviews5–7 and the references quoted in them). parallel magnetic field not only results in Zeeman splitting, During electron scattering on a magnetic impurity at tem- but also significantly changes the energy spectrum (see for 8 perature T below the Kondo temperature TK, there appears in example Refs. 34 and 35). By changing the amplitude and the FO of the electron density n(ϵF, r) an additional shift of the direction of the magnetic field, it is possible by a change in phase of the oscillations depending on r/rK (rK = ħνF / TK is the the spectrum to control all electron characteristics of a 2D Kondo length) which, however, is absent in oscillations ρ(ϵF, conductor with R-D SOI. r)oftheLDSasmeasuredbymeansofSTM.9 The magnetic In this article we studied FO oscillations around the defect, along with oscillations of the electron density, results in point of a magnetic defect in 2D electron gas with R-D SOI oscillations of the local density of magnetization (LDM) of located in a parallel magnetic field. General expressions for electron gas m(ϵF, r), which is a magnetic analogue of the the LDS and LDM and their asymptotic expressions at large FO10 (hereinafter we will use for brevity the term FO of the distances from the defect were obtained in the Born approxi- LDM). Spatial oscillations of the LDM are investigated by mation. The dependences of the FO of the LDS and LDM on means of a spin-polarized scanning tunnel microscope having the amplitude and direction of the magnetic field were ana- a contact (tip) with a magnetic covering.11 lyzed. The effect of the occurrence of oscillations of the The asymmetrical confinement electric potential (con- density of magnetization during scattering by a non-magnetic finement potential) that limits the motion of electrons along defect and the dependence of the oscillations of the density the normal to the edge (for surface states) or to the border of of states on the magnitude of the magnetic moment of the the heterotransition with a quantum well, as well as the defect were predicted. 2. Statement of the problem problems:36 2.1. The Hamiltonian 1 D(r) ¼ γσ0 þ Jσ δ(r), (7) Using the calibration A = (0, 0, Bxy – Byx), we write the 2 Hamiltonian of the 2D electron gas with R-D SOI in a paral- lel magnetic field B =(B , B ,0) in the absence of defects, as where γ > 0 is the constant of potential interaction of elec- x y ^ σ^ σ^ σ^ σ^ J a linear approximation on the wave vector operator k ¼ trons with the defect; =( x, y, z) is the Pauli vector; and irr (see, for example, Refs. 34 and 35): is the effective magnetic moment of the defect which differs from its true value S (S ≥ 1) due to the Kondo effect, resulting S h2(^k2 þ ^k2) in partial screening of the spin by conductivity electrons. H^ ¼ x y σ þ α(σ ^k σ ^k ) þ β(σ ^k σ ^k ) We consider the direction of the vector J to be fixed, and will 0 2m 0 x y y x x x y y not consider the processes of revolution and precession of the þ g μ σ þ σ B(Bx x By y), (1) defect spin. 2 Temperature T is assumed equal to zero. Such an approach is quite justifiable since the quantum interferen- where m is the effective mass of an electron, B =(B , B ,0) is x y tial phenomena to which FO pertain are usually observed the induction of the magnetic field, σ are Pauli matrices, x,y,z at low temperatures, when scattering of electrons on σ^ is the 2 × 2 identity matrix, α and β are constants of the 0 phonons is sufficiently small. For T =0,theLDS ρ(ϵ , r) Rashba (α) and Dresselhaus (β) SOI, μ is the Bohr magne- F B and LDM m(ϵ , r) can be calculated with the aid of the ton, and g* is the effective g-factor of the 2D system, which F retarded Green function G^ R(E, r ,r) in the coordinate rep- can significantly differ from the value g = 2 for free elec- 1 2 0 resentation trons.12 For definiteness we