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Magnetic stability for the Hartree-Fock ground state in two dimensional Rashba-Gauge electronic systems
Journal of Magnetism and Magnetic Materials xxx (xxxx) xxxx
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Magnetic stability for the Hatree-Fock ground state in two dimensional Rashba-Gauge electronic systems H. Vivas C. Departamento de Física, Universidad Nacional de Colombia, Sede Manizales, A.A. 127, Colombia
A R T I C LE I N FO
A B S T R A C T
Keywords: Rashba interaction Spin gauge vector potential Transverse spin polarization Helmholtz free energy
The magnetic Hartree Fock ground state stability for a two-dimensional interacting electron system with Rashbatype coupling is studied by implementing the standard many body Green’s function formalism. The externally applied electrical field E enters into the Hamiltonian model through a local gauge-type transformation ∼ Ej Aj (k) , with Aj (k) as the spin gauge vector potential. Phase diagrams associated to the average spin polarization, the Fermi energy, electron density and energy band gap at zero temperature are constructed. The magnetic polarization state as a function of E is obtained by minimizing the Helmholtz energy functional F with respect to the average Z-spin: σẐ . We have found that the electric field might reverse the magnetic ground state, ̂ k × E) . unlike the characteristic decreasing behavior associated to the linear term σ ·(
1. Introduction The control of magnetization states on nanoscaled multiferroic devices via externally applied electric field constitutes a technological challenge with multiple and interesting applications [1–10]. The general consensus for the cross-coupling mechanism of electric/magnetic polarization due to magnetic/electric sources relies on the very first sight upon several phenomena: i) the elastic strain on surfaces or interfaces, ii) the exchange-type Dzyaloshinskii-Moriya interaction and iii) the charge-driven magnetoelectric effects generated by charge density accumulation [11,12]. Underlying Physics shall be understood from local symmetry properties associated to the spin-momentum coupling in the framework of the Rashba approximation: H R = αk y σX̂ − αk x σŶ , with σ ̂ representing the set of Pauli matrices. An additional term on the non-interacting Rashba Hamiltonian emerges as a result of relativistic electric field-spin-momentum interdependence. This energy contribution might be described only by including appropriate local gauge transformations intrinsically related to the electron corresponds to certain unitary spin rotations in momentum space. If U −1H transformation upon HR ̂ = −γσ ·̂ BR in the form U R U = − αkσẐ , 2 2 1/2 with k = (k x + k y ) , the induced gauge term under an applied electric −1∂k U ) ≡ Ej Aj , where Aj is a field E may be written as [13,14]: ∼ Ej (iU j
2 × 2 matrix in k space. In the particular case E = (Ex , 0, 0) , and U given by: = 1 U 2
⎛1
⎜ (−k y
⎝
1 ⎞ + ik x )/ k (k y − ik x )/ k ⎟, ⎠
(1)
x Ax U −1 leads to ∼ −σẐ Ex k y / k 2 , corresponding to the the product UE supplementary piece on the Z component for the effective Zeeman-like magnetic field, in the original basis of H R [15–20]. Attempts for the calculation of the Hartree Fock ground state have been performed through different techniques in terms of the Wigner-Seitz radius rs ∼ 1/ ns , (ns as the particle density), predicting partial spin polarization due to the Coulomb exchange-driven on semiconducting structures [21,22]. In this scenario, we study the magnetic ground state stability in two dimensional interacting electron gas at zero temperature. Electron charge interaction is taken into account via Coulomb contact-type exchange, while the energy band gap is biased by the external electric field for the linear and non-linear (gauge) momentum form. An average out-of-plane spin polarization σẐ rises in a finite range of the Coulomb exchange strength. Analytical results for asymptotic behavior of σẐ at zero applied field and negligible kinetic energy are also discussed. Magnetic phase stability is described in terms of positive to zero transition line for the compressibility factor κ −1, as well as its electron density parameter. 2. Hartree Fock self-consistent formulation for transverse magnetization Single electron dynamics in two-dimensional systems might be described by using a Zeeman-type Hamiltonian HΣ ̂ = −γσ ·̂ BΣ , for γBΣ taken as the effective magnetic field containing the Rashba constant α , the electric field intensity Ex and the electron momentum k: γBΣ = (−αk y, αk x , −Δk ) . The last component contains the spin gauge
E-mail address: [email protected]. https://doi.org/10.1016/j.jmmm.2019.166113 Received 9 April 2019; Received in revised form 17 September 2019; Accepted 4 November 2019 Available online 06 November 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: H. Vivas C., Journal of Magnetism and Magnetic Materials, https://doi.org/10.1016/j.jmmm.2019.166113
