Intervalley scattering by charged impurities in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Intervalley scattering by charged impurities in graphene
L.S. Braginsky and M.V. Entin ∗ Institute of Semiconductor Physics, Siberian Branch,Russian Academy of Sciences, Novosibirsk, 630090 RussiaNovosibirsk State University, Novosibirsk, 630090 Russia (Dated: August 10, 2018)Intervalley charged-impurity scattering processes are examined. It is found that the scatteringprobability is enhanced due to the Coulomb interaction with the impurity by the Sommerfield factor F Z ∝ ǫ √ − g − , where ǫ is the electron energy and g is the dimensionless constant of the Coulombinteraction. PACS numbers: 73.50.Bk, 73.20.Fz, 73.20.Jc, 72.25.Dc
Introduction
The presence of multiple valleys is an ordinary situ-ation in semiconductors; the intervalley scattering, e.g.in Si and Ge, has been studied since 1950th. Largedistance between the valleys makes the transitions be-tween them difficult, as compared with the intravalleyprocesses. Therefore, the valley population becomes awell-conserving quantity that determines different prop-erties of such semiconductors. By analogy with the ordi-nary spin, the valley number can be treated as a newquantum number ”pseudospin”, which determines thelong-living electron states in semiconductors. The pro-cesses caused by different population of the equivalentvalleys, in particular, surface photocurrent and polarizedphotoluminescence were studied long ago (see, e.g., [1, 2]and references therein).The processes involving a different valley populationgave birth of the promising new electronic device appli-cations called valleytronics [3–5]. The valley-polarizedcurrent can be emerged in the graphene point contactwith zigzag edges, [4], the graphene layer with brokeninversion symmetry, [6] or under illumination of the cir-cularly polarized light [7].The study of the valley dynamics attracted attention tothe intervalley relaxation that controls the valleys pop-ulation. Recent interest to graphene has been mainlyfocused on its conic electron spectrum. However, thepresence of two different valleys has been remained outof interest for a long time. Meanwhile, just low DOSnear the cone point supposedly suppresses the intervalleytransitions and makes the valley population long-living.The non-equilibrium between two graphene valleysmeans the violation of both spatial and time reversibility.As far as the spatial irreversibility determines the valleyphotocurrents [9], the time reversibility is responsible forweak localization [8], thereby the valley relaxation timeis an important electronic parameter of graphene.The valley relaxation is determined by the processes ∗ Electronic address: [email protected], [email protected] with a large momentum transfer, and, therefore, its scat-tering length is of the order of the lattice constant. Atthe same time, the Coulomb impurity determines the in-teraction on the large distances. Consequently, the prob-ability of the electron penetration to the short-scale im-purity core, where it experiences intervalley scattering, isdetermined by the large-scale wave function behavior andstrongly depends on the electron energy. In particular,the Coulomb attraction or repulsion to impurity shouldessentially affect this process.The purpose of the present paper is to study theintervalley charge-impurity scattering in the monolayergraphene. We consider the problem in the envelope-function approximation. The solution of the impurityscattering problem will be found in the Born approxima-tion. Then, the Coulomb solution will be applied to therenormalization of the Born short-range scattering result.
