Theory of inverse Faraday effect in disordered metal in THz regime
aa r X i v : . [ c ond - m a t . o t h e r] A ug Theory of inverse Faraday effect in disordered metal in THzregime
Katsuhisa Taguchi and Gen Tatara
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan (Dated: September 3, 2018)
Abstract
We calculate the magnetization dynamics induced by the inverse Faraday effect in disorderedmetals in THz regime by using the diagrammatic method. We find that the induced magnetizationis proportional to the frequency of circularly polarized light. . INTRODUCTION Photoinduced magnetization by circularly polarized light (the inverse Faraday effect) hasbeen studied since its theoretical prediction in the 1960s . Recently, experiments usingintense laser have been conducted, and the induced effective magnetic field was shown to beas strong as a few Tesla . The fast magnetization reversal realized is expected to be usefulfor application to magnetic devices .The mechanism of the inverse Faraday effect was discussed by Pitaevskii based on asymmetry argument . He noted that the free energy of the electromagnetic field in solids isgenerally given by F = X ij Re( ε ij E i E ∗ j ) , (1)where ε ij is a 3 × E i is thecomplex representation of the electric field. The magnetization induced by light is given bythe functional derivative of F with respect to the external magnetic field, H ex , as M k = X ij Re (cid:18) ∂ε ij ∂H ex k E i E ∗ j (cid:19) | H ex =0 . (2)Because of the Onsager relation, the asymmetric components of ε ij are linear or higher oddpower in the external magnetic field, while the symmetric components are an even powerof the magnetic field . Therefore, expanding up to the linear order in the magnetic field,the dielectric tensor is ε ij = ε δ ij + iχ ǫ ijk H ex k , where ε and χ are constants independentof H ex . The induced magnetization is therefore written generally as M = iχ E × E ∗ . (3)It is thus proportional to the helicity vector of the circularly polarized light, i ( E × E ∗ ).( E × E ∗ is parallel to light propagation and is pure imaginary) . Because the helicityvector, i ( E × E ∗ ), of the oscillate phase factor is canceled out for Ee i Ω t and E ∗ e − i Ω t , M is a static and rectified magnetization. This magnetization is converted from the opticalangular momentum of circularly polarized light, E × E ∗ , and this conversion efficiencyis determined by χ . The coefficient χ was explicitly calculated in the visible light caseconsidering plasma oscillation . It was found to strongly depend on the angular frequency Ωas χ ∝ Ω − . Another mechanism of the inverse Faraday effect is spin-orbit interaction, as2iscussed by Pershan and coworkers . They also showed that the spin-orbit contributionalso results in χ ∝ Ω − .Besides the effect of the visible light lasers, the effect of terahertz lasers on magneticsystems has recently been attracting interest. For example, the spin structure and excitationin ferroelectric magnets were explored by utilizing use of the magnetoelectric effect due toa terahertz laser .In this paper, we will theoretically investigate the magnetization induced by the inverseFaraday effect in a THz regime. The magnetization is calculated using the diagrammaticmethod in disordered metals considering the spin-orbit interaction. The induced magneti-zation is found to linearly depend on the frequency in the THz regime. II. HAMILTONIAN
