Impurity states in graphene with intrinsic spin-orbit interaction
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Impurity states in graphene with intrinsic spin-orbitinteraction
M Inglot ‡ and V K Dugaev , Department of Physics, Rzesz´ow University of Technology, Al. Powsta´nc´owWarszawy 6, 35-959 Rzesz´ow, Poland Department of Physics and CFIF, Instituto Superior T´ecnico, Technical Universityof Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, PortugalE-mail: [email protected]
Abstract.
We consider the problem of electron energy states related to stronglylocalized potential of a single impurity in graphene. Our model simulates the effect ofimpurity atom substituting the atom of carbon, on the energy spectrum of electronsnear the Dirac point. We take into account the internal spin-orbit interaction, whichcan modify the structure of electron bands at very small neighborhood of the Diracpoint, leading to the energy gap. This makes possible the occurrence of additionalimpurity states in the vicinity of the gap.PACS numbers: 73.22Pr,73.20Hb ‡ Corresponding author: [email protected] mpurity states in graphene with intrinsic spin-orbit interaction
1. Introduction
Graphene attracted a lot of attention recently due to very unusual properties of electronenergy spectrum and transport properties, including both the transport of electrons andphonons [1, 2, 3, 4, 5]. The most striking properties of graphene are related to theenergy spectrum near the Dirac points, where this spectrum is linear as a function ofmomentum, and the Hamiltonian of free electrons can be described by the relativistictwo-dimensional Dirac model [6].Naturally, the impurities and defects can strongly affect the energy spectrum ofgraphene. Especially important is the effect of impurities on the spectrum near theDirac point. The impurity states and the corresponding variation of the electron densityof states have been already discussed in several papers [7, 8, 9, 10] without takinginto account the spin-orbit (SO) interaction. It was found that the localized impuritypotential gives the resonant states in the spectrum of graphene. They can be locatednear the Dirac point in the case of relatively strong impurity potential, and this isquite unusual for the semiconductor physics. In the case of carbon vacancy, thereappears a local energy level at the Dirac point, with E = 0. In all of these works, themain attention has been paid to a finite density of impurities and defects leading tomodification of the density of states in graphene.In this paper we mostly concentrate on the problem of single impurity taking intoaccount the internal SO interaction. The SO interaction opens a gap in the electronenergy spectrum [11]. However, it was found that the magnitude of SO-induced gap isvery small in graphene [12, 13, 14, 15], and therefore it would be very difficult to observethis gap experimentally. Nevertheless, the problem exists: how this small gap wouldaffect the behavior of impurity states in the vicinity of Dirac point?We demonstrate that the SO gap induces appearance of additional impurity statescorresponding to very weak impurity potential. As a result, the SO gap in graphene canbe experimentally unobservable due to a large number of adsorbed light atoms at thegraphene surface creating the impurity states in the gap.
2. Model
We use the following Hamiltonian, which describes electrons near the Dirac point K ,with the intrinsic SO interaction [11]ˆ H k = σ z ∆ vk − vk + − σ z ∆ ! , (1)where ∆ is the band splitting related to the SO interaction, v is the velocity parameter,and we denote k ± = k x ± ik y . The matrix form of (1) is due to the choice of wavefunctionbasis corresponding to different sublattices in the lattice of graphene. The basis functionsof Hamiltonian (1) are | k (1 , σ i = X s ∈ ( A,B ) e i ( k + k ) · r i ψ s ( r ) | σ i , (2) mpurity states in graphene with intrinsic spin-orbit interaction ψ s ( r ) is the tight-binding electron state at site s belonging to sublattice A or B, k is the wave vector corresponding to the chosen Dirac point, and σ = ↑ , ↓ refers to spinup and down states, respectively. The eigenvalues of Hamiltonian (1) are E k , = ± ε k ,where ε k = (∆ + v k ) / , so that the value of 2 | ∆ | is the energy gap. The Hamiltonianfor the other Dirac point K ′ differs from (1) by the opposite sign in the diagonal terms.Thus, the results for the point K ′ are can be found by using calculations with theHamiltonian (1) and reverting the sign of ∆.Since the SO interaction in Equation (1) does not mix the spins, one can considerseparately spin up and down channels. For the spin up electrons the Hamiltonian canbe presented as it gives the Hamiltonian for up-spin electronsˆ H k ↑ = τ z ∆ + v τ · k , (3)where τ i are the Pauli matrices acting in the space of sublattices A and B. For thedown-spin Hamiltonian, the sign of ∆ in Equation (3) is opposite. We consider first thespin up Hamiltonian (3).