will assume that the spin-orbit interaction constants are positive. h i ρ ϵ r ¼ 1 ^ R ϵ r r The eigenvalues and eigenfunctions of the Hamiltonian ( F, ) π Im Sp G ( F, , ) , (8) in (1) are written as h i 1 R m ϵ r ¼ σ^G^ ϵ r r : h 2k2 ( F, ) π Im Sp ( F, , ) (9) ϵ1,2(k) ¼ + A(kx, ky), q2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We will take account of the effect of electron scattering at 2 2 A ¼ (hx þ αky þ βkx) þ (hy þ αkx þ βky) , (2) a defect in the Born approximation (see, for example, Ref. 37 from the potential of scattering (7), after presenting the Green function in the form of a decomposition 1 ikr 1 1 ikr ψ (r) ¼ pffiffiffi e θ ; e f(θ ), (3) 1,2 π ei 1,2 π 1,2 ^ R ϵ r r ^ R ϵ r r þ ^ R ϵ r r ^ R ϵ r 2 2 2 G ( , 1, 2) G0 ( , 1 2) G0 ( , 1)D( )G0 ( , 2), (10) where the f(θ1, 2) are the spin parts of the wave functions of (3), in which the retarded Green function in the absence of ^ R ϵ r – r h αk βk defects, G0 ( F, 1 2), depends only on the difference of sin θ (k) ¼ y x y ,cosθ (k) r r – r 1 A 1 coordinates = 1 2 : h αk βk 1ð ¼ x y x θ ¼ θ þ π: 2 ikr , 2 1 (4) ^ R ϵ r ¼ 1 d ke ϵ [ R: A G0 ( , ) 2 ^ ; (11) (2π) ϵ H0 þ i0 1 In order to reduce writing in later formulas, we introduce the ^ designation The Hamiltonian H0 is defined by expression (1). Naturally, fi g formula (10) can be used to describe the FO only at suf ciently h ¼ μ B: (5) 2 B large distances from the defect r > rD when the term associated with scattering is small. The value of rD with respect to the The spin orientation for each of the branches of the spectrum order of magnitude of size can be estimated as the Fermi wave- in (2) is defined by the mean length rD ∼ λF ∼ ħ /pF for potential scattering, and as the Kondo length rD – rK=ħvF /TK for magnetic scattering. y s1,2(θ) ¼ f (θ1,2)σf(θ1,2) ¼ (cos θ1,2, sin θ1,2, 0), (6) 3. The Green function and, in accordance with the formulas in (4), this depends not 3.1. General ratios only on the direction of the wave vector, but also on its We derived in Ref. 38 exact analytical expressions for value, for combined R-D SOI in the presence of a parallel the equilibrium Green function at temperature zero for a 2D magnetic field. system of electrons with R-D SOI, and their asymptotes for large values of the spatial variable. Here we will provide 2.2. Scattering by a defect certain relationships based on earlier-obtained results38 We will model the interaction of electrons with a point which will be necessary for further calculations magnetic defect at the point r = 0 using the two-dimensional The equilibrium retarded Green function (11) can be δ-potential, often used in the examination of various physical presented in the form of a decomposition on the Pauli matrices the spectrum, in which the energy is h2(k2 þ k2 ) α2 þ β2 þ αβ w G^ R(ϵ, r) ¼ g (ϵþ i0, r)σ^ þ g (ϵþ i0, r)σ^ 0x 0y 2 2 2 sin2 h 0 0 0 x x ϵ1,2(k0) ¼ ¼ h h 2m α2 β2 2 þ gy(ϵþ i0, r)σ^y, ϵ, [ R, (12) 2m( ) ¼ ϵ0 . 0, (18) where ð X and the angle wh specifies the direction of the magnetic field 1 kr 1 g (ϵ, r) ¼ d2kei , (13) h = h(cos w , sin w , 0). The wave vector k =(k , k , 0 π 2 ϵ ϵ k h h 0 x0 y0) 2(2 ) j¼1,2 j( ) corresponding to a point with energy ϵ0, is determined from A k k X ð the condition ( x0, y0) = 0 in the expressions in (2). g ϵ r 1 kr θ k 1 ϵ x( , ) ¼ 2 i cos j( ) In the variables in (16), the dependence of energy 1, 2 2 d ke , ~ gy(ϵ, r) π sin θj(k) ϵ ϵ k 2(2 ) j¼1,2 j( ) on k and f is written as 2 (14) h 2~k h2~k ϵ (~k, f ) ¼ λ ( f ) þ ϵ , (19) 1,2 2m 2 1,2 0 θj is the angle defining the direction of spin of an electron (6). Its dependence on the pulse is defined by formula (3). where Each of the terms with j = 1, 2 contributes to the Green func- α sin ( f w ) β cos ( f þ w ) λ(1,2)( f ) ¼ h h h tion of one of the branches of the energy spectrum ϵ (2). α2 β2 1,2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The components g0, x,y of the Green function (12) satisfy the m symmetry relationships + α2 þ β2 þ 2αβ sin (2f ): (20) h 2 In the new coordinates of (16), the point of contact of the g0(r, h) ¼ g0( r, h), gx,y(r, h) ¼ gx,y( r, h): branches of the spectrum corresponds to ^k =0. (15) The angles defining the direction of spin, θ1, 2 ( f), Further, we will assume α ≠ β in all calculations, except for depend only on the direction of the wave vector (angle f) and special cases when the equality of the constants of SOI is the constants of SOI, after the replacement in (16): stipulated separately. For α ≠ β, it is convenient to transition ^ α cos f þ β sin f to new variables of integration k and f. sin θ1,2( f ) ¼ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , α2 þ β2 þ 2αβ sin 2 f ¼ þ ~ ¼ þ ~ α sin f þ β cos f kx kx0 k cos f , ky ky0 k sin f , (16) cos θ1,2( f ) ¼ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (21) α2 þ β2 þ 2αβ sin 2 f where Hence, for each branch of the spectrum, the directions of elec- α sin w þ β cos w α cos w þ β sin w ^ h h h h tron spin s1,2 ( f) are antisymmetric relative to the point k =0. kx0 ¼ h ; ky0 ¼ h α2 β2 α2 β2 For each angle f, the directions of spin of electron belonging to (17) different branches of the spectrum are strictly opposite. Substituting expressions (16) and (21) into formulas are the coordinates of the point of contact of the branches of (13) and (14), after integration on ^k we derive
þ X X (j) m k+ g (ϵ, r) ¼ exp [i(k cos w þ k sin w )þ] df Fk(j), r cos ( f w ) , ϵ [ C, (22) 0 π 2 x0 r y0 r (j) (j) + r (2 h) j¼1,2 + k+ k+
þ ϵ r X θ X k(j) gx( , ) ¼ m w þ w cos j( f ) + (j) w ϵ [ C 2 exp [i(kx0 cos r ky0 sin r)r] df (j) (j) Fk+, r cos ( f r) , , (23) gy(ϵ, r) π sin θj( f ) (2 h) j¼1,2 + k+ k+
^ (1, 2) ϵ 1ð where k = k± ( ) are the roots of the equations ~ π ¼ dk i~kr ¼ ikr j j þ þ i F(k, r) ~ e e Ci( k r ) iSi(kr) sgn r , ϵ ~ ¼ ϵ k k 2 1,2(k, f ) , (24) 0 r [ R, k [ C, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (26) 2 2m(ϵ ϵ ) w k(1,2) ¼ λ(1,2) + (λ(1,2)) þ 0 , (25) the angle r in formulas (22) and (23) determines the direc- + 2 h tion of the vector r = r(cos wr, sin wr, 0). (n) In the specific case of equality of the constants of SOI H0 (x) is the Hankel function, and K0(x) is the modified α = β and the directions of the magnetic field along the axis Bessel function of the second kind. y = –x and h = h/√ 2(–1, 1, 0), the integrals in formulas (13) and (14) may be expressed through the Bessel functions: 3.2. Asymptotic formulas For large r, the stationary phase method39 makes it pos- 1 g0(ϵþ i0, r) ¼ (Gþ(ϵ, r) þ G (ϵ, r)), ϵ [ R, (27) sible to derive very simple asymptotic expressions of formu- 2 ( j) las (22) and (23). Then the stationary phase f = fst should be determined from the equation gx(ϵþ i0, r) 1 ¼ + pffiffiffi(Gþ(ϵ, r) G (ϵ, r)), ϵ [ R, (28) g (ϵþ i0, r) ÀÁ y 2 2 d j w j ¼ k+ cos ( f r) f ¼f (j) 0, (31) df st where pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi in which the k( j) are the positive real solutions (25) of α α ± 2m R 2m ϵ ∈ R G+(ϵ, r) ¼ exp +i (x þ y) G ϵþ + h, r , Eq. (24) for . h2 2D h2 As a result of standard calculations,38 we obtain the fol- (29) lowing asymptotes of the components of the Green function in (12) R ϵ and G2D( , r) is the retarded Green function of free 2D elec- trons, i X X 1 g0(ϵ þ i0, r) ≃ pffiffiffiffiffi pffiffiffiffiffiffiffi π (j) j j 8 ÀÁpffiffiffiffiffiffiffiffiffi 2 2 j¼1,2 s hv Kj r < (1) ϵj j= ϵ . m iH0 2m r h ; 0 iπ R pffiffiffiffiffiffiffiffiffiffiffi j (j) ϵ [ R G (ϵ, r) ¼ 2 , (30) exp iSjr sgn Kj f ¼f , , (32) 2D 2 h2 : jϵjj j= ϵ , 4 st π K0 2m r h ; 0
ϵþ r X X θ π gx( i0, ) ≃ piffiffiffiffiffi cos j p1 ffiffiffiffiffiffiffiffiffi i j ϵ [ R exp iSjr sign Kj f ¼f (j) , , (33) gy(ϵþ i0, r) π sin θj (j) j j st 2 2 j¼1,2 s hv Kj r 4