Journal of Magnetism and Magnetic Materials xxx (xxxx) xxxx
H. Vivas C.
̂ k × E) . For effect plus the usual linear momentum-field interaction σ ·( this particular symmetry: eEx k y ⎛ k 02 ⎞ ⎜1 − ⎟, k2 ⎠ k 02 ⎝
Δk = Δ0 +
(2)
m☆α/ℏ2
with k 0 = [23,24]. The parameter Δ0 represents the band gap energy at zero field. In the framework of the Hartree–Fock approximation, the interacting contact (short range) electron–electron Coulomb energy may be written by using the decoupling scheme suggested in references [25,26]:
J H σẐ Z = − J σẐ σẐ + 2
2
I ,̂ (3)
where I ̂ is the 2 × 2 unit matrix and σẐ is taken as the average spin polarization on Z direction. The minimum Rashba-Hartree–Fock Hamiltonian in our model takes the form:
̂ = HK ̂ + HZ ̂ + HΣ,̂ HRHF
(4)
̂ generates two energy with HK ̂ = (ℏ2k 2/2m☆) I .̂ The eigenvalues of HRHF subbands: Ek ±
ℏ2k 2 J σẐ = + 2m☆ 2
Fig. 1. Normalized Helmholtz free energy (in units of αk 0 ) as a function of σẐ ¯ 0 = 0.1. (a) and different electric field intensities for T = 0, J = 2, μ¯ = 0.5 and Δ Ex = 0 , (b) Ex = 0.05, (c) Ex = 0.11, (d) Ex = 0.15, (e) Ex = 0.2 , (f) Ex = 0.3. Dotted lines characterize different exchange couplings at zero field. Minimumshift is observed for increasing values of J > 0 . Calculations have been performed under the constraint Ek + < μ¯ . Inset: Normalized inverse compressibility κ −1 ( σẐ )/ κ −1 (0) , with ζ = 1. Dash-dotted: J = 0.5, dashed: J = 1, solid: J = 2 , dotted: J = 5.
2
± εk0,
(5)
Δ2J k )1/2
+ and ΔJ k = Δk − J σẐ . Average spin polarizawith = tion is obtained by calculating the Green’s propagator G 0 (k ) for H RHF , under the prescription σẐ = (ℏβ )−1 ∑k Tr{σẐ G 0 (k )} [27]. Specifically: εk0
(α 2k 2
Gij0 (k, iωn) =
Mijσ (k)
∑ σ = {±}
iωn − ℏ−1 (Ekσ − μ)
,
functional δF¯/ δ σẐ = 0 , with F¯ = ∑ks (Ekσ − μ¯) integrated under the domain D : Ekσ ⩽ μ¯ . Fig. (1) shows the minimum-shift for the normalized Helmholtz energy F¯ as a function of order parameter σẐ at zero temperature. Stronger exchange coupling J depletes the average magnetization on Z direction. Increasing field effects over the local minimum of F¯ are represented by continuous lines (a)-(f). Inversion in the sign of σẐ is possible for some specific values of Ex . Magnetic phase stability might also be described by considering the electron gas density n μ¯ and first order derivatives discontinuities of F . The 2D compressibility factor κ , defined in terms of the variations of F with respect to the density factor rs , can be written as [31]:
(6)
ωn = (2n + 1)/ℏβ , (n = 0, 1, 2…) as the set of fermionic Matsubara frê quencies and Mσ (k) = ukσ ukσ corresponds to the matrix of HRHF ̂ eigenstates. Normalized eigenvectors of HRHF are given by ukσ ≡ (−iσeiϕ k Fkσ , 1)/(1 + Fk2σ )1/2 , while Mijσ (k) is defined through:
Mijσ (k) =
Fk2σ iσFkσ e−iϕ k ⎞ ⎛ 1 , iϕ ⎟ 1 (1 + Fk2σ ) ⎜− iσFkσ e k ⎝ ⎠
(7)
1/2
and tanϕ k = k y / k x . From Eq. (6), with αkFkσ = σ ΔJ k + (α 2k 2 + Δ2J k ) the average spin polarization takes the self consistent form:
σẐ =
∑ k⊆D
ΔJ k α 2k 2 + Δ2J k
ρS (k),
κ −1 = (8)
n μ¯ rs ⎛ dF d 2F ⎞ + rs ⎜− ⎟, 4 ⎝ drs drs2 ⎠
(9)
Inverse compressibility profile at zero field is with = shown in Fig. 1 (inset). Explicit dependence of F (rs ) is calculated by recasting the electron momentum k on the energy bands Ek ±, in the form: k¯ = a0 rs k , with a0 = 4πε0/ m☆e 2 as the Bohr’s radius, and ε0 as the dielectric constant for the vacuum. Normalized electron band energy Ek ± = Ek¯ ±/ αk 0 in Eq. (5) transforms into:
rs2
ρS (k) = sinh(βεk0)/[cosh(βμJ k ) + cosh(βεk0)] μJ k = and with μ − (J /2) σẐ 2 − ℏ2k 2/2m☆ as the modified chemical potential. At zero temperature and zero field, the distribution function ρS (k) evolves towards the step function, with ρS (k) = 1 inside the circular domain D of ¯ 2J 0)1/2)]1/2 and ρS (k) = 0 , otherradii k¯ ∓ = [2(1 + μ¯J 0 ∓ (1 + 2μ¯J 0 + Δ wise. The domain D gets along into complex geometry as Ex ≠ 0 , and its breaching is closely related with strong fluctuations on the electronic Density Of States (DOS). The factors ΔJ 0 , J , αk and μJ 0 have been normalized (and labeled with an overlying bar) to the reference Rashba energy αk 0 . Also, Ex = eEx /αk 02 . For instance, with α = 4.62 × 10−12 eV·m, which is typical for semiconducting InGaAs/InP asymmetric quantum wells, k 0−1 falls into the range of 44.63 nm with m☆ = 0.37m 0 and 76.38 nm for m☆ = 0.04m 0 in 2DEG Inx Ga1 − x As [28,29]. For Bix Pb1 − x /Ag (111) alloys, α reaches the value 3 × 10−10 eV·m, with m☆ = 0.3m 0 and 1.2 < k 0−1 < 2.5 nm [30]. In the former case, the electrical field reaches a maximum value of 4.6 V·cm−1 for Ex ≈ 0.2 , used as the upper limit for our estimations.