3. Nonmagnetic impurity
Let us consider the impurity state in the case of a single impurity, described bythe perturbation localized in one of the sublattices. In the continuous model underconsideration it corresponds to the perturbation at r = 0 in sublattice Aˆ V ↑ ( r ) = V δ ( r ) 00 0 ! . (4)The matrix of perturbation (4) in the basis functions of Hamiltonian ˆ H k ↑ isˆ V kk ′ ↑ = V
00 0 ! ≡ ˆ V ↑ . (5)The effect of perturbation ˆ V kk ′↑ on the energy spectrum in all orders of magnitude canbe described using the T -matrix method [16]. In the general case, the equation for the T -matrix is ˆ T kk ′ ( ε ) = ˆ V kk ′ + X k ˆ V kk ˆ G k ( ε ) ˆ T k k ′ ( ε ) , (6)where ˆ G k ( ε ) = (cid:16) ε − ˆ H k (cid:17) − is the Green’s function of Hamiltonian ˆ H k . Using (3) wefind the Green’s function for spin up electronsˆ G k ↑ ( ε ) = ε + τ z ∆ + v τ · k ε − ε k . (7)In the following we assume that the energy parameter includes a small imaginary part, ε → ε + iδ sgn ε , which corresponds to the choice of retarded Green’s function. Using(5) and (7) we findˆ T ↑ ( ε ) = " − ˆ V ↑ X k ˆ G k ↑ ( ε ) − ˆ V ↑ ≡ h ˆ1 − ˆ V ↑ ˆ F ↑ ( ε ) i − ˆ V ↑ , (8) mpurity states in graphene with intrinsic spin-orbit interaction F ↑ ( ε ) ≡ X k ˆ G k ↑ ( ε ) = − ε + τ z ∆4 πv ln v k m + ∆ − ε ∆ − ε ≃ − ε + iδ sgn ε + τ z ∆4 πv ln v k m + ∆ − ε − i | ε | δ ∆ − ε − i | ε | δ , (9)where the upper limit (cutoff) of integration over momentum k m is introduced. Itcorresponds to the region of linearity of the spectrum near the Dirac point. Assuming v k m ≫ | ∆ − ε | we obtainˆ F ↑ ( ε ) ≃ − ε + iδ sgn ε + τ z ∆4 πv ln v k m h (∆ − ε ) + 4 ε δ i / + i ( ϕ − ϕ ) , (10)where ϕ , are the angles related to the phase of complex function ˆ F ( ε ), which can bemade analytical in the whole complex plane of ε after proper choice of cuts in this plane.We assume that te cut is made along the real axis from −| ∆ | to + | ∆ | . Then the phasescan be chosen ϕ = − tan − | ε | δv k m , (11) ϕ = π + tan − | ε | δ | ε − ∆ | , ε > ∆ , − tan − | ε | δ | ε − ∆ | , ε < ∆ . (12)The real part of Equation (9)Re ˆ F ↑ ( ε ) ≃ − ε + τ z ∆4 πv ln v k m h (∆ − ε ) + 4 ε δ i / . (13)In the limit of δ → F ↑ ( ε ) ≃ − ε + τ z ∆4 πv ln v k m | ∆ − ε | , (14)Im ˆ F ↑ ( ε ) ≃ ε + τ z ∆4 v , ε > ∆ , , ε < ∆ , . (15)The functions Re F , ↑ ( ε ) and Im F , ↑ ( ε ) are presented in figure 1 and figure 2.The matrices ˆ V ↑ and ˆ F ↑ ( ε ) are both diagonal. Therefore, ˆ T ( ε ) calculated fromEquation (8) is diagonal, tooˆ T ↑ ( ε ) = diag ( V ↑ − V ↑ Re F , ↑ ( ε ) − iV ↑ Im F , ↑ ( ε ) , ) . (16)The location of impurity level is determined by the pole of T -matrix1 − V ↑ Re F , ↑ ( ε ) = 0 . (17) mpurity states in graphene with intrinsic spin-orbit interaction Figure 1.
The dependence Re F ( ε ) for spin-up and spin down states, related to theDirac point K in the case of non-magnetic impurity. Figure 2.