w ¼ (j) w Sj( f , r) k+( f ) cos ( f r), (34) The values of the phase kr of rapidly oscillating functions as → ∞ 43 ≫ r in the integrals in (13) and (14) can be interpreted in which are valid for Sj r 1. All functions must be calculated 40 j) terms of the support function of the isoenergy contour ϵ at the points of the stationary phase f = f ( . The summation 1, 2 st (k)=ϵ. on s takes account of the possibility of the existence, on a non-convex isoenergy contour belonging to a branch of the (j) Sj(k) ¼ kn (k); k [ ϵi(k) ¼ ϵ,(37) spectrum ϵ2 (k), of several points of the stationary phase cor- v responding to the given direction of the vector r. In formulas knowing which, it is possible to restore the contour and find its (32) and (33), K1,2 ( f) ≠ 0 is the curvature of the isoenergy curvature at any point. curve ϵ1, 2 ( f)=ϵ. k(j)( f )2 þ 2k(j)( f )2 k(j)( f )k(j)( f ) ¼ + + + + Kj( f ) = , (35) 4. Friedel oscillations (j) (j) 2 3 2 (k+( f ) þ k+( f ) ) 4.1. Fermi contours and vj ≠ 0 is the absolute value of the velocity of an electron It was shown in Refs. 41–43 that the geometry of the ( j) v = ∇k ϵj /ħ. Hereinafter, a dot over a function signifies dif- constant phase lines of the FO oscillations of the LDS and ferentiation on the angle f of the direction of a wave vector in their period depend on the local geometry of the Fermi the displaced coordinates (16). Solutions of Eq. (31) satisfying surface. Therefore, in this section we will introduce some the inequality Sj( f) [see formula (34)] correspond to the condi- information that will be needed later regarding the energy tion of parallelism of vectors r and v( j) (see also Ref. 43) spectrum of the system being studied. In the case of 2D electron gas with R-D SOI placed in a v(j) rn(j)j ¼ n(j) ¼ parallel magnetic field, the energy spectrum contains two v ¼ (j) r; v ( f ) f fst jv(j)j ϵ k ≠ ϵ –k 0 1 branches 1, 2 ( ) 1, 2 ( ) (2), not possessing central symmetry. The surface ϵ = ϵ (k) is always convex, at the (j) (j) (j) (j) 1 B k+ sin f þ k+ cos f k+ cos f k+ sin f C ϵ ϵ k fi ¼ +@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: same time that the surface = 2 ( ) for a de ned region of (j)2 (j)2 (j)2 (j)2 values of SOI constants and magnetic field contains saddle k+ þ k+ k+ þ k+ points and areas of negative Gaussian curvature (see for (36) example Refs. 34 and 35. There exists a critical value of the ( j) magnetic field trum ϵ0 (18) and the functions λ ( f)(20)onh, it is possi- ble by varying the magnitude and direction of the magnetic 2 (α2 β2) field to change smoothly the energy spectrum and therefore hc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (38) α4 þ α2β2 þ β4 þ αβ α2þ β2 w the geometry of isoenergy contours. 6 4 ( ) sin 2 h for values less than which, for h < hc2 the function ϵ = ϵ1 (k) 4.2. LDS oscillations has no extrema and takes the smallest value ϵ = ϵ0 (18) at the point of contact of the branches of the spectrum, and for h > Using expressions (8) and (10), we will present LDS in hc, ϵ = ϵ1 (k) has an absolute minimum. the form of the sum of two components: In the coordinate system of (16), the equations of Fermi (1,2) ρ ϵ r ¼ ρ ϵ þ Δρ ϵ r contours k (ϵF, f)=k± are defined by the formulas (25) for ( F, ) 0( F) ( F, ), (39) (1, 2) ϵ = ϵF in the field of values of parameters for which k± ≥ 0. ρ ϵ In the case when the inequality ϵF > ϵ is satisfied, the func- where 0 ( F) is the density of states of two-dimensional (1, 2) 34,38 tions k+ (ϵF, f) > 0 for any values of f while at the same degenerate gas from R-D SOI in the absence of defects: (1, 2) time the roots k – (ϵF, f) < 0 are always negative. To each 8 m branch of the spectrum there corresponds one Fermi contour > ; ϵ ϵ ϵ (1, 2) ϵ ϵ < π 2 F 0 k ( F, f)=k+ .For F < 0 the real roots of Eq. (24) exist h þ ρ ϵ ¼ X (j) when the inequality is satisfied 0( F) > m λ :> df pffiffiffiffiffiffi Θ(λ(j))Θ(ξ(j)); ϵ ϵ , π2 2 (j) F 0 2 h j¼1,2 ξ 2m(ϵ0 ϵF) (λ(j))2, (40) h 2 ( j) and both roots k± take positive values in some interval of ϵ ϵ (1,2) (1,2) 2 2m( F 0) angles for which λj ≥ 0. In this case the Fermi contours do ξ ¼ (λ ) þ : (41) h2 not cover the coordinate origin (kx0, ky0) (17), and the two ( j) points k = k± on the same Fermi contour correspond to the The part of LDS Δρ (ϵF, r) that depends on scattering same direction of the angle f. Due to the dependence of the describes the FO. Using decomposition (12) of the Green energy at the point of contact of the branches of the spec- function over Pauli matrices, we derive
Δρ r ¼ 2 γ r r þ r r þ r r þ r r þ r r ( ) π Im{ [g0( )g0( ) gx( )gx( ) gy( )gy( )] Jx[g0( )gx( ) gx( )g0( )]
þ Jy[g0(r)gy( r) þ gy(r)g0( r)] þ iJz[gy(r)gx( r) gx(r)gy( r)]}: (42)
The lack of a center of inversion of the electron scattering law (2) results in the appearance in Δρ in (42) of a component pro- portional to the components Jx, y, z of the magnetic moment of the defect, which goes to zero for h =0. Using asymptotic expressions (32) and (33) for the Green functions at large distances from the defect, we derive
X X θ θ À Á À Á Δρ r ¼ γ 2 i j þ θ w w þ w θ ( ) Qij 2 cos J sin J cos J i cos J j i,j¼1,2 s 2 À Á À Á À Á þ þ f θ θ θ þ þ f sin Si Sj r ij J cos J sin i J cos Si Sj r ij }, (43)
π 1 (1,2) ¼ f ¼ þ : directed opposite to the direction of the vector r, and nv Qik 2 (i) (j) ; ij (sign Ki sign Kj) 2π hv v jKiKjjr 4 ^ (1,2) (1,2) (1,2) (kst )=–nv (kst ). Hence, each of the components in (44) the sum (43) takes account of the contribution to LDS of electron back-scatter with a transition between two points w θ The angles J and J specify the direction of the vector of with opposite direction of the velocity on the same (i = j)or the magnetic moment of the defect. different (i ≠ j) Fermi contours. The asymptotic formula in (43) makes it possible to J ¼ w θ w θ θ : J(cos J sin J, sin J sin J, cos J ) (45) easily interpret the reason for the occurrence of a magnetic contribution to the density of states: due to the connection The line over a function signifies that its value is taken at the between the directions of spin and the wave vector, the mag- ^ (1, 2) (1, 2) point k = kst for which the velocity v of an electron is netic scattering that results in spin flip at some angle Δθi = θi – θj ≠ 0, ±π and that corresponds to the change of velocity m0 and the oscillatory additive Δm(r) fl of an electron to its opposite, changes the ux of electrons m r ¼ m þ Δm r : propagated in the opposite direction. ( ) 0 ( ) (46) It is clear that due to the common relationships (12) and (9), 4.3. LDM oscillations the component of the density of magnetization m0z =0. We present an expression for LDM m(r) in the form of However, the components m0 x, y may be distinct from zero at a sum not dependent on the coordinates of the component the Fermi energy ϵF ≤ ϵ0:
8 <> 0, ϵF ϵ0, X þ ¼ m cos θ λ(j) mx,y > df i pffiffiffiffiffiffi Θ(λ(j))Θ(ξ(j)), ϵ ϵ : (47) : π2 2 sin θ (j) F 0 2 h j¼1,2 i ξ
Using a representation of the Green function in the form of decomposition on Pauli matrices (12), we will write expressions for the component of the vector Δm(r) in the following form:
Δ r ¼ 2 r r þ r r r r þ r r þ r r mx( ) π Im{Jx[g0( )g0( ) gx( )gx( ) gy( )gy( )] Jy[gx( )gy( ) gy( )gx( )]
iJz[g0(r)gy( r) þ gy(r)g0( r)] þ γ[g0(r)gx( r) þ gx(r)g0( r)]}, (48)
Δ r ¼ 2 r r þ r r r r þ r r þ r r my( ) π Im{Jy[g0( )g0( ) gx( )gx( ) gy( )gy( )] Jx[gy( )gx( ) gx( )gy( )]
iJz[g0(r)gx( r) þ gx(r)g0( r)] þ γ[g0(r)gy( r) þ gy(r)g0( r)]}, (49)
Δ r ¼ 2 r r þ r r r r þ r r þ r r my( ) π Im{Jz[g0( )g0( ) gx( )gx( ) gy( )gy( )] iJx[g0( )gy( ) gy( )g0( )]
