RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
RKKY interaction in graphene
Eugene Kogan
Bar-Ilan University
December 15, 2011
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene 2004
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene 2010
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene sublattices and Brillouin zone
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Bonding and anti-bonding
In isolated form, carbon has six electrons in the orbital
configuration 1s 2 2s 2 2p 2 . When arranged in the honeycomb
crystal, two electrons remain in the core 1s orbital, while the other
orbitals hybridize, forming three sp 2 bonds and one p z orbital. The
sp 2 orbitals form the σ band, which contains three localized
electrons. The bonding configuration among the p z orbitals of
different lattice sites generates a valence band, or π-band,
containing one electron, whereas the antibonding configuration
generates the conduction band (π ∗ ), which is empty.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Dirac points and dispersion law
Figure: Brillouin zone centered around the Γ point and Dirac cone
resulting from the linearization of tight-binding spectrum around the K
points (blue circles).
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Nearest and next–nearest neighbor hopping model
H =
∣
−t ′ ∑ e ik·a −t ∑ δ i
e ik·δ ∣
i ∣∣∣
−t ∑ δ i
e −ik·δ i
−t ′ ∑ e ik·a ,
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Dispersion law
E ± (k) = ±t √ 3 + f (k) − t ′ f (k),
f (k) = ∑ (√ )
(√ ) ( )
e ik·δ i 3 3
= 2 cos 3ky a + 4 cos
2 k y a cos
2 k y a .
δ i
At the vertices of the Brillouin zone
( ) ( 2π
K =
3a , 2π
2π
3 √ , K ′ =
3a
3a , − 2π )
3 √ 3a
f (K) = f (K ′ ) = −3,
hence the Brillouin zones merge there.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
The general tight–binding Hamiltonian
H =
∣
− ∑ a t′ (a)e ik·a − ∑ ∣
a
t(a + δ)eik·(a+δ) ∣∣∣
− ∑ a t∗ (a + δ)e −ik·(a+δ) − ∑ a t′ (a)e ik·a ,
where a is an arbitrary lattice vector, and δ is any (fixed) vector
connecting two sites of different sub-lattices. The dispersion law
thus is given by equation
where
F (E, k) =
[
E + ∑ a
F (E, k) = 0,
] 2 ∣ t ′ (a)e ik·a ∑
∣∣∣∣
2
−
t(a + δ)e ik·(a+δ)
∣
a
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Merging of the zones
∑
t(a + δ)e iK·(a+δ) = 0
a
t(a + δ) realizes the A 1 (unit) representation of the point
symmetry group C 3v , and e iK·(a+δ) , e iK′·(a+δ) realize E
representation of the group.
On the other hand, generally
∑
t ′ (a)e iK·a ≠ 0. (1)
a
This is because t ′ (a) realizes the A 1 representation of the point
symmetry group C 6v , and e iK·a , e iK′·a realize representation
A 1 + B 1 .
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Linear spectrum
In mathematics the Dirac points are called conical points of the
surface; if the surface is given by
F (E, k) = 0,
the conditions for the conical points are
∂F
∂E = 0,
∂F
∂k = 0.
F (E, k) =
[
E + ∑ a
] 2 ∣ t ′ (a)e ik·a ∑
∣∣∣∣
2
−
t(a + δ)e ik·(a+δ)
∣
a
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
History of RKKY
RKKY stands for Ruderman-Kittel-Kasuya-Yosida and refers to a
coupling mechanism of nuclear magnetic moments or magnetic
ions in a normal metal by means of an exchange interaction with
the conduction electrons. The RKKY interaction was originally
proposed as a means of explaining unusually broad nuclear spin
resonance lines that had been observed in natural metallic silver.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Physical system
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
RKKY interaction in metals
Total Hamiltonian
H T = H − JS i·s i − JS j·s j ,
where H is the Hamiltonian of free electrons, S i is the spins of the
impurity and s i is the spin of itinerant electrons at site i.