The imaginary part Im F ( ε ) for spin-up and spin down states, related tothe point K in the case of non-magnetic impurity. The dependence Re F , ↑ ( ε ) presented in figure 1 can be used for the graphical solutionof Equation (17). Using (14) we find1 + V ↑ ε + ∆4 πv ln v k m | ∆ − ε | = 0 . (18)Thus, the equation for the impurity level is ε ↑ = − ∆ − πv V ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↑ (cid:12)(cid:12)(cid:12) . (19)This equation has several solutions for the same potential V ↑ .The Hamiltonian describing the spin down statesˆ H k ↓ = − τ z ∆ + v τ · k , (20) mpurity states in graphene with intrinsic spin-orbit interaction V ↓ ( r ) = ˆ V ↑ ( r ). Performing the same calculations as beforewith the substitution ∆ → − ∆ we find the equation for impurity level correspondingto the spin-down state ε ↓ = ∆ − πv V ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↓ (cid:12)(cid:12)(cid:12) . (21)The spin up and down states corresponding to solution of Equation (19) and Equation(21), respectively, are split in energy. The magnitude of splitting is 2 | ∆ | , which ofcourse is very small as was discussed in the Introduction. Nevertheless, considering theimpurity states in the model with one Dirac point we come to a weakly magnetizedstate at the nonmagnetic impurity. This nonequivalence of spin up and down states isexactly compensated by the states related to another Dirac point, K ′ , for which theHamiltonians of spin up and down states differ by the sign of ∆ from those in Equation(3) and Equation (21) [11]. Thus, the magnetization of the localized state is absent ifwe take into account both nonequivalent Dirac points. Figure 3.
The location of impurity levels as a function the impurity strengthparameter V (a) and the schematic presentation of the impurity levels (b) in thecase of non-magnetic impurity. The numerical solutions of Equation (19) and Equation (21) describing the statesrelated to the Dirac point K , as well as the corresponding solutions related to theDirac point K ′ are presented in Figure 3,a. The schematic representation of thelevels is presented in Figure 3,b. All the levels are spin degenerate due to theoverlapping solutions of equations, related to the nonequivalent Dirac points. We usedthe parameters with much larger SO gap ∆ to visualize better the character of solutions.If the state is located within the gap, the impurity level is discrete. For the level withenergy | ε ↑ | > ∆, it is a resonant state of width Im F ↑ ( ε ↑ ). It should be noted that mpurity states in graphene with intrinsic spin-orbit interaction F ↑ ( ε ) is not the density of states in graphene because Equation (9) does not includethe trace over sublattices.It should be noted that both impurity states with energies ε ↑ and ε ↓ are mostlylocalized at the site of A sublattice in accordance with the assumption that the impuritypotential (3) is located on the A-site. However, the real spread of the wavefunction canbe much larger than the distance between nearest A and B sites.
4. Magnetic impurity
In the case of magnetic impurity we choose perturbation, which is different in sign forthe spin up and down electrons, V kk ′ ↓ = − V kk ′ ↑ , and V kk ′ ↑ is described by Equation (5).The resulting impurity levels related to the Dirac points K and K ′ are to be found fromthe following equations ε K ↑ , ↓ = ∓ ∆ ∓ πv V ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↑ , ↓ (cid:12)(cid:12)(cid:12) . (22) ε K ′ ↑ , ↓ = ± ∆ ∓ πv V ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↑ , ↓ (cid:12)(cid:12)(cid:12) . (23)There is the spin splitting of the states related to the K point, which does not vanishas ∆ →
0, and the resulting magnetization of impurity states is not compensated byanother point K ′ . Corresponding numerical solutions of equations (22) and (23) arepresented in Figure 4. Figure 4.
The same dependences as in Figure 3 in the case of magnetic impurity. mpurity states in graphene with intrinsic spin-orbit interaction
5. Nonmagnetic impurity with strong SO interaction
Using the same formalism, one can consider the impurity, which locally enhances theinternal SO interaction. In this case the corresponding perturbation for K point hasthe form ˆ V ∆ = σ z ∆
00 0 ! , (24)where ∆ is the local After substitution of V → ∆ , the solution for the K point doesnot differ from the case of magnetic impurity considered above. However, for the point K ′ we have to change the sign of ∆ . As a result we obtain ε K ↑ , ↓ = ∓ ∆ ∓ πv ∆ ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↑ , ↓ (cid:12)(cid:12)(cid:12) . (25) ε K ′ ↑ , ↓ = ± ∆ ± πv ∆ ln v k m (cid:12)(cid:12)(cid:12) ∆ − ε ↑ , ↓ (cid:12)(cid:12)(cid:12) . (26)In this case the spin splitting of the states related to the point K does not vanish as∆ → K ′ .The solutions of equations (25) and (26) are presented in figure 5. Figure 5.