iJy[g0(r)gx( r) gx(r)g0( r)] þ iγ[gx(r)gy( r) gy(r)gx( r)]}: (50)
Using the derived formulas (48)–(50) and asymptotic expressions (32)–(33) for the component of the Green function (12) for large r,wefind the LDM components that are oscillatory with distance from the defect:
X X θ þ θ Δ r ¼ θ w 2 i j þ w θ þ θ þ θ θ θ mx( ) Qij J sin j cos J cos sin J sin ( i j) J cos J (sin i sin j) i,j¼1,2 s 2 þ þ f þ γ θ þ θ þ θ þ θ þ þ f cos ((Si Sj)r ij) (cos i sin i cos j sin j)] sin ((Si Sj)r ij)}, (51) X X θ þ θ Δ r ¼ θ w 2 i j þ w θ þ θ θ θ θ my( ) Qij J sin j 2 sin J sin cos J sin ( i j) J cos J (sin i sin j) i,j¼1,2 s 2 þ þ f þγ θ þ θ þ θ þ θ þ þ f cos ((Si Sj)r ij) (cos i sin i cos j sin j)] sin ((Si Sj)r ij)}, (52) X X θ þ θ Δ r ¼ θ 2 i j þ þ þ f θ w θ θ mz( ) Qij 2J cos j sin sin ((Si Sj)r ij) [J sin J(sin J(sin i sin j) i,j¼1,2 s 2 w θ θ þ γ θ θ þ þ f : cos J(cos i cos j)) sin ( i j)] cos ((Si Sj)r ij)} (53)
The derived results (51)–(53) define the dependence of the FO back-scatter by a non-magnetic defect is accompanied by a of the LDM on an external magnetic field. As in the case of transition to a state with the spin direction turned relative to the LDS, the non-magnetic contribution to LDM can be easily initial at an angle Δθi = θi – θj ≠ 0, ±π, which results in a interpreted on the basis of the asymptotic formulas (51)–(53): change of the local density of magnetization around the defect. Fig. 1. (a) Typical form of Fermi con- tours for ϵF > ϵ0 in a parallel magnetic field. (b) Oscillatory part of LDS Δρ ( x, y) for scattering by a non- magnetic defect (J = 0). The following values of parameters are used: α = 0.7, β = 0.3, h = 0.6, wh = 2.0.
4.4. The special case α = β and hx = – hy. Analytical solution part of the LDS: In the specific case that α = β and h = h√2(–1, 1, 0), Jy Jx Jy Jx two branches of the spectrum are crossed along the parabola Δρ(r) ¼ γþ pffiffiffi Rþ þ γ pffiffiffi R ,(56) 2 2 2 2
2 2 2 2 where h ky1 h h kx þ ky h kx ky ϵ ¼ þ , kx1 ¼ pffiffiffi ¼ , ky1 ¼ pffiffiffi : 2m 8mα2 2α 2 2 2 m (+) (+) R+(r) ¼ J0(k r)Y0(k r)Θ(ϵþ), (57) (54) 4π h4
where J (x)andY (x) are the Bessel functions of the first and ϵ ħ2 2 α2 0 0 For F > h /8m , the Fermi contours have two common second kind, respectively. For large values of the arguments points and each of them consists of two arcs of circles of k(±) r ≫ 1, we have (±) radius k = √(2mϵ±)/ħ (see Fig. 5): 2 m (+) R+(r) ≃ cos (2k r): (58) 4π2h 4k(+)r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α2 (+) 2m k ¼ 2mjϵ+j=h , ϵ+ ¼ ϵF þ + h: (55) h 2 The components of the oscillatory part of the LDM, Δm(r), are defined by the following expressions: The spin directions on each of the arcs that comprise one γ γ θ π θ –π Jy Jx Jz contour are opposite: + =3 /4 or _= /4, and the arcs Δmx(r) ¼ pffiffiffi Rþ þ pffiffiffi R R,(59) 2 2 2 2 with the identical spin directions θ± form complete circles [see Fig. 5(a)]. γ Jy Jx γ Jz Using the expressions for the Green functions in (27) and Δmy(r) ¼ pffiffiffi þ Rþ pffiffiffi R R,(60) (28), we derive the following expression for the oscillatory 2 2 2 2
Fig. 2. FO of the LDMqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for scattering by a a non-magnetic defect with ϵF > ϵ0.(a) Distribution of the component Δm z normal to the plane. (b) Distribution of Δ 2 þ Δ 2 Δm ¼ Δ Δ the absolute value mx my of the LDM component in the plane. Arrows indicate the direction of the vector ( mx, my). Values of the parameters coincide with those given in Fig. 1. Fig. 3. (a) Fermi contours for ϵ0 = ϵF (18), h > hc (38), wh =3π/4. The black point shows k0, the point of contact of the branch of the range of (17).(b) LDS for scattering by magnetic defect J =(J, 0, 0) and γ =0 (7). The following values of parameters are used: α = 0.5, β = 0.2, h = 1.4.