Ek¯ ± =
1/ πn μ¯ a02 .
1 k¯ 2 J + 2 ζ2 2
(κ −1)
2
σẐ
± ε¯k¯0, α /e2
(10)
as a dimensionless scale factor assowith ζ = ζR rs and ζR = 4πε0 ciated to the relative Rashba-Coulomb strength interaction. There are two main states for the electron gas defined under conditions κ −1 ⩾ 0 . Instability region κ −1 = 0 might be attached for the regime J ⩾ JC in the range σẐ ⩽ 1/2 . Main frame in Fig. (2) depicts a magnetic phase diagram [ σẐ , rs] for the particular case J = 2 at zero applied field. It is shown that the electron gas is magnetically stable for positive compressibility (κ −1 > 0 on the inner frame) inside the tuning fork-shape ¯ 0 , μ¯ line, whose curvature and vertex offset are involved functions of Δ and J . Zero magnetization condition rs0 is calculated as: ¯ 20)1/2)1/4 . Therefore, our results are valid rs−01 = ζ R1/2 (1 + μ¯ − (1 + 2μ¯ + Δ in the regime for which the density factor rs ⩾ rsC , taking ζ = 1, where rsC satisfies the relationship δrs / δ σẐ = 0 in the domain σZ ⩽ 1/2 .
3. Results The average spin polarization value σẐ (Eq. 8) at zero temperature might also be obtained by minimizing the Helmholtz free energy 2
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¯ 0 , μ¯] for σẐ < 1/2 at zero electric field and Fig. 3. Phase space correlation [Δ zero exchange interaction. Inset: Z-spin polarization at J = 0 and Ex = 0 as a ¯ 0 = 0.25, Dashed: ¯ 0 = 0.1. Dotted: Δ function of the Fermi energy. Solid: Δ ¯ 0 = 0.457039 . Δ
Fig. 2. Spin average behavior σZ as a function characteristic density rs in the ¯ 0 = 0.1 and J = 2 . Inset: Phase dialimit κ −1 = 0, ζR = 1, Ex = 0, μ¯ = 0.5 and Δ gram showing magnetic stability region for (positive) inverse compressibility κ −1 > 0 , and the same set of chosen parameters as in the main frame. rs → rs0 at σẐ = 0 . In this case rs0 ≈ 1.87 .
Asymptotic values for spin average in the low density regime (rs → ∞) ± ¯ 0)/ J ]1/2 ∓ 1. In general, rsC < rs0 for = ± [1 + 2(μ¯ ± Δ are given by σZ ∞ ¯ 0 ≠ 0 . Higher values of J dwindles the boundaries with positive Δ compressibility, indicating that the electron gas tends to be magnetically unbiased as the relative ratio exchange-Rashba interactions, augments. The critical condition κ −1 = 0 is not found for externally applied fields within the specific parameters studied. Electron density is also defined in terms of the Green’s propagator as n μ¯ = (ℏβ )−1 ∑k Tr{G 0 (k )} = ∑k ⊆ D ρN (k) , with the density function number defined by: ρN (k) = 1 + sinh(βμJ k )/[cosh(βμJ k ) + cosh(βεk0 )]. ¯ 0 = 0 ), without exchange ( J = 0 ), nor applied In the gapless limit (Δ field (Ex = 0 ) and μ¯ > 0 , we get n μ¯ = (k 02/ π )[1 + μ¯ − (1 + 2μ¯)1/2]. n μ¯ is ¯ 0 increases at fixed μ¯ [32,33]. In the low-density electron reduced as Δ gas approximation, Ek ≈ εk0 , the energy band takes the Dirac cone-like structure, and in the case in which the gauge interaction coupling is exclusively considered in Eq. (2) (i.e., Δk ≈ −eEx k y / k 2 ), the electron density reads: n μG¯ (Ex ) = (k 02 μ¯ 2 / π 2) E (4E x2/ μ¯ 4 ) . The function E (4E x2/ μ¯ 4 ) is the complete elliptic integral of the second kind, with real values in the interval 0 < 4E x2/ μ¯ 4 < 1, or Ex restrained into {0, μ¯ 2 /2} , range for continuous but not simply connected Fermi surfaces. Electron density lies on the interval k 02 μ¯ 2 / π 2 < n μG¯ (Ex ) < k 02 μ¯ 2 /2π [34]. An estimation for the density of states (DOS) at zero temperature can be directly obunder the definition tained from n μG¯ G D μ¯ (Ex ) = ∂n μ¯ /∂μ¯ = (2μ¯ /π ) DF K (4E x2/μ¯ 4 ) , where K (4E x2/μ¯ 4 ) represents the complete elliptic integral of first kind, and DF = m☆/ π ℏ2 corresponds to the DOS for two dimensional non interacting electron systems. In the limit μ¯ 4 ≫ 4E x2 (or Ex = 0 ), DOS behaves in a linear form with D μ¯ (0) ≈ μ¯ DF , while it reveals a strong peak at μ¯ = (2Ex )1/2 . Under the same set of conditions and Δk ≈ eEx k y / k 02 , the density is given by n μ¯D (Ex ) = k 02 μ¯ 2 /2π (1 + E x2)1/2 . The carrier density n μG¯ decreases more rapidly than n μ¯D in the range 0 < Ex ⩽ μ¯ 2 /2 . Fig. (3) shows ¯ 0 , μ¯] at J = 0 (shaded area) for physically possible the phase space [Δ spin range σẐ ⩽ 1/2 . Higher band gap values restrict the allowed Fermi energy (Inset graph). Fig. (4) portrays the main result in this paper: the numerical solutions for σẐ as a function of the external applied field Ex and several coupling strength J for conducting electrons dwelling on the positive helicity band Ek + [35]. Comparative results for the usual non-gauge contribution Δk = Δ0 + eEx k y / k 02 are also shown for 0 < J < 2.0 . In this case, Ex -scale is reduced in one order of magnitude without inversion in the magnetic orientation [36]. Inset
¯ 0 = 0.1. (a) J = 0.0 , (b) Fig. 4. Numerical solutions for σẐ , at μ¯ = 0.5, Δ J = 0.25, (c) J = 0.5, (d) J = 1.0 , (e) J = 2.0 . Dashed line describes the rê k × E) with sponse for a linear spin-electric field coupling in the form ∼ σ ·( J = 0 . The horizontal scale might be adjusted by a factor of 10 in this case. Inset: Phase diagram [J , σẐ ] for different gap magnitudes at zero field, ¯ 0 = 0.2 , (Squares) Δ ¯ 0 = 0.4 , ¯ 0 = 0.1, (Solid) Δ σẐ > 0 and μ¯ = 0.5. (Dotted) Δ ¯ 0 ≈ 0.5. (Dashed) Δ