H RKKY =
J 2
µ 0 (gµ B ) 2 2k F R ij cos(2k F R ij ) − sin(2k F R ij )
(2k F R ij ) 4 S i·S j
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Adatom
Figure: Graphene honeycomb latice with an adatom sitting on top of
carbon atom.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
The correction to the thermodynamic potential
H T = H + H int
{
}
∆Ω = −T ln ⟨S⟩ ≡ −T ln tr S · e −H/T /Z
where the S–matrix is
S = exp
{
−
∫ 1/T
0
H int (τ)dτ
}
H int (τ) = e τ(H−µ ˆN) H int e −τ(H−µ ˆN)
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Second order term
H int = −JS i·s i − JS j·s j ,
s i = 1 2 c† iα σ αβc iβ ,
the second order term of the expansion is
∆Ω = J2 T
4
∫ 1/T ∫ 1/T
0
0
∑
S i·σ αβ S j·σ γδ
αβγδ
dτ 1 dτ 2
⟨
T τ
{c † iα (τ 1)c iβ (τ 1 )c † jγ (τ 2)c jδ (τ 2 )
}⟩
.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Wick theorem
For non-interacting electrons
{
}⟩
⟨T τ c † iα (τ 1)c iβ (τ 1 )c † jγ (τ 2)c jδ (τ 2 )
{
}⟩ {
}⟩
= −
⟨T τ c iβ (τ 1 )c † jγ (τ 2)
⟨T τ c jδ (τ 2 )c † iα (τ 1)
{
}⟩
G βγ (i, j, τ 1 − τ 2 ) = −
⟨T τ c iβ (τ 1 )c † jγ (τ 2)
⟨
}⟩
= −δ βγ T τ
{c i (τ 1 )c † j (τ 2)
is the Matsubara Green’s function.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Spin indices summation
∑
αβγδ
S i·σ αβ S j·σ γδ δ βγ δ δα = ∑ αβ
S i·σ αβ S j·σ βα = S i·S j ,
∆Ω = −J 2 χ ij S i·S j ,
where
χ ij = − 1 4
∫ 1/T
0
G(i, j; τ)G(j, i; −τ)dτ
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
RKKY interaction
Thus we obtain
H RKKY = −J 2 χ ij S i·S j ,
where χ ij is the free electrons static real space spin susceptibility.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Matsubara Green’s function
The Green’s function can be easily written down using
representation of eigenvectors and eigenvalues of the operator H
It is
(H − E n ) u n = 0.
G(i, j; τ) = ∑ n
u ∗ n(i)u n (j)e −ξ nτ
{ − (1 − nF (ξ n )) , τ > 0
n F (ξ n ), τ < 0 ,
where ξ n = E n − µ, and n F (ξ) = ( e βξ + 1 ) −1 is the Fermi
distribution function.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Hamiltonian of the bipartite lattice
Bipartite lattice we’ll understand in the sense, that all the sites can
be divided in two sublattices, and there is only inter–sublattice
hopping (no intra–sublattice hopping). Thus the Hamiltonian H in
matrix representation is
H =
( 0 T
T † 0
)
,
where T is some matrix N × M.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene lattice
Graphene lattice consists of two sublattices
H = ∑
t ij c † i c j.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Mathematic preliminaries
Secular equation
˜H =
( 0 B
C 0
∣ −EI
C
)
.
B
−EI ∣ = 0.
Determinant of the block matrix
∣ A B
C D ∣ = |A| ∣ ∣D − CA −1 B ∣ .
∣
∣E 2 I M×M − CB ∣ ∣ = 0.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Symmetry of the spectrum and the wave functions
The spectrum is symmetric, that is non-zero eigenvalues of the
matrix H are present in pares (E, −E).
Eigenfunctions equation
( −En I T
T † −E n I
Hence
u n (i) = ±u n (i),
)
u n = 0,
where u n is the eigenfunction corresponding to E n and u¯n is the
eigenfunction corresponding to −E n , and in the r.h.s. there is + if
the site i belongs to one sublattice, and there is - if the site
belongs to the opposite sublattice.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Symmetry of the Green’s function
G(i, j; τ) = ∑ n
{
un(i)u ∗ n (j)e −ξnτ − (1 − nF (ξ
×
n )) , τ > 0
n F (ξ n ), τ < 0 ,
u n (i) = ±u n (i),
For µ = 0
n F (ξ m ) = 1 − n F (ξ m ).