Jz Jy Jx where the designations R±(r)aredefined by expression (57) Δmz(r) ¼ (Rþ R ) pffiffiffi R,(61) 2 2 2 and
pffiffiffi 2 α m 2 2m (þ) ( ) ( ) (þ) R(r) ¼ sin (x þ y) J0(k r)Y0(k r) þ J0(k r)Y0(k r) Θ(ϵþ)Θ(ϵ ), (62) 2πh 4 h 2
2 α π α π r ≃ p4mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (þ) þ ( ) þ 4m w þ (þ) þ ( ) þ 4m w þ (+) : R( ) sin k k sin r r sin k k sin r r ; k r 1 (63) π2h 4r k(þ)k( ) h 2 4 h 2 4
In the case being studied, the FO of the LDS (56) are isotro- depending on the direction wr in the coordinate space. It pic, and at large distances from the defect contain two har- follows from formula (63) that the lines of constant phase for monics with periods Δr = π /k(±), associated with back-scatter oscillations of the LDM with the period in (64) are ellipses between states belonging to different Fermi contours, while (for k(+) + k(–) >4mα/ħ2) or hyperbolas (for k(+) + k(–) at the same time in LDM oscillation (59)–(61), a contribution <4mα/ħ2), and the point r = 0 coincides with one of the foci. is made by the transitions between states of the same Fermi contour that result in the appearance of harmonics of the FO, with periods 5. Discussion of results The effect of a parallel magnetic field the FO of a 2D 2π Δr ¼ , (64) electron gas with Rashba-Dresselhaus SOI results from two 4mα π fi B k(þ) þ k( ) + sin w þ main causes. First, the magnetic eld breaks the central h 2 r 4 symmetry of the Fermi contours and changes their local
Fig. 4. FO of the LDS Δρ ( r) with scattering by a magnetic defect, J = (1, –1, 0)J /√2 and γ = 0, near the value 0 of the magnetic field hmin1 = 1.36, where the Fermi level passes through the point of the minimum of the branch of the range ϵ .(a) h < h = 1 min1 1.4. (b) hmin1 > h = 1.28. The following values of parameters are used: α = 0.5, β = 0.1, wh =3π/4. Fig. 5. FO of the LDS ρ in the case of equality of constants of R-D SOI for scattering by a non-magnetic defect, γ ≠ 0 and J =0. (a) Fermi contours, (b) oscillations of the z-component of the LDM, m z, for scattering by a mag- netic defect, γ = 0 and J = (0, 0, J). The following values of parameters were used in constructing the graphs: α = β = 0.3, h =1,wh =3π/4.