graph depicts the average spin at zero field and different band gap energies. The change in the curvature of the function σẐ is possible for ¯ 0 > 0.33 and it exhibits a maximum at J ≈ 0.27 for Δ ¯ 0 = 0.4 . At zero Δ field and J = 0 , integral (8) is analytically computable with the result 4π 2/ S ): (up to a normalization factor ¯ 0 [(1 + 2μ¯ + Δ ¯ 20)1/2 − 1 − Δ ¯ 0]. σẐ exhibits a fluctuating σẐ = 4πk 02 Δ behavior in the studied interval of electric field strengths with non symmetrical amplitudes and decaying tendency for stronger intensities. These curves resembles the typical butterfly-shape associated to the strain intermediate magnetoelastic interaction in ferroelectric/ferromagnetic hybrid structures, in the direct polarization cycle [37,38]. Zspin polarization magnitude and its direction are symmetrical under electric field inversion in our model. Parameter J weights the repulsive electron–electron contact interaction against the characteristic Rashba coupling αk 0 , indicating that, for systems under strong repulsion field, 3
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the magnetic ordering is depleted for Ex < Exc , with Exc as the coercive field value that suppress the magnetic polarization. For Ex > Exc , σẐ changes the sign and its amplitude as well as their lowest values are also affected under variations of J . The coercive field magnitude augments ¯ 0 grows, and its cutoff value Exc0 also increases with μ¯ , but it is as Δ independent of J . A reliable model might be proposed in terms of an 2 2 ¯ 0 = μ¯ (1 − exp[−η (E xc − E xc exponential-like behavior: Δ 0 )]) , with η and Exc0 as adjusting parameters, with Exc ⩾ Exc0 . The relationship Exc0 (μ¯) is fairly linear in the range of parameters studied.
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4. Concluding remarks Possible spin magnetization control under in-plane applied electric field has been discussed for two dimensional electron systems with Rashba interaction. The parameter σẐ is obtained by implementing a minimization algorithm on the Helmholtz energy functional at zero temperature under topologically constrained Fermi surfaces. The contact electron-electron repulsive effect is considered in the model by using the Hartree–Fock approximation, while the electrical field couples the spin degree of freedom via emerging spin-gauge vector potential. Phase diagrams in the electron density, the average spin polarization and Fermi energy are constructed in the particular case Ex = 0 , and taking into account the positive compressibility condition. We have found in our model that for those systems in which the gauge coupling predominates over the linear spin-momentum-field, it is possible to induce a controlled spin shift inversion for smaller intensities of the external electric field. Non switch orientation in the average magnetization is obtained in the last scenario. Further investigations on additional spin-gauge vector terms might be taken into account, since Aj (k) shall be reconsidered in a higher order expansion scheme when the effective magnetic field BΣ emerges in the framework of the nonadiabatic Zeeman-like coupling [39]. This model must also be reexamined beyond the plain contact electron–electron approach. Finally, anisotropic energy associated to surface and magnetoelastic strain stimulated by longitudinally applied voltages shall also be included into the formalism [40,41]. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author acknowledges availability and technical support at Bloque-W Computer Lab, UN Manizales. References [1] N.A. Spaldin, R. Ramesh, Nature Materials 18 (2019) 203, https://doi.org/10. 1038/s41563-018-0275-2. [2] S.T.B. Goennenwein, Europhys. News 41 (4) (2010) 17–20, https://doi.org/10. 1051/epn/2010401. [3] J.T. Heron, Nature 516 (2014) 370 doi:10.1038/nature14004.. [4] N.A. Spaldin, M. Fiebig, Science 309 (2005) 391, https://doi.org/10.1126/science. 1113357. [5] V. Krivoruchko, A. Savchenko, Acta Physica Polonica A 133 (2018) 463, https:// doi.org/10.12693/APhysPolA.133.463. [6] S.E. Barnes, J. Ieda, S. Maekawa, Sci. Rep. 4 (2014) 4105, https://doi.org/10.1038/ srep04105. [7] C.H. Li, F. Wang, Y. Liu, X.Q. Zhang, Z.H. Cheng, Y. Sun, Phys. Rev. B 79 (2009)
4
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Magnetic stability for the Hatree-Fock ground state in two dimensional Rashba-Gauge electronic systems H. Vivas C. Departamento de Física, Universidad Nacional de Colombia, Sede Manizales, A.A. 127, Colombia
A R T I C LE I N FO
A B S T R A C T
Keywords: Rashba interaction Spin gauge vector potential Transverse spin polarization Helmholtz free energy
The magnetic Hartree Fock ground state stability for a two-dimensional interacting electron system with Rashbatype coupling is studied by implementing the standard many body Green’s function formalism. The externally applied electrical field E enters into the Hamiltonian model through a local gauge-type transformation ∼ Ej Aj (k) , with Aj (k) as the spin gauge vector potential. Phase diagrams associated to the average spin polarization, the Fermi energy, electron density and energy band gap at zero temperature are constructed. The magnetic polarization state as a function of E is obtained by minimizing the Helmholtz energy functional F with respect to the average Z-spin: σẐ . We have found that the electric field might reverse the magnetic ground state, ̂ k × E) . unlike the characteristic decreasing behavior associated to the linear term σ ·(
