Pis'ma v ZhETF, vol. 101, iss. 5, pp. 355-359
© 2015 March 10
Intervalley scattering by charged impurities in graphene
L. S. Braginsky, M. V. Entin^ Institute of Semiconductor Physics SB of the RAS, 630090 Novosibirsk, Russia
Novosibirsk State University, 630090 Novosibirsk, Russia
Submitted 29 December 2014
Intervalley charged-impurity scattering processes are examined. It is found that the scattering probability is enhanced due to the Coulomb interaction with the impurity by the Sommerfield factor Fz oc e2"^1-4®2-2, where e is the electron energy and g is the dimensionless constant of the Coulomb interaction.
DOI: 10.7868/S0370274X15050082
Introduction. The presence of multiple valleys is an ordinary situation in semiconductors; the intervalley scattering, e.g. in Si and Ge, has been studied since 1950th. Large distance between the valleys makes the transitions between them difficult, as compared with the intravalley processes. Therefore, the valley population becomes a well-conserving quantity that determines different properties of such semiconductors. By analogy with the ordinary spin, the valley number can be treated as a new quantum number "pseudospin", which determines the long-living electron states in semiconductors. The processes caused by different population of the equivalent valleys, in particular, surface photocur-rent and polarized photoluminescence were studied long ago (see, e.g., [1, 2] and references therein).
The processes involving a different valley population gave birth of the promising new electronic device applications called valleytronics [3-5]. The valley-polarized current can be emerged in the graphene point contact with zigzag edges [4], the graphene layer with broken inversion symmetry [6], or under illumination of the circularly polarized light [7].
The study of the valley dynamics attracted attention to the intervalley relaxation that controls the valleys population. Recent interest to graphene has been mainly focused on its conic electron spectrum. However, the presence of two different valleys has been remained out of interest for a long time. Meanwhile, just low DOS near the cone point supposedly suppresses the intervalley transitions and makes the valley population long-living.
The non-equilibrium between two graphene valleys means the violation of both spatial and time reversibility. As far as the spatial irreversibility determines the valley photocurrents [8], the time reversibility is response-mail: entin@isp.nsc.ru
sible for weak localization [9], thereby the valley relaxation time is an important electronic parameter of graphene.
The valley relaxation is determined by the processes with a large momentum transfer, and, therefore, its scattering length is of the order of the lattice constant. At the same time, the Coulomb impurity determines the interaction on the large distances. Consequently, the probability of the electron penetration to the short-scale impurity core, where it experiences intervalley scattering, is determined by the large-scale wave function behavior and strongly depends on the electron energy. In particular, the Coulomb attraction or repulsion to impurity should essentially affect this process.
The purpose of the present paper is to study the intervalley charge-impurity scattering in the monolayer graphene. We consider the problem in the envelope-function approximation. The solution of the impurity scattering problem will be found in the Born approximation. Then the Coulomb solution will be applied to the renormalization of the Born short-range scattering result.
Problem formulation. We use the two-atom basis of graphene |a) and |6). The tight-binding Hamiltonian for the ideal graphene in the momentum representation reads
Here = tl^P«"1 + 2e~iPya/2 cos(pxa^/2)], a = 0.246 nm is the lattice constant, and t is the tunnel amplitude. The energy is counted from the permitted band center.
The long-range Coulomb interaction with an impurity V should be situated on the diagonal of the matrix, while the sort-range interaction with the impurity core gives the off-diagonal operator U:
H = H00+V+U- (2)
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L. S. Braginsky, M. V. Eutin
Here V and U represent the long- and short-range interactions with the impurity. In the case of the Coulomb impurity in the envelope-function representation V(r) = = e11Xr, where x is a half-sum of the dielectric constants of surrounding media, r is a 2D radius-vector in the graphene plane. The short-range part U of the interaction acts over the atomic distance at the impurity. It is specific for the type of impurity.
Consider now the states of free electrons. Near the conic points p = ±K, K = (2n/ay/3,0) the Hamilto-nian Hqq can be transformed to Eq. (1) with f= = s(—vkx + iky), where s = 3io/2, p = ±K + k, v = ±. The corresponding wave functions near the point K can be written as (1, —sign(e)ei^k)ei(K+k^r/v/^4, where eifk = _)_ iky)/k, A is the patten area, the wave functions are normalized to the full surface. Below we assume e > 0 corresponding to the case of electrons.