geometry. The back-scatter of electrons, which provides the creation of the diagrams: chief contribution to the FO, corresponds to transitions between states with opposite direction of velocity which in mα mβ h 2mh α ¼ , β ¼ , h ¼ ¼ , turn depends on the field B. As a result, a change to the mag- 2 2 2 2 h k h k ϵF h k nitude and direction of the vector B changes both the period F F F k ϵ of the FO [as a consequence of the change to the magnitude ¼ ¼ ϵ ¼ k , r kFr, ϵ ,(65) of the wave vector corresponding to a point of a stationary kF F phase, see (34)] and their amplitude [as a consequence of the change of the curvature of the Fermi contour, see (35)]. The π2h 4 π2h 4 Δρ ( r) ¼ Δρ(r); Δm ( r) ¼ Δm(r), second main circumstance is the change of electron spin m2(γ þ J) m2(γ þ J) fi under the effect of the direction of the eld [see (6)]. Since (66) with SOI the spin of an electron and its wave vector are inter- connected, the matrix elements of transitions between two where kF = √(2mϵF)/ħ. The arrows placed on the dia- quantum states with back-scatter depend on the direction grams on the Fermi contours show the direction of the of the spins before and after scattering. For B =0,replace- vector of velocity (arrows directed perpendicular to the line ment of the direction of a wave vector by its opposite of the Fermi contour) and the direction of electron spin. results in a spin-flip for states on the same Fermi contour, Fig. 1(a) demonstrates a violation of the central sym- and to preserving its direction upon transition to another metry of the Fermi contours and a change in direction of Fermi contour. As a result, due to the selection rules for the spins under the effect of a parallel magnetic field spin, there are no components in the FO of the LDS that under conditions of R-D SOI. With scattering by a non- depend on the magnetic defect moment, and there are nat- magnetic defect, the FO of the LDS Δρ ( r)[Fig.1(b)] pre- urally no FO of the LDM during scattering by a non- serve the central symmetry, but no longer have symmetry magnetic defect (see Refs. 23, 26,and33). In a parallel relative to axes x = y and x = –y, whichexistsinthe magnetic field during back-scatter, states corresponding to absence of field B =0.33 The dashed lines in Fig. 1(b) show a spin-flip to some angle depending on the field B are per- the directions of the maximum amplitude of the FO that missible. As a result, new harmonics appear in the FO of coincide with the direction of velocity at the inflection the LDS, whose periods depend only on the characteristics points of Fermi contour belonging to branch ϵ2.Eachtwo of one of the Fermi contours, as well as components pro- lines (with identical length of a stroke) limit the “fan” of the portional to the magnetic defect moment J.Forthesame directions, in which the FO have more than two harmonics. reason, the FO of the LDM contain harmonics propor- In the zero field, both “fans” coincide. tional to the constant γ of the potential interaction of an Fig. 2 illustrates the effect of the emergence of a non- electron with a defect. Note that the listed conclusions uniform distribution of the density of magnetization existing remain valid in the presence of only one SOI type (Rashba only in a magnetic field with scattering by a non-magnetic or Dresselhaus), and all our analytical results make it pos- defect. It is interesting to note that potential scattering leads sible to set one of the constants of SOI equal to zero. to nonzero density of magnetization not only in the plane 2D Asymptotic formulas (43) and (51)–(53) provide a com- of electrons [Fig. 1, Eq. (6)] in which the vector B lies, but plete qualitative description of all harmonics of the FO of also in the direction perpendicular to this plane [Fig. 1(a)]. the LDM and LDS, and the dependences of their periods In Fig. 3, the FO of the LDS caused by magnetic scatter- and amplitudes on the field B. ing for the special case when the Fermi energy coincides The spatial distributions of local densities of states with the energy ϵ0 at the point of a contact of the branches and magnetization in Figs. 1–5, obtained by means of the of the range in (18), and a magnetic field greater than the general expressions (42) and (48)–(50), are only several critical hc (38), i.e., a branch of the range ϵ = ϵ1 (k) are given min specific examples illustrating the variety of the nature of has an absolute minimum with energy ϵ1 < ϵ0. the anisotropy of Friedel oscillations in the system under Figure 4 visually demonstrates the essential change in study. We use the following dimensionless values in the the nature of the FO of the LDS associated with magnetic scattering in a narrow interval of magnetic fields near the 7Lan Chen, Peng Cheng, and Kehui Wu, J. Phys. Condens. Matter 29, value 103001 (2017). 8J. Kondo, Prog. Theor. Phys. 32, 37 (1964). m 3π 9I. Affleck, L. Borda, and H. Saleur, Phys. Rev. B 77, 180404 (2008). h ¼ h(1) ¼ (α þ β)2 þ ϵ , w ¼ , (67) 10 106 min 2 F h K. Yosida, Phys. Rev. , 893 (1957). 2h 4 11M. Bode, Rep. Prog. Phys. 66, 523 (2003). 12 corresponding to the decreases in the minimum of the branch R. Winkler, Spin-Orbit Coupling effects in Two-Dimensional Electron and ϵ ϵ (1) Hole Systems, Springer-Verlag, Berlin Heidelberg (2003). of range 1 with Fermi level F.Forh > hmin there is only 13D. Berciouxand and P. Lucignano, Report Progress Phys. 78, 106001 one Fermi contour belonging to branch ϵ2 [inset in Fig. 4(a)] (2015). and defining the unique harmonic of the FO in Fig. 4(a).For 14J. I. Pascual, G. Bihlmayer, Yu. M. Koroteev, H.-P. Rust, G. Ceballos, magnetic fields lower than the minimum field h