1. Introduction The control of magnetization states on nanoscaled multiferroic devices via externally applied electric field constitutes a technological challenge with multiple and interesting applications [1–10]. The general consensus for the cross-coupling mechanism of electric/magnetic polarization due to magnetic/electric sources relies on the very first sight upon several phenomena: i) the elastic strain on surfaces or interfaces, ii) the exchange-type Dzyaloshinskii-Moriya interaction and iii) the charge-driven magnetoelectric effects generated by charge density accumulation [11,12]. Underlying Physics shall be understood from local symmetry properties associated to the spin-momentum coupling in the framework of the Rashba approximation: H R = αk y σX̂ − αk x σŶ , with σ ̂ representing the set of Pauli matrices. An additional term on the non-interacting Rashba Hamiltonian emerges as a result of relativistic electric field-spin-momentum interdependence. This energy contribution might be described only by including appropriate local gauge transformations intrinsically related to the electron corresponds to certain unitary spin rotations in momentum space. If U −1H transformation upon HR ̂ = −γσ ·̂ BR in the form U R U = − αkσẐ , 2 2 1/2 with k = (k x + k y ) , the induced gauge term under an applied electric −1∂k U ) ≡ Ej Aj , where Aj is a field E may be written as [13,14]: ∼ Ej (iU j
2 × 2 matrix in k space. In the particular case E = (Ex , 0, 0) , and U given by: = 1 U 2
⎛1
⎜ (−k y
⎝
1 ⎞ + ik x )/ k (k y − ik x )/ k ⎟, ⎠
(1)
x Ax U −1 leads to ∼ −σẐ Ex k y / k 2 , corresponding to the the product UE supplementary piece on the Z component for the effective Zeeman-like magnetic field, in the original basis of H R [15–20]. Attempts for the calculation of the Hartree Fock ground state have been performed through different techniques in terms of the Wigner-Seitz radius rs ∼ 1/ ns , (ns as the particle density), predicting partial spin polarization due to the Coulomb exchange-driven on semiconducting structures [21,22]. In this scenario, we study the magnetic ground state stability in two dimensional interacting electron gas at zero temperature. Electron charge interaction is taken into account via Coulomb contact-type exchange, while the energy band gap is biased by the external electric field for the linear and non-linear (gauge) momentum form. An average out-of-plane spin polarization σẐ rises in a finite range of the Coulomb exchange strength. Analytical results for asymptotic behavior of σẐ at zero applied field and negligible kinetic energy are also discussed. Magnetic phase stability is described in terms of positive to zero transition line for the compressibility factor κ −1, as well as its electron density parameter. 2. Hartree Fock self-consistent formulation for transverse magnetization Single electron dynamics in two-dimensional systems might be described by using a Zeeman-type Hamiltonian HΣ ̂ = −γσ ·̂ BΣ , for γBΣ taken as the effective magnetic field containing the Rashba constant α , the electric field intensity Ex and the electron momentum k: γBΣ = (−αk y, αk x , −Δk ) . The last component contains the spin gauge
E-mail address: [email protected]. https://doi.org/10.1016/j.jmmm.2019.166113 Received 9 April 2019; Received in revised form 17 September 2019; Accepted 4 November 2019 Available online 06 November 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: H. Vivas C., Journal of Magnetism and Magnetic Materials, https://doi.org/10.1016/j.jmmm.2019.166113
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̂ k × E) . For effect plus the usual linear momentum-field interaction σ ·( this particular symmetry: eEx k y ⎛ k 02 ⎞ ⎜1 − ⎟, k2 ⎠ k 02 ⎝
Δk = Δ0 +
(2)
m☆α/ℏ2
with k 0 = [23,24]. The parameter Δ0 represents the band gap energy at zero field. In the framework of the Hartree–Fock approximation, the interacting contact (short range) electron–electron Coulomb energy may be written by using the decoupling scheme suggested in references [25,26]:
J H σẐ Z = − J σẐ σẐ + 2
2
I ,̂ (3)
where I ̂ is the 2 × 2 unit matrix and σẐ is taken as the average spin polarization on Z direction. The minimum Rashba-Hartree–Fock Hamiltonian in our model takes the form:
̂ = HK ̂ + HZ ̂ + HΣ,̂ HRHF
(4)
̂ generates two energy with HK ̂ = (ℏ2k 2/2m☆) I .̂ The eigenvalues of HRHF subbands: Ek ±
ℏ2k 2 J σẐ = + 2m☆ 2
Fig. 1. Normalized Helmholtz free energy (in units of αk 0 ) as a function of σẐ ¯ 0 = 0.1. (a) and different electric field intensities for T = 0, J = 2, μ¯ = 0.5 and Δ Ex = 0 , (b) Ex = 0.05, (c) Ex = 0.11, (d) Ex = 0.15, (e) Ex = 0.2 , (f) Ex = 0.3. Dotted lines characterize different exchange couplings at zero field. Minimumshift is observed for increasing values of J > 0 . Calculations have been performed under the constraint Ek + < μ¯ . Inset: Normalized inverse compressibility κ −1 ( σẐ )/ κ −1 (0) , with ζ = 1. Dash-dotted: J = 0.5, dashed: J = 1, solid: J = 2 , dotted: J = 5.