In view of ±K states, the Hamiltonian Hqo is split-ted into two independent Hamiltonians referred to the points ±I\:
H{
o—
/ 0
iky
0
\ 0
k>X iky 0 0 0
0 0 0
- %}V<ji
\
ik„
/
(3)
The elements of the wave function (column of four terms) are |o, K), \b, K), |o, — K), \b, — K), respectively.
In the envelope-function approximation, the coordinate representation of the short-range interaction potential can be expressed as
U = Ü6(r), Ü=(U0 + S<r).
(4)
In the tight-binding model, the components Uq = = S0(eA + eB)/2, Sz = S0(eA - eB)/2, were S0 = a2/2 is the graphene unite cell area, are determined by the levels of A(ca) and B(cb) atoms of the cell in the origin, while the components Sx and Sy are determined by the perturbation of the amplitude of transition between these atoms. This amplitude is mapped onto the vector S as (Sx + iSv) = tSo■ In principle, the Hamiltonian U describes both the monomer and dimer impurities. Below we deal with the case of a single impurity at the A site with the perturbation of energy level SeA = 2Uq, Sz = Uq without transition amplitude (Sx = Sy = 0) perturbation.
In the envelope-function approximation, the longrange Coulomb interaction mixes the states within one cone. It is located on the diagonal in the 4x4 form of the Hamiltonian. Keeping in mind the divergency of the final result at the small distances; later on, we should cut off this divergency at the lattice constant a.
The short-range Hamiltonian of interaction contains the matrix elements between the states |±K,a) and |±K, b). The blocks in the left-up and right-down from the diagonal yield the intravalley mixing, while the blocks in the right-up and left-down from the diagonal relate to the 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 order of magnitude. Roughly, these elements are the Fourier harmonics of Coulomb potential at the momentum K.
Short-range potential. In this case V = 0. The scattering amplitude is described by the i-matrix satisfying the equation
t = U + UGt, (5)
where G is the Green function, with the formal solution
t = T6(r), T = (1 - UR)-lU, (6)
where is the projection of the Green func-
tion onto the origin lattice cell (e.g., (00)) populated by the impurity atom. The scattering probability is
Graphene with substitution impurity (large circle), which energy differs from that of the host atoms
Eq. (6) gives the symbolic solution of the short-gange scattering problem. Let us apply it to the Hamiltonian (4) with use of R
R
d2p
I
4tt2 e2 - |Qp|2
Qt
(7)
where the integration runs over the Brillouin zone. This integration gives a finite result even if e —> 0:
Ri
o —
fp ( o I/ÎÎ;
4tt2 \ i/Qp 0
0[e log(ea/s)]. (8)
The amplitudes of the intra- and inter-valley transitions in the Born approximation (U —> 0) are
A
K,k;K,k'
U0 + Sz- (Sx + iSy)é+
+ (U0 - - (Sx - iSy)e-^' /A, (9)
A
K,k;-K,k'
U0 + Sz- (Sx + iSyy+
+ (Sz - U0)^+^') + (Sx - iSy)ei^'\/A
It should be emphasized that the transition probability has the essential angular dependence on the angles 4>k and fa • Besides, this dependence concerns not only the relative angle fa — fa, but also the absolute angles. This dependence originates from the degeneracy of the states near the cone points and possible asymmetry of the defect. Note that such a dependence is absent for the ¿-potential in the envelope-function approximation. In the specific case of the monomer impurity, ^K,k;-K,k' = ^K,k;K,k' = Uq/A.
If ¿7 —> 00, T —> 1/R and ceases to depend on U.
Electron states in the Coulomb potential. The long-range Coulomb scattering does not change the valley. To find the transition amplitude, we should use the intervalley block of Hamiltonian U. The Coulomb interaction in the "final" state corrects the amplitude. The Coulomb corrections to the wave function are formed at the distances much exceed the lattice constant. In that case U should be multiplied by the limit of the Coulomb wave function at a low distance from the impurity. This limit is determined by the zero-momentum projection component of the wave function. The Coulomb wave function should be matched with the free solution of the equation without any potential.
The equation with long-range Coulomb potential for two-component envelope wave function (</>, x) in the polar coordinates (r, y>) reads
e-g/r + ia,
- ah
0.
We search for the solution of this equation with the substitution
<t>
x
„1/21
ei(M-l/2)v(f>
'M
■i(M+l/2)v
Here M = m, + 1/2, and m is an integer. Then
(r<t>M)' + (er - q)xm ~ M(j>M = 0, (í"Xm)' - (er - g)4>M + M\m = 0.
(10)
The wave function diverges at small distances and has a divergent phase ("falling down the center"). The integral of the electron density converges, while the potential and the kinetic energies diverge. This divergence is connec
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