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Symmetry of different terms in the Green’s function
G(i, j; τ) = ∑ n
{
un(i)u ∗ n (j)e −ξnτ − (1 − nF (ξ
×
n )) , τ > 0
n F (ξ n ), τ < 0 ,
The terms with non-zero energy are pairwise antisymmetric (with
respect to simultaneous transformation τ → −τ, i ⇄ j and
complex conjugation) for the sites i and j belonging to the same
sublattice, and pairwise symmetric for the sites i and j belonging
to opposite sublattices. The term (terms) with E = 0 is
antisymmetric with respect to the above mentioned
transformation, no matter which sublattices the sites belong to.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Symmetry of the Green’s function
For the sites i and j belonging to the same sublattice
G(j, i; −τ) = −G ∗ (i, j; τ).
For the sites i and j belonging to different sublattices
G(j, i; −τ) = G ∗ (i, j; τ),
provided there are no zero energy states, or we can neglect there
contribution to the Green’s function.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Symmetry of the RKKY interaction
G(j, i; −τ) = ∓G ∗ (i, j; τ).
χ ij = − 1 4
∫ 1/T
0
G(i, j; τ)G(j, i; −τ)dτ
The RKKY interaction on the bipartite lattice at half filling is
ferromagnetic between impurities on the same sublattice and is
antiferromagnetic between impurities on opposite sublattices
(under the above stated restriction).
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene wave functions
The wave function for the momentum around Dirac points
ψ ν,K (k) = 1 √
2
( e
−iθ k /2
νe iθ k/2
)
, (2)
ψ ν,K ′(k) = 1 √
2
( e
iθ k /2
νe −iθ k/2
)
, (3)
where α = ±1 corresponds to electron and hole band respectively.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene Green’s functions
G(i, j; τ) = ∑ n
u ∗ n(i)u n (j)e −ξ nτ
{ − (1 − nF (ξ n )) , τ > 0
n F (ξ n ), τ < 0 ,
the ∑ n
turns into
a2
∫
(2π) d 2 p
2
u n (i) = e ip·R i
ψ p ,
where ψ p is the appropriate component of spinor electron
wave-function (depending upon which sublattice the magnetic
adatom belongs to) in momentum representation.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Graphene Green’s functions
For i and j belonging to the same sublattice
G AA (i, j; τ > 0) = − 1 2
[e iK·R ij
+ e iK′·R ij
]
a 2 ∫
(2π) 2
d 2 ke ik·R ij −E +(k)τ ,
and for i and j belonging to different sublattices
G AB (i, j; τ > 0) = 1 a 2 ∫
2 (2π) 2 d 2 ke −E +(k)τ
]
×
[e i(K+k)·R ij −iθ k
+ e i(K′ +k)·R ij +iθ k
.
For τ < 0 we should change the sign of the Green’s functions and
substitute E − for E + .
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Angle integration
For isotropic dispersion law E(k) = E(k)
∫
∫ ∞
d 2 ke ik·R ij −E(k)τ = dkkJ 0 (kR)e −E(k)τ
0
∫
d 2 ke ik·R ij ±iθ k −E(k)τ =
∫ ∞
0
dkkJ 1 (kR)e −E(k)τ
(J 0 and J 1 are the Bessel function of zero and first order
respectively.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Linear dispersion E ± (k) = ±v F k
Using mathematical identity
(√ ) ν
∫ ∞
x n−1 e −px J ν (cx)dx = (−1) n−1 −ν ∂n−1 p 2 + c 2 − p
c
∂p n−1 √ ,
p 2 + c 2
0
after simple calculus we obtain
χ AA (R ij ) =
a 4 [
1 + cos((K − K ′
256v F R 3 )·R ji ) ]
χ AB (R ij ) = −
3a4 [
1 + cos((K − K ′
256v F R 3 )·R ji − 2θ R ) ] ,
where θ R is the angle between the vectors K − K ′ and R ij ).
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Bilayer graphene: Bernal (Ã − B) stacking.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Bilayer graphene: Bernal (Ã − B) stacking.
a) Top view of a graphene bilayer; white and black circles: top
layer carbon atoms; gray and red: bottom layer.
b) four-band spectrum of the bilayer, ±E γ (p), with γ = ± near the
corner of the Brillouin zone.
c) Brillouin zone with high symmetry points.
d) Illustration of the four band spectrum around the K point.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Low–energy modes
The low–energy modes are localized on A and ˜B sites, we consider
RKKY interaction of the magnetic adatoms siting on top of carbon
atom in A and/or ˜B sites.
and wave functions
E ± (k) = ± k2
2m
ψ ν,K (k) =
ψ ν,K ′(k) =
1 √
2
( e
−iθ k
νe iθ k
1 √
2
( e
iθ k
νe −iθ k
)
)
.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
RKKY interaction in bilayer graphene
∫ ∞
0
J 0 (x) exp(−px 2 )xdx = 1 (
2p exp − 1 )
.