2
± εk0,
(5)
Δ2J k )1/2
+ and ΔJ k = Δk − J σẐ . Average spin polarizawith = tion is obtained by calculating the Green’s propagator G 0 (k ) for H RHF , under the prescription σẐ = (ℏβ )−1 ∑k Tr{σẐ G 0 (k )} [27]. Specifically: εk0
(α 2k 2
Gij0 (k, iωn) =
Mijσ (k)
∑ σ = {±}
iωn − ℏ−1 (Ekσ − μ)
,
functional δF¯/ δ σẐ = 0 , with F¯ = ∑ks (Ekσ − μ¯) integrated under the domain D : Ekσ ⩽ μ¯ . Fig. (1) shows the minimum-shift for the normalized Helmholtz energy F¯ as a function of order parameter σẐ at zero temperature. Stronger exchange coupling J depletes the average magnetization on Z direction. Increasing field effects over the local minimum of F¯ are represented by continuous lines (a)-(f). Inversion in the sign of σẐ is possible for some specific values of Ex . Magnetic phase stability might also be described by considering the electron gas density n μ¯ and first order derivatives discontinuities of F . The 2D compressibility factor κ , defined in terms of the variations of F with respect to the density factor rs , can be written as [31]:
(6)
ωn = (2n + 1)/ℏβ , (n = 0, 1, 2…) as the set of fermionic Matsubara frê quencies and Mσ (k) = ukσ ukσ corresponds to the matrix of HRHF ̂ eigenstates. Normalized eigenvectors of HRHF are given by ukσ ≡ (−iσeiϕ k Fkσ , 1)/(1 + Fk2σ )1/2 , while Mijσ (k) is defined through:
Mijσ (k) =
Fk2σ iσFkσ e−iϕ k ⎞ ⎛ 1 , iϕ ⎟ 1 (1 + Fk2σ ) ⎜− iσFkσ e k ⎝ ⎠
(7)
1/2
and tanϕ k = k y / k x . From Eq. (6), with αkFkσ = σ ΔJ k + (α 2k 2 + Δ2J k ) the average spin polarization takes the self consistent form:
σẐ =
∑ k⊆D
ΔJ k α 2k 2 + Δ2J k
ρS (k),
κ −1 = (8)
n μ¯ rs ⎛ dF d 2F ⎞ + rs ⎜− ⎟, 4 ⎝ drs drs2 ⎠
(9)
Inverse compressibility profile at zero field is with = shown in Fig. 1 (inset). Explicit dependence of F (rs ) is calculated by recasting the electron momentum k on the energy bands Ek ±, in the form: k¯ = a0 rs k , with a0 = 4πε0/ m☆e 2 as the Bohr’s radius, and ε0 as the dielectric constant for the vacuum. Normalized electron band energy Ek ± = Ek¯ ±/ αk 0 in Eq. (5) transforms into:
rs2
ρS (k) = sinh(βεk0)/[cosh(βμJ k ) + cosh(βεk0)] μJ k = and with μ − (J /2) σẐ 2 − ℏ2k 2/2m☆ as the modified chemical potential. At zero temperature and zero field, the distribution function ρS (k) evolves towards the step function, with ρS (k) = 1 inside the circular domain D of ¯ 2J 0)1/2)]1/2 and ρS (k) = 0 , otherradii k¯ ∓ = [2(1 + μ¯J 0 ∓ (1 + 2μ¯J 0 + Δ wise. The domain D gets along into complex geometry as Ex ≠ 0 , and its breaching is closely related with strong fluctuations on the electronic Density Of States (DOS). The factors ΔJ 0 , J , αk and μJ 0 have been normalized (and labeled with an overlying bar) to the reference Rashba energy αk 0 . Also, Ex = eEx /αk 02 . For instance, with α = 4.62 × 10−12 eV·m, which is typical for semiconducting InGaAs/InP asymmetric quantum wells, k 0−1 falls into the range of 44.63 nm with m☆ = 0.37m 0 and 76.38 nm for m☆ = 0.04m 0 in 2DEG Inx Ga1 − x As [28,29]. For Bix Pb1 − x /Ag (111) alloys, α reaches the value 3 × 10−10 eV·m, with m☆ = 0.3m 0 and 1.2 < k 0−1 < 2.5 nm [30]. In the former case, the electrical field reaches a maximum value of 4.6 V·cm−1 for Ex ≈ 0.2 , used as the upper limit for our estimations.