Problem Formulation
We use the two-atom basis of graphene | a > and | b > .The tight-binding Hamiltonian for the ideal graphene inthe momentum representation readsˆ H = (cid:18) p Ω ∗ p (cid:19) . (1)Here Ω p = t ( e ip y a + 2 e − ip y a/ cos( p x a √ / , a = 0 . t is the tunnel amplitude.The energy is counted from the permitted band center.The long-range Coulomb interaction with an impurityˆ V should be situated on the diagonal of the matrix, whilethe sort-range interaction with the impurity core givesthe off-diagonal operator ˆ U :ˆ H = ˆ H + ˆ V + ˆ U . (2)Here ˆ V and ˆ U represent the long- and short-range inter-actions with the impurity. In the case of the Coulombimpurity in the envelope-function representation V ( r ) = e /χr , where χ is a half-sum of the dielectric constantsof surrounding media, r is a 2D radius-vector in thegraphene plane. The short-range part ˆ U of the inter-action acts over the atomic distance at the impurity. Itis specific for the type of impurity.Consider now the states of free electrons. Near theconic points p = ± K , K = (2 π/a √ ,
0) the Hamilto-nian H can be transformed to Eq. (1) with Ω ( ν ) = s ( − νk x + ik y ), where s = 3 ta/ p = ± K + k , ν = ± .The corresponding wave functions near the point K can be written as (1 , − sign( ǫ ) e iϕ k ) e i ( K + k ) r / √A , where e iϕ k = ( k x + ik y ) /k , A is the patten area, the wave func-tions are normalized to the full surface. Below we assume ǫ > ± K states, the Hamiltonian H is splittedinto two independent Hamiltonians referred to the points ± K :ˆ H = − k x + ik y − k x − ik y k x + ik y k x − ik y . (3)The elements of the wave function (column of four terms)are | a, K > , | b, K > , | a, − K > , | b, − K > , respectively.In the envelope-function approximation, the coordi-nate representation of the short-range interaction poten-tial can be expressed asˆ U = ˜ U δ ( r ) , ˜ U = ( U + S σ ) . (4)In the tight-binding model, the components U = S ( ǫ A + ǫ B ) / S z = S ( ǫ A − ǫ B ) /
2, were S = a / A ( ǫ A ) and B ( ǫ B ) atoms of the cell in the origin,while the components S x and S y are determined by theperturbation of the amplitude of transition between theseatoms. The amplitude of transition between these atomsis mapped onto the vector S as ( S x + iS y ) = tS . In prin-ciple, the Hamiltonian ˆ U describes both the monomerand dimer impurities. Below we deal with the case ofa single impurity at the A site with the perturbation ofenergy level δǫ A = 2 U , S z = U without transition am-plitude ( S x = S y = 0) perturbation.In the envelope-function approximation, the long-range Coulomb interaction mixes the states within onecone. It is located on the diagonal in the 4 × a .The short-range Hamiltonian of interaction containsthe matrix elements between the states |± K , a i and |± K , b i . The blocks in the left-up and right-down fromthe diagonal yield the intravalley mixing, while the blocksin the right-up and left-down from the diagonal relate tothe intervalley matrix elements of the impurity potential.Although, these blocks are identical in the model Eq.(4), K → − K blocks have, generally speaking, a lower orderof magnitude. Roughly, these elements are the Fourierharmonics of Coulomb potential at the momentum K . Short-range potential a xy FIG. 1: Graphene with substitution impurity (large circle),which energy differs from that of the host atoms.
In this case ˆ V = 0. The scattering amplitude is de-scribed by the t-matrix satisfying the equation t = U + U Gt (5)with the formal solution t = T δ ( r ) , T = (1 − ˜ U R ) − ˜ U , (6)where R ν,µ is the projection of the Green function ontothe origin lattice cell (e.g., (00)) populated by the impu-rity atom.The scattering probability is 2 π |h ν, k | t | ν ′ , k ′ i| δ ( ǫ ν, k − ǫ ν ′ , k ′ ).Eq. (6) gives the symbolic solution of the short-gangescattering problem. Let us apply it to the Hamiltonian(4) with use of RR = Z d p π ǫ − | Ω p | (cid:18) ǫ Ω p Ω ∗ p ǫ (cid:19) , (7)where the integration runs over the Brillouin zone. Thisintegration gives a finite result even if ǫ → R = Z d p π (cid:18) / Ω ∗ p / Ω p (cid:19) + O ( ǫ log( ǫa/s )) . (8)The amplitudes of the intra- and inter-valley transi-tions in the Born approximation ( ˆ U →