The same dependences as in Figure 3 in the case of spin-orbit impurity.
6. Wave function of the localized impurity state
Using the Schr¨odinger equation for the wave function, one can find the equation forimpurity state [16] ψ i ( r ) = Z d r ′ ˆ G ( r , r ′ ; ε ) ˆ V ( r ′ ) ψ i ( r ′ ) , (27) mpurity states in graphene with intrinsic spin-orbit interaction (cid:16) ε − ˆ H r (cid:17) ˆ G ( r , r ′ ; ε ) = δ ( r − r ′ ) . (28)For the impurity located at r = 0 we obtain from (27) ψ i ( r ) = ˆ G ( r , ε ) ˆ T ( ε ) ψ (0) , (29)where ψ ( r ) is the eigenfunction of unperturbed Hamiltonian (1) (cid:16) ε − ˆ H r (cid:17) ψ ( r ) = 0 . (30)The Green function for Hamiltonian (3) has been calculated before [17]ˆ G ↑ ( r , r ′ ; ε ) = − i ( ε − ∆ τ z )4 v H (1)0 | r − r ′ |√ ε − ∆ v ! +( r − r ′ ) · τ √ ε − ∆ v | r − r ′ | H (1)1 | r − r ′ |√ ε − ∆ v ! . (31)Thus, for the spin-up impurity state with energy ε ↑ | ψ i ↑ ( r ) | ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (1)0 r q ε ↑ − ∆ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (1)1 r q ε ↑ − ∆ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ e − r/R , (32)where H (1) ν ( z ) are the Hankel functions [18] and R ↑ = v/ (cid:12)(cid:12)(cid:12) ε ↑ − ∆ (cid:12)(cid:12)(cid:12) / is thecharacteristic radius of the impurity wavefunction.The function ψ ( r ) in Equation (29) can be calculated using (30). Denoting thebispinor components ψ T ( r ) = (cid:16) ϕ T ( r ) , χ T ( r ) (cid:17) we find the relation χ ( r ) = − iv ( ∂ x + i∂ y )∆ + ε ϕ ( r ) . (33)The equation for the function ϕ ↑ with the Hamiltonian (3) in polar coordinates ( r, α )reads ∂ ∂r + 1 r ∂∂r + 1 r ∂ ∂α − κ ↑ ! ϕ ↑ ( r, α ) = 0 , (34)which has the following solutions decaying at large rϕ ↑ ( r, α ) = K m ( κ ↑ r ) e imα , (35)where κ ↑ = (cid:16) ∆ − ε ↑ (cid:17) /v , K m ( z ) is the modified Bessel function and m ∈ Z . Then thecorresponding χ ↑ -component can be found from (33) and (35) χ ↑ ( r, α ) = − ivκ ↑ ∆ + ε ↑ K m +1 ( κ ↑ r ) e i ( m +1) α . (36)Thus, up to the normalization we have ψ ↑ m ( r, α ) = (∆ + ε ↑ ) K m ( κ ↑ r ) e imα − ivκ ↑ K m +1 ( κ ↑ r ) e i ( m +1) α ! (37)It should be noted that for ψ (0) in Equation (29) we have to use the cutoff at smalldistance r c ∼ k − max ≫ a /π , where a is the lattice constant and k max is the upper limitfor the Dirac model in graphene. It means that using the Dirac model we can find theimpurity wave function at distances much larger than a . On the other hand, as we seefrom (32), the radius of the impurity states R near the Dirac point is much larger thanthis limit. mpurity states in graphene with intrinsic spin-orbit interaction
7. Conclusions
We have calculated the energies and wave functions of impurity states near the Diracpoints in graphene taking into account the SO interaction. The calculations showthe SO-induced spin splitting of these states. The existence of two nonequivalentDirac points in the Brillouin zone leads to the spin degeneracy of the states withdifferent spin. It should be noted that in principle the valley degeneracy can bebroken by inhomogeneous deformations, which would result in the appearance of localmagnetization.
Acknowledgments
This work was supported by the Polish Ministry of Science and Higher Education as aresearch project in years 2007 – 2010.
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