Ek¯ ± =
1/ πn μ¯ a02 .
1 k¯ 2 J + 2 ζ2 2
(κ −1)
2
σẐ
± ε¯k¯0, α /e2
(10)
as a dimensionless scale factor assowith ζ = ζR rs and ζR = 4πε0 ciated to the relative Rashba-Coulomb strength interaction. There are two main states for the electron gas defined under conditions κ −1 ⩾ 0 . Instability region κ −1 = 0 might be attached for the regime J ⩾ JC in the range σẐ ⩽ 1/2 . Main frame in Fig. (2) depicts a magnetic phase diagram [ σẐ , rs] for the particular case J = 2 at zero applied field. It is shown that the electron gas is magnetically stable for positive compressibility (κ −1 > 0 on the inner frame) inside the tuning fork-shape ¯ 0 , μ¯ line, whose curvature and vertex offset are involved functions of Δ and J . Zero magnetization condition rs0 is calculated as: ¯ 20)1/2)1/4 . Therefore, our results are valid rs−01 = ζ R1/2 (1 + μ¯ − (1 + 2μ¯ + Δ in the regime for which the density factor rs ⩾ rsC , taking ζ = 1, where rsC satisfies the relationship δrs / δ σẐ = 0 in the domain σZ ⩽ 1/2 .
3. Results The average spin polarization value σẐ (Eq. 8) at zero temperature might also be obtained by minimizing the Helmholtz free energy 2
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¯ 0 , μ¯] for σẐ < 1/2 at zero electric field and Fig. 3. Phase space correlation [Δ zero exchange interaction. Inset: Z-spin polarization at J = 0 and Ex = 0 as a ¯ 0 = 0.25, Dashed: ¯ 0 = 0.1. Dotted: Δ function of the Fermi energy. Solid: Δ ¯ 0 = 0.457039 . Δ
Fig. 2. Spin average behavior σZ as a function characteristic density rs in the ¯ 0 = 0.1 and J = 2 . Inset: Phase dialimit κ −1 = 0, ζR = 1, Ex = 0, μ¯ = 0.5 and Δ gram showing magnetic stability region for (positive) inverse compressibility κ −1 > 0 , and the same set of chosen parameters as in the main frame. rs → rs0 at σẐ = 0 . In this case rs0 ≈ 1.87 .
Asymptotic values for spin average in the low density regime (rs → ∞) ± ¯ 0)/ J ]1/2 ∓ 1. In general, rsC < rs0 for = ± [1 + 2(μ¯ ± Δ are given by σZ ∞ ¯ 0 ≠ 0 . Higher values of J dwindles the boundaries with positive Δ compressibility, indicating that the electron gas tends to be magnetically unbiased as the relative ratio exchange-Rashba interactions, augments. The critical condition κ −1 = 0 is not found for externally applied fields within the specific parameters studied. Electron density is also defined in terms of the Green’s propagator as n μ¯ = (ℏβ )−1 ∑k Tr{G 0 (k )} = ∑k ⊆ D ρN (k) , with the density function number defined by: ρN (k) = 1 + sinh(βμJ k )/[cosh(βμJ k ) + cosh(βεk0 )]. ¯ 0 = 0 ), without exchange ( J = 0 ), nor applied In the gapless limit (Δ field (Ex = 0 ) and μ¯ > 0 , we get n μ¯ = (k 02/ π )[1 + μ¯ − (1 + 2μ¯)1/2]. n μ¯ is ¯ 0 increases at fixed μ¯ [32,33]. In the low-density electron reduced as Δ gas approximation, Ek ≈ εk0 , the energy band takes the Dirac cone-like structure, and in the case in which the gauge interaction coupling is exclusively considered in Eq. (2) (i.e., Δk ≈ −eEx k y / k 2 ), the electron density reads: n μG¯ (Ex ) = (k 02 μ¯ 2 / π 2) E (4E x2/ μ¯ 4 ) . The function E (4E x2/ μ¯ 4 ) is the complete elliptic integral of the second kind, with real values in the interval 0 < 4E x2/ μ¯ 4 < 1, or Ex restrained into {0, μ¯ 2 /2} , range for continuous but not simply connected Fermi surfaces. Electron density lies on the interval k 02 μ¯ 2 / π 2 < n μG¯ (Ex ) < k 02 μ¯ 2 /2π [34]. An estimation for the density of states (DOS) at zero temperature can be directly obunder the definition tained from n μG¯ G D μ¯ (Ex ) = ∂n μ¯ /∂μ¯ = (2μ¯ /π ) DF K (4E x2/μ¯ 4 ) , where K (4E x2/μ¯ 4 ) represents the complete elliptic integral of first kind, and DF = m☆/ π ℏ2 corresponds to the DOS for two dimensional non interacting electron systems. In the limit μ¯ 4 ≫ 4E x2 (or Ex = 0 ), DOS behaves in a linear form with D μ¯ (0) ≈ μ¯ DF , while it reveals a strong peak at μ¯ = (2Ex )1/2 . Under the same set of conditions and Δk ≈ eEx k y / k 02 , the density is given by n μ¯D (Ex ) = k 02 μ¯ 2 /2π (1 + E x2)1/2 . The carrier density n μG¯ decreases more rapidly than n μ¯D in the range 0 < Ex ⩽ μ¯ 2 /2 . Fig. (3) shows ¯ 0 , μ¯] at J = 0 (shaded area) for physically possible the phase space [Δ spin range σẐ ⩽ 1/2 . Higher band gap values restrict the allowed Fermi energy (Inset graph). Fig. (4) portrays the main result in this paper: the numerical solutions for σẐ as a function of the external applied field Ex and several coupling strength J for conducting electrons dwelling on the positive helicity band Ek + [35]. Comparative results for the usual non-gauge contribution Δk = Δ0 + eEx k y / k 02 are also shown for 0 < J < 2.0 . In this case, Ex -scale is reduced in one order of magnitude without inversion in the magnetic orientation [36]. Inset