We consider the conduction electrons in the presence of spin-orbit interaction and im-purity scattering. The applied THz light is represented by the vector potential A em . TheHamiltonian of the system is H ≡ H + H em + H so + H i , where H = Z d x ~ m ∇ c † · ∇ c, (4) H em = − Z d x c † e m ( p · A em + A em · p − eA ) c, (5) H so = − λ ~ Z d x c † σ · [( e ˙ A em − ∇ u i ) × ( p − e A em )] c, (6) H i = Z d x u i c † c, (7)and c , c † are the conduction electron’s annihilation and creation operators, respectively. Weconsider two origins for spin-orbit interaction: one from the applied electromagnetic fieldand the other induced by random impurities. The coefficient λ ≡ ~ m c is the strength ofspin-orbit coupling, and u i = u P ℓ δ ( x − R ℓ ) is the impurity potential, where R ℓ representsthe randomly distribution impurity positions. (The average of λu represents the strength ofthe spin-orbit interaction in the material.) The scattering of the electron by the impuritiesis represented by H i . We consider the elastic lifetime of the electron, τ , arising from thisscattering.We consider a monochromatic light with a frequency Ω and wave vector q (The unit vectorof electromagnetic field, ˆ q = q / | q | , is parallel to E × E ∗ ). The vector potential is related3o the electric field by E em = − ˙ A em . We express the vector potential and electric field byutilizing the complex amplitudes E and A as E em = 12 ( E e i ( q · x − Ω t ) + E ∗ e − i ( q · x − Ω t ) ) , (8) A em = 12 ( A e i ( q · x − Ω t ) + A ∗ e − i ( q · x − Ω t ) ) . (9)The amplitude thus satisfies A = − i E Ω and A ∗ = i E ∗ Ω . III. CALCULATION OF THE MAGNETIZATION DUE TO THE INVERSEFARADAY EFFECT
The spin density induced by the applied circularly polarized light (inverse Faraday ef-fect) is calculated by estimating the expectation value s α ≡ h c † σ α c i ( α = x, y, z representsthe spin direction, and h i denotes expectation values). Using the lesser Green’s function, G < ( x, x ; t, t ), the spin density is expressed s α ( x , t ) = − i ~ Tr [ σ α G < ( x , x ; t, t )] . (10)The calculation of the spin density is performed by treating H em and H so perturbatively.From the spin density, the magnetization of inverse Faraday effect is given by M α = − gµ B s α ,where µ B is the Bohr magneton, and g is the g -factor. In the following section, we only con-sider the contribution proportional to E × E ∗ , because we are interested in the magnetizationinduced by the inverse Faraday effect. A. Spin-orbit interaction from the electric field
We first estimate the spin-orbit interaction arising from the applied electric field, whoseHamiltonian we represent as H (1)so , i.e., H (1)so ≡ − λ ~ Z d x c † σ · [ e ˙ A em × ( p − e A em )] c. (11)The contribution to the spin density, s (1) α , due to H (1)so are shown in Fig. 1, and are calculatedas s (1) α = − i e λǫ αγℓ (cid:0) E γ A ∗ ℓ + E ∗ γ A ℓ (cid:1) K (Ω) , (12)4 IG. 1: Diagrams contributing to the spin density s (1) induced by circularly polarized light. Theelectron’s Green’s functions including the impurity average here are denoted by thick lines. Thewavy line and the dashed arrow represent the gauge field of light ( A em ) and electric field ( ˙ A em ),respectively. λ is spin-orbit coupling constant accompanied with ˙ A em . The vertex shows the Paulimatrix ( σ ). where K (Ω) ≡ X k,ω (cid:20) [ g k,ω g k,ω ] < + ~ m [ I ℓζ (Ω) + I ℓζ ( − Ω)] (cid:21) , (13) I ℓζ (Ω) ≡ ~ m X k,ω k ℓ k ζ [ g k,ω g k,ω +Ω g k,ω ] < , and g k,ω is the free Green’s function on a Keldysh contour. The lesser component of the freeGreen’s function is g Next, we consider the inverse Faraday effect from spin-orbit interaction caused by therandom impurity potential: H (2)so ≡ λ ~ Z d x c † σ · [ ∇ u i × ( p − e A em )] c. (16)The random potential is treated by averaging, h u i ( q ) u i ( q ′ ) i i = n i u δ ( q + q ′ ) and h u i ( q ) u i ( q ′ ) u i ( q ′′ ) i i = n i u δ ( q + q ′ + q ′′ ), where q , q ′ , q ′′ are wave numbers, n i is the concen-tration of the impurity, and h i i is the random impurity average.Contributions to the spin density proportional to E × E ∗ are diagrammatically shownin Figs. 2 and 3. Processes shown in Fig. 2 and Fig. 3 differ in impurity averaging; theaveraging in Fig. 2 is third order in the impurities as in the case of the anomalous Halleffect . We first estimate the contribution (a) of Fig. 2, defined as s (2) α a ) . It is s (2) α a ) = − e ~ λnu m ǫ αℓγ E ℓ E ∗ γ (Γ Ω − Γ − Ω ) . (17)In the above equation, we have used the relation, A em = (cid:0) E i Ω e − i Ω t − E ∗ i Ω e i Ω t (cid:1) . Γ Ω is writtenasΓ Ω = 2Ω X k,k ′ ,k ′′ ,ω k ( f ω +Ω − f ω ) g rk,ω g ak,ω ( g ak,ω +Ω g ak ′ ,ω +Ω g ak ′′ ,ω +Ω − c.c ) − g rk,ω +Ω g ak,ω +Ω ( g ak,ω g ak ′ ,ω g ak ′′ ,ω − c.c ) + δ Γ Ω = 2Ω X k,k ′ ,k ′′ ,ω k h(cid:16) ( f ω +Ω − f ω ) g rk,ω g ak,ω ( g ak,ω +Ω g ak ′ ,ω +Ω g ak ′′ ,ω +Ω − c.c ) (cid:17) + (Ω → − Ω) i + δ Γ Ω , (18)and δ Γ is given by δ Γ Ω = 2Ω X k,k ′ ,k ′′ ,ω k f ω (cid:2)(cid:0) g ak,ω g ak,ω +Ω ( g ak ′ ,ω g ak ′′ ,ω ( g ak,ω + g ak ′ ,ω + g ak ′′ ,ω ) − g ak,ω g ak ′ ,ω +Ω g ak ′′ ,ω +Ω ) (cid:1) − c.c (cid:3) . (19)The first term of Γ Ω is even function respect with Ω, and therefore Γ Ω − Γ − Ω = δ Γ Ω − δ Γ − Ω .The Green’s function can expanded as g ak,ω +Ω = g ak,ω − ~ Ω( g ak,ω ) +( ~ Ω) ( g ak,ω ) − ( ~ Ω) ( g ak,ω ) +6 IG. 2: Diagrams contributing to the spin density in the spin-orbit interaction due to randomimpurities. Dashed lines denote the impurity averaging between the spin-orbit interaction(thecombination between the wavy line and the open circle) and the impurity scattering (open circle).The contribution of diagrams (a) and (b) indicates s (2) α a ) and s (2) α b ) , respectively. o (Ω ), and thus δ Γ Ω − δ Γ − Ω is reduced to δ Γ Ω − δ Γ − Ω = 4 ~ Ω X k,k ′ ,k ′′ ,ω k f ω +( g ak,ω ) ( g ak ′ ,ω ) g ak ′′ ,ω + ( g ak,ω ) g ak ′ ,ω ( g ak ′′ ,ω ) +( g ak,ω ) ( g ak ′ ,ω ) g ak ′′ ,ω + ( g ak,ω ) g ak ′ ,ω ( g ak ′′ ,ω ) +( g ak,ω ) ( g ak ′ ,ω ) ( g ak ′′ ,ω ) + ( g ak,ω ) ( g ak ′ ,ω ) ( g ak ′′ ,ω ) +( g ak,ω ) ( g ak ′ ,ω ) ( g ak ′′ ,ω ) − c.c + o (Ω ) . (20)This contribution at low frequency contains only the terms of considering of only the retardedor advanced Green’s functions, and it turns out to be negligibly small compared with J Ω ofEq. (22).The contribution of Fig. 2(b), s (2) α b ) , is s (2) α b ) = 2 e ~ λnu m ǫ αℓm E ∗ ℓ E m ( J Ω − J − Ω ) , (21)7 IG. 3: Diagrammatic representation of spin density. where J Ω ≡ ~ m Ω X k,k ′ ,k ′′ ω ( k ′ ) ( k ′′ ) [ g k,ω g k ′′ ,ω g k ′′ ,ω +Ω g k ′ ,ω +Ω g k ′ ,ω g k,ω ] < + k ( k ′′ ) [ g k,ω g k,ω +Ω g k ′′ ,ω +Ω g k ′′ ,ω g k ′ ,ω g k,ω ] < + k ( k ′′ ) [ g k,ω g k ′′ ,ω g k ′′ ,ω +Ω g k ′ ,ω +Ω g k,ω +Ω g k,ω ] < + k ( k ′ ) [ g k,ω g k ′′ ,ω g k ′ ,ω g k ′ ,ω +Ω g k,ω +Ω g k,ω ] < + k ( k ′ ) [ g k,ω g k,ω +Ω g k ′′ ,ω +Ω g k ′ ,ω +Ω g k ′ ,ω g k,ω ] < . The dominant contribution of J Ω is estimated as J Ω = ~ m Ω X k,k ′ ,k ′′ ,ω k ( k ′′ ) ( f ω +Ω − f ω ) | g rk ′ ,ω | g ak,ω g rk ′′ ,ω ( g ak,ω +Ω g ak ′′ ,ω +Ω − c.c )+ | g rk,ω | ( g ak ′ ,ω g ak ′′ ,ω + g rk ′ ,ω g rk ′′ ,ω )( g ak,ω +Ω g ak ′′ ,ω +Ω − c.c )+ | g rk,ω | ( g ak ′′ ,ω + g rk ′′ ,ω )( g ak,ω +Ω g ak ′ ,ω +Ω g ak ′′ ,ω +Ω − c.c ) + δJ Ω , (22)where δJ Ω ≡ ~ m Ω X k,k ′ ,k ′′ ,ω k ( k ′′ ) f ω ( g ak ′ ,ω ) g ak,ω g ak ′′ ,ω g ak,ω +Ω g ak ′′ ,ω +Ω +2( g ak,ω ) g ak ′ ,ω g ak ′′ ,ω g ak,ω +Ω g ak ′′ ,ω +Ω +2( g ak,ω ) g ak ′′ ,ω g ak,ω +Ω g ak ′ ,ω +Ω g ak ′′ ,ω +Ω − c.c. (23)Because δ Γ Ω and δJ Ω have only the contribution only retarded or advanced Green’s func-tions, Eqs. (19) and (23) are both negligibly small compared with Eq. (22) (by the order of ~ ǫ F τ ≪ 1, where τ ≡ ~ πνnu is a elastic electron lifetime). From these estimations, the spin8ensity contribution shown in Fig. 2, s (2) α = s (2) α a ) + s (2) α b ) , is finally obtained as s (2) α ≃ − i e ~ λn i u ǫ αℓγ Ω E ∗ ℓ E γ "X k | g rk | = i π e ν λ (cid:16) uτ ~ (cid:17) Ω τ ǫ αℓγ E ℓ E ∗ γ (24)The contribution of Fig. 3 is calculated similarly as s (2) α = e ~ λn i u m ǫ αℓγ E ∗ ℓ E γ [ I Ω − I − Ω ] , (25)where I Ω ≡ X k,k ′ ω ( k ′ ) [ g k,ω g k ′ ,ω g k ′ ,ω +Ω g k,ω + g k,ω g k ′ ,ω +Ω g k ′ ,ω g k,ω ] < + 1Ω X k,k ′ ω k [ g k,ω g k ′ ,ω g k,ω +Ω g k,ω + g k,ω g k,ω +Ω g k ′ ,ω g k,ω ] < + 1Ω X k,k ′ ω k [ g k,ω g k ′ ,ω +Ω g k,ω +Ω g k,ω + g k,ω g k,ω +Ω g k ′ ,ω +Ω g k,ω ] < . (26)The odd function of Ω, I Ω − I − Ω , is calculated as I Ω − I − Ω = 2Ω X k,k ′ ω k f ω (cid:2) ( g ak,ω ( g ak ′ ,ω ) g ak,ω +Ω + ( g ak,ω ) g ak,ω +Ω ( g ak ′ ,ω − g ak ′ ,ω +Ω )) − c.c (cid:3) +( f ω +Ω − f ω ) g ak,ω +Ω g rk,ω +Ω ( g ak,ω g ak ′ ,ω − g rk,ω g rk ′ ,ω ) − ( f ω +Ω − f ω ) g ak,ω g rk,ω ( g ak,ω +Ω g ak ′ ,ω +Ω − g rk,ω +Ω g rk ′ ,ω +Ω ) = 4 ~ Ω X k,k ′ ω k f ω (cid:2) ( g ak,ω ) ( g ak ′ ,ω ) ( g ak,ω + g ak ′ ,ω ) − c.c (cid:3) + o (Ω )= m Ω iπ ν η ǫ F + o (Ω ) (27)From Eqs. (25) and (27), s (2)2 reads s (2) α = iπ e λν ~ Ω ǫ (cid:18) ~ ǫ F τ (cid:19) E ℓ E ∗ γ , (28)Equation (28) is small compared with Eq. (24) by the order of ǫ F u (cid:16) ~ ǫ F τ (cid:17) ≪ H (2)so is given by s (2)3 .We summarize the spin density contributed from the spin-orbit interaction H (1) and H (2) . Since s (1) are negligible small compared with s (2)3 from Eq. (15) and (28), the total9agnetization of the inverse Faraday effect in spin-orbit interaction, M ≃ − gµ B s (2)3 isobtained as M ≃ iχ E × E ∗ , (29)where χ = − i gµ B π e ν λ (cid:16) uτ ~ (cid:17) Ω τ (30)The magnetization is proportional to the strength of spin-orbit coupling, intensity of light,frequency, and electron’s relaxation time. IV. SUMMARY AND DISCUSSION We have shown that the spin density induced by the inverse Faraday effect in the THzregime is proportional to E × E ∗ . From the spin density induced by circularly polarizedlight, the magnitude of the magnetic field generated in the medium, B eff can be estimated as B eff = µ M , where µ are magnetic susceptibility in the medium. Here, we chose the magneticsusceptibility in the vacuum. The magnitude of spin-orbit coupling is λk = 0 . − , and we choose here λk ≈ . . Wechoose the amplitude and frequency of applied electromagnetic field as | E | = 10 V / m andΩ = 1THz, respectively. In metals with ǫ F ≈ uǫ F = 0 . 1, and ~ ǫ F τ ≈ . 01, the magneticfield can be estimated by | B eff | ≈ − T. We have theoretically studied the spin densityinduced by circularly polarized light in THz regime in metals spin-orbit interaction. Theinduced spin is proportional to spin-orbit interaction, frequency of applied THz light, andalso depends on the electrons relaxation time. Acknowledgments This work was supported by a Grant-in-Aid for Scientific Research (B) (Grant No.22340104) from Japan Society for the Promotion of Science and UK-Japanese Collabo-ration on Current-Driven Domain Wall Dynamics from JST. This work is also financially10upported by the Japan Society for the Promotion of Science for Young Scientists. L. P. Pitaevskii, Sov. Phys. JETP , 1008 (1961). R. Hertel, J. Magn. Magn. Mater. , L1 (2006). P. S. Pershan, Phys. Rev. , 919 (1963). J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, Phys. Rev. Lett. , 190 (1965). P. S. Pershan, J. P. van der Ziel, and L. D. 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