4p
After simple calculus we obtain
χ AA (R ji ) =
ma2
16π 2 R 2 [
1 + cos((K − K ′ )·R ji ) ] .
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Frequency representation of Matsubara Green’s function
χ ij = T 4
∑
un(i)u ∗ m (i)u n (j)um(j) ∑ ∗
n,m
ω
1 1
iω − ξ n iω − ξ m
where ω = πT (2l + 1) (l is an integer) is Matsubara frequency.
χ ij = 1 4
∑
n,m
u ∗ n(i)u m (i)u n (j)u ∗ m(j) n F (ξ m ) − n F (ξ n )
ξ n − ξ m
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
The way the analytic calculations shouldn’t be done
χ(q) = ∑ νν ′ ,p
M ν,ν ′ ,p,q
n F [E ν ′(p + q)] − n F [E ν (p)]
,
E ν (p) − E ν ′(p + q)
where E ν (p) is the graphene spectrum, and ν = ±1 labels the
conduction and valence band.
χ (R ij ) =
∫
a2
(2π) 2
d 2 qχ (q) e iq·R ij
.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
The way the analytic calculations can be done
χ AA a 4 [
(R ij ) = 1 + cos((K − K ′
4π 2 v F R 3 )·R ij ) ]
∫ ∞
0
dxxJ 0 (x)
∫ ∞
0
dx ′ x ′ J 0 (x ′ 1
)
x + x ′
χ AB (R ij ) = −
a4 [
1 + cos((K − K ′
4π 2 v F R 3 )·R ij ) ]
∫ ∞
0
dxxJ 1 (x)
∫ ∞
0
dx ′ x ′ J 1 (x ′ 1
)
x + x ′ .
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Convergence of the integrals
∫ ∞
0
x ν
x + z J ν(cx)dx =
πzν
2 cos νπ [H −ν(cz) − Y −ν (cz)],
H ν (z) is the Struve function and Y ν (z) is the Neumann function
∫
π ∞
[
dxx 2 J 0 (x) Y 0 (x) − H 0 (x) + 2 ]
2 0
πx
∫
π ∞
dxx 2 J 1 (x)[Y −1 (x) − H −1 (x)].
2
0
The divergence of the integrals is guaranteed by the asymptotics
1
( x
) ν−1 (
H ν (x) − Y ν (x) → √ ( )
πΓ ν +
1 + O (x/2)
ν−3 ) .
2
2
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
Adatom positions
For the case of substitutional impurities
χ S AS A
(R ij ) = X S AS A
v F R 7 [
1 + cos((K − K ′ )·R ij ) ]
χ S AS B
(R ij ) = − X S AS B
v F R 7 [
1 − cos((K − K ′ )·R ij − 6θ R ) ] .
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
From Anderson to Kondo model
Figure: Intersection of the Dirac cone spectrum, E(k) = ±v|k|, with the
localized level E f = ε 0 : (b) ε 0 > 0, (c) ε 0 < 0.
H hyb =
∑ (
)
vi λ c † iα f λα + h.c. ,
λ,α,i∈P
H pd = −
∑
λ,α,β,i,j∈P
Jv λ
i vj
λ
∗
S · σαβ c † iα c jβ.
Eugene Kogan
RKKY interaction in graphene

Introduction
RKKY interaction in general
Symmetry of the RKKY interaction on the bipartite lattice
Analytic calculation of the RKKY interaction in graphene
Discussion
Conclusions
The RKKY interaction on the bipartite lattice at half filling is
ferromagnetic between impurities on the same sublattice and is
antiferromagnetic between impurities on opposite sublattices. The
latter is true, provided there are no zero energy states..
H RKKY = −J 2 χ ij S i·S j ,
χ AA (R ij ) =
a 4 [
1 + cos((K − K ′
256v F R 3 )·R ji ) ]
χ AB (R ij ) = −
3a4 [
1 + cos((K − K ′
256v F R 3 )·R ji − 2θ R ) ] .
Eugene Kogan
RKKY interaction in graphene