¯ 0 = 0.1. (a) J = 0.0 , (b) Fig. 4. Numerical solutions for σẐ , at μ¯ = 0.5, Δ J = 0.25, (c) J = 0.5, (d) J = 1.0 , (e) J = 2.0 . Dashed line describes the rê k × E) with sponse for a linear spin-electric field coupling in the form ∼ σ ·( J = 0 . The horizontal scale might be adjusted by a factor of 10 in this case. Inset: Phase diagram [J , σẐ ] for different gap magnitudes at zero field, ¯ 0 = 0.2 , (Squares) Δ ¯ 0 = 0.4 , ¯ 0 = 0.1, (Solid) Δ σẐ > 0 and μ¯ = 0.5. (Dotted) Δ ¯ 0 ≈ 0.5. (Dashed) Δ
graph depicts the average spin at zero field and different band gap energies. The change in the curvature of the function σẐ is possible for ¯ 0 > 0.33 and it exhibits a maximum at J ≈ 0.27 for Δ ¯ 0 = 0.4 . At zero Δ field and J = 0 , integral (8) is analytically computable with the result 4π 2/ S ): (up to a normalization factor ¯ 0 [(1 + 2μ¯ + Δ ¯ 20)1/2 − 1 − Δ ¯ 0]. σẐ exhibits a fluctuating σẐ = 4πk 02 Δ behavior in the studied interval of electric field strengths with non symmetrical amplitudes and decaying tendency for stronger intensities. These curves resembles the typical butterfly-shape associated to the strain intermediate magnetoelastic interaction in ferroelectric/ferromagnetic hybrid structures, in the direct polarization cycle [37,38]. Zspin polarization magnitude and its direction are symmetrical under electric field inversion in our model. Parameter J weights the repulsive electron–electron contact interaction against the characteristic Rashba coupling αk 0 , indicating that, for systems under strong repulsion field, 3
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the magnetic ordering is depleted for Ex < Exc , with Exc as the coercive field value that suppress the magnetic polarization. For Ex > Exc , σẐ changes the sign and its amplitude as well as their lowest values are also affected under variations of J . The coercive field magnitude augments ¯ 0 grows, and its cutoff value Exc0 also increases with μ¯ , but it is as Δ independent of J . A reliable model might be proposed in terms of an 2 2 ¯ 0 = μ¯ (1 − exp[−η (E xc − E xc exponential-like behavior: Δ 0 )]) , with η and Exc0 as adjusting parameters, with Exc ⩾ Exc0 . The relationship Exc0 (μ¯) is fairly linear in the range of parameters studied.
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4. Concluding remarks Possible spin magnetization control under in-plane applied electric field has been discussed for two dimensional electron systems with Rashba interaction. The parameter σẐ is obtained by implementing a minimization algorithm on the Helmholtz energy functional at zero temperature under topologically constrained Fermi surfaces. The contact electron-electron repulsive effect is considered in the model by using the Hartree–Fock approximation, while the electrical field couples the spin degree of freedom via emerging spin-gauge vector potential. Phase diagrams in the electron density, the average spin polarization and Fermi energy are constructed in the particular case Ex = 0 , and taking into account the positive compressibility condition. We have found in our model that for those systems in which the gauge coupling predominates over the linear spin-momentum-field, it is possible to induce a controlled spin shift inversion for smaller intensities of the external electric field. Non switch orientation in the average magnetization is obtained in the last scenario. Further investigations on additional spin-gauge vector terms might be taken into account, since Aj (k) shall be reconsidered in a higher order expansion scheme when the effective magnetic field BΣ emerges in the framework of the nonadiabatic Zeeman-like coupling [39]. This model must also be reexamined beyond the plain contact electron–electron approach. Finally, anisotropic energy associated to surface and magnetoelastic strain stimulated by longitudinally applied voltages shall also be included into the formalism [40,41]. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author acknowledges availability and technical support at Bloque-W Computer Lab, UN Manizales. References [1] N.A. Spaldin, R. Ramesh, Nature Materials 18 (2019) 203, https://doi.org/10. 1038/s41563-018-0275-2. [2] S.T.B. Goennenwein, Europhys. News 41 (4) (2010) 17–20, https://doi.org/10. 1051/epn/2010401. [3] J.T. Heron, Nature 516 (2014) 370 doi:10.1038/nature14004.. [4] N.A. Spaldin, M. Fiebig, Science 309 (2005) 391, https://doi.org/10.1126/science. 1113357. [5] V. Krivoruchko, A. Savchenko, Acta Physica Polonica A 133 (2018) 463, https:// doi.org/10.12693/APhysPolA.133.463. [6] S.E. Barnes, J. Ieda, S. Maekawa, Sci. Rep. 4 (2014) 4105, https://doi.org/10.1038/ srep04105. [7] C.H. Li, F. Wang, Y. Liu, X.Q. Zhang, Z.H. Cheng, Y. Sun, Phys. Rev. B 79 (2009)
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