0) are A K , k ; K , k ′ = (cid:16) U + S z − ( S x + iS y ) e iφ k +( U − S z ) e i ( φ k − φ k ′ ) − ( S x − iS y ) e − iφ k ′ (cid:17) / A A K , k ; − K , k ′ = (cid:16) U + S z − ( S x + iS y ) e iφ k +( S z − U ) e i ( φ k + φ k ′ ) + ( S x − iS y ) e iφ k ′ (cid:17) / A (9)It should be emphasized that the transition probabilityhas the essential angular dependence on the angles φ k and φ k ′ . Besides, this dependence concerns not onlythe relative angle φ k ′ − φ k , but also the absolute an-gles. This dependence originates from the degeneracyof the states near the cone points and possible asymme-try of the defect. Note that such a dependence is ab-sent for the δ -potential in the envelope-function approx-imation. In the specific case of the monomer impurity, A K , k ; − K , k ′ = A K , k ; K , k ′ = U / A .If ˜ U → ∞ , T → /R and ceases to depend on ˜ U . Electron states in the Coulomb potential
The long-range Coulomb scattering does not changethe valley. To find the transition amplitude, we shoulduse the intervalley block of Hamiltonian ˆ U . The Coulombinteraction in the ”final” state corrects the amplitude.The Coulomb corrections to the wave function are formedat the distances much exceed the lattice constant. In thatcase ˆ U should be multiplied by the limit of the Coulombwave function at a low distance from the impurity. Thislimit is determined by the zero-momentum projectioncomponent of the wave function. The Coulomb wavefunction should be matched with the free solution of theequation without any potential.The equation with long-range Coulomb potential fortwo-component envelope wave function ( φ, χ ) in the polarcoordinates ( r, ϕ ) reads ǫ − g/r e − iϕ ( i∂ r + r ∂ ϕ ) e iϕ ( i∂ r − r ∂ ϕ ) ǫ − g/r ! φχ ! = 0 . We search for the solution of this equation with the sub-stitution φχ ! = r / e i ( M − / ϕ φ M ie i ( M +1 / ϕ χ M ! . Here M = m + 1 /
2, and m is an integer. Then( rφ M ) ′ + ( ǫr − g ) χ M − M φ M = 0 , ( rχ M ) ′ − ( ǫr − g ) φ M + M χ M = 0 (10)The wave function diverges at small distances and hasa divergent phase (”falling down the center”). The inte-gral of the electron density converges, while the potentialand the kinetic energies diverge. This divergence is con-nected with falling down the center. In fact, the conicapproximation fails at small distances from the center.The problem can be resolved by introduction of a short-range cutoff.Eq.(10) corresponds to the Eq.(35.5) with solutionEq.(36.11) from [10] at m = 0. Using these equations,we have the finite at r = 0 solutionΞ m = φ M χ M o = (2 εr ) γ − × ImRe n e i ( εr + ξ ) F ( γ − ig, γ + 1 , − iεr ) o . (11)Here γ = p M − g , and the real value ξ satisfies theequation e − iξ = ( γ − ig ) /M . The asymptotics of (11) areΞ m = ImRe ( Γ(2 γ + 1) i − γ + ig √ γ + ige irǫ (2 rǫ ) − ig Γ( ig + γ + 1) ) , at ǫr → ∞ (12)Ξ m = (2 εr ) γ − (cid:18) sin ξ cos ξ (cid:19) , at ǫr → k in absence of the Coulomb potential can be expandedinto the radial the waves asΨ k = 1 √ k X m i m e im ( ϕ − ϕ k ) J m ( kr ) ie iϕ J m +1 ( kr ) ! (13)Here ϕ k and ϕ are the polar angles of k and r , corre-spondingly. The solution (12) is normalized to the uniteflux in the plane wave. At r → ∞ Ψ k = 1 √ πkr X m i m e im ( ϕ − ϕ k ) cos (cid:0) πm − kr + π (cid:1) − ie iϕ sin (cid:0) πm − kr + π (cid:1) ! (14)To find the cross-section of the process, one should re-late the solutions (12) and (14) at the infinity so thatthe coefficients at the divergent (or convergent) wavesin each solutions coincide for the incoming (or outgoing)solutions. The presence of the Coulomb potential at thelarge distance leads to a logarithmic phase change; thischange should be neglected, because it has no effect onthe flux.The angular-momentum expansion of the trueCoulomb wave function with a given momentum k readsΨ d,c k = X m c d,cm e imϕ (cid:18) e iϕ (cid:19) Ξ m , (15)where superscripts d and c refer to the wave functionsobtained by equating the coefficients at the divergent ( ∝ e ikr ) or convergent ( ∝ e − ikr ) parts of the standing radialwaves in Ψ k in Eq.(14). Equating the asymptotics X m c d,cm e imϕ (cid:18) e iϕ (cid:19) Ξ d,cm = Ψ k at r ∼ r , where r is some distance from the center largerthan 1 /ǫ (latter on r will disappear from the result), weobtain c cm = (1 − i )( − m (2 r ǫ ) ig e − imϕ k + iα m √ πR m ,c dm = − (1 + i )( − m (2 r ǫ ) − ig e − i ( α m − mϕ k ) √ πR m , (16) R m e iα m = Γ(2 γ + 1) i − γ + ig √ γ + ig Γ( ig + γ + 1) . We utilized the facts that the divergent wave in our casebelongs to the − K point and according to symmetry re-lations, the corresponding wave function can be found bytransformation ϕ k → − ϕ k . Intervalley scattering
Consider now the probability of the intervalley tran-sitions caused by ˆ U taking into account the finite-stateCoulomb interaction. The amplitude of the intervalleytransitions is h Ψ d k ′ | ˆ U | Ψ c k i , where Ψ d,c k are determined byEq. (15). Note that only the terms with m = m ′ and m = m ′ ± d,c k have not been vanished after theangular integration. Subsequent radial integration leavesthe only divergent term m = m ′ = 0. As a result, | A K , k ; − K , k ′ | = 1 A π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( γ + 1 + ig )Γ(2 γ + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 aǫ ) √ − g − × U − sin (2 ξ ) cos (cid:18) ϕ k + ϕ k ′ (cid:19) ! (17)The intervalley relaxation time is found by summationof all transitions in a box with n i A impurities in it.The averaging over ϕ k ′ gives the intervalley relaxationtime 1 τ v = 16 n i ǫπ ~ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( γ + 1 + ig )Γ(2 γ + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Z U − g ! ,F Z = (2 aǫ/s ~ ) √ − g − . (18)We revived ~ in this final formula. Eq. (18) is notexact. Its accuracy is logarithmic: log h Ψ d k ′ | ˆ U | Ψ c k i| =2 p − g − / aǫ ) + o (log(1 / aǫ )). At 2 aǫ ) ∼ g ≪ g < / g ≪ /
4, the expansion yields F Z =(2 aǫ ) − g . One may express the intervalley scatteringrate in the Coulomb field 1 /τ v via the perturbative one1 /τ v, as 1 /τ v = F Z /τ v, .While the Born intervalley relaxation rate drops whenthe energy goes down, the factor F Z = (2 aǫ ) √ − g − plays the role of the enhancement factor due to theCoulomb interaction. At a small g this enhancement isweak, but essential.It should be emphasized that the enhancement is in-dependent of the sign of g , e.g. in the case of a repulsivepotential the enhancement also takes place! The source of this phenomenon lies in the conversion of the electronsto hole states under the barrier surrounding the repulsiveimpurity with subsequent attraction to the impurity core.At | g | > √ / g = 2 e / ~ s ( κ + 1)for graphene on the substrate with the dielectric con-stant κ = 14 has the value 0 .
3. In this case, thepower (2 p − g −
2) = − . ǫ = 1 mV, F Z = 20 .
4. For U = 1 eV S and impurity concentration n = 10 cm − , we have τ v = 1 . × − s. Discussion and conclusions
We have found the intervalley scattering rate withand without Coulomb interaction. While the short-range interaction gives rise to the intervalley amplitudethat is independent of the electron energy, the long-range Coulomb interaction also contributes to the pro-cess via the Sommefield prefactor power-like dependingon the electron energy. Our consideration is limitedby the weak enough electrostatic interaction constant | g | < /
2. Strictly speaking, this is not the case ofthe free-suspended graphene, but in most cases of thegraphene on the semiconductor substrate this conditionis fulfilled. The case of | g | > / Acknowledgements
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