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Particle physics Condensed-matter physics
K. Shizuya
"Graphene"
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"Graphene" : monolayer of C atoms"relativistic" condensed-matter system
Electrons and holes ~ massless Dirac fermions
half-integer QHE
fermion # fractionalization , chiral anomalyzero-energy Landau levels
Exotic phemonena
How Dirac fermions appear in a cond-mat. system●
●
Test ●
Novoselov et al, Nature ('05)
Zhang, et al, Nature ('05)
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Honeycomb LatticeThe Bravais lattice is triangular andthe unit cell contains two sitesBasis vectors area1 = (
!3
2 , 12)a
a2 = ( 0 , 1 )a
The sublattices are connected byb1 = ( 1
2!
3, 1
2 )a
b2 = ( 12!
3,!1
2)a
b3 = (! 1!3, 0 )a
In the tight-binding approximation where only nearest neighbor interactions are retained,the Hamiltonian is
H = !!
A,i
"U †(A) V (A + bi) + V †(A) U(A + bi)
#
+"!
A
"U †(A) U(A)! V †(A + b1) V (A + b1)
#
where A = n1a1 + n2a2.
U : annihilation operator for electrons localized on site A; V for electrons on site B.! : hopping A" B": di!erence of energy of electrons localized on A and B.
Fourier transform
U(A) =$
!B
d2k
(2#)2eik·A U(k), V (B) =
$
!B
d2k
(2#)2eik·B V (k),
H =!
k
(U †(k), V †(k) )
%
&'" ! $(k)
! $"(k) !"
(
)*
%
&'U(k)
V (k)
(
)*
$(k) = eik·b1 + eik·b2 + eik·b3
energy: E(k) = ±+
"2 + !2|$(k)|2
one electron per unit cll per spin deg. freedom
The negative-energy states (valence band) are filled andthe positive-energy states (conduction band) are empty.
** The separation of the conduction and valence bands minimized at zeros of
1
Honeycomb LatticeThe Bravais lattice is triangular andthe unit cell contains two sitesBasis vectors area1 = (
!3
2 , 12)a
a2 = ( 0 , 1 )a
The sublattices are connected byb1 = ( 1
2!
3, 1
2 )a
b2 = ( 12!
3,!1
2)a
b3 = (! 1!3, 0 )a
In the tight-binding approximation where only nearest neighbor interactions are retained,the Hamiltonian is
H = !!
A,i
"U †(A) V (A + bi) + V †(A) U(A + bi)
#
+"!
A
"U †(A) U(A)! V †(A + b1) V (A + b1)
#
where A = n1a1 + n2a2.
U : annihilation operator for electrons localized on site A; V for electrons on site B.! : hopping A" B": di!erence of energy of electrons localized on A and B.
Fourier transform
U(A) =$
!B
d2k
(2#)2eik·A U(k), V (B) =
$
!B
d2k
(2#)2eik·B V (k),
H =!
k
(U †(k), V †(k) )
%
&'" ! $(k)
! $"(k) !"
(
)*
%
&'U(k)
V (k)
(
)*
$(k) = eik·b1 + eik·b2 + eik·b3
energy: E(k) = ±+
"2 + !2|$(k)|2
one electron per unit cll per spin deg. freedom
The negative-energy states (valence band) are filled andthe positive-energy states (conduction band) are empty.
** The separation of the conduction and valence bands minimized at zeros of
1
Tight-binding Hamiltonian
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!(k) = ei k·b1 + ei k·b2 + eik·b3
These occur at
q1 =4!
3a(!
32
,12) and q2 = "q1
(and all other equivalent points on the corners of the Brillouin zone).
The separation of the conduction and valence bandsminimized at zeros of
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In the continuum (low energy) limit (a! 0),only electron state near q1 and q2 participate in the dynamics.
(1) Near k = !q1,
H = vF
!
k
(U †, V † )"
ie!i!/3(kx!iky)!iei!/3(kx+iky)
# "U(k!q1)V (k!q1)
#
= vF
!
k
(U †e!i!/6, V †ei!/6 )"
i(kx!iky)!i(kx+iky)
# "ei!/6U(k!q1)e!i!/6V (k!q1)
#
vF =!
3!/2Dirac Hamiltonian
with !µ = ("3, i"1, i"2), and ki = (kx, ky).
!i(kx!iky)
!i(kx+iky)
"=!kx!2 + ky!1 = "0(!k1"
1 ! k2"2)
! H1 = "!0/k for "1 # ei !6 !3(U(k"q1), V (k"q1))t.
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Similarly, (2) Near k = q1,
Similarly, (2) Near k = q1,
H = vF
!
k
(U †, V † )"
iei!/3(kx+iky)!ie!i!/3(kx!iky)
# "U(k+q1)V (k+q1)
#
One has!
i(kx+iky)!i(kx!iky)
"= !kx!2 ! ky!1 = !2(!kx!2 + ky!1)!2
= !2!3(!k1"1 ! k2"2)!2
Let !2 = "2e!i !6 !3(U(k+q1), V (k+q1))t..
! H2 = "#0/k
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!i = !i!i + eAi
H =!
d2x"!†#vF "k!k + m"3 ! eA0
$!
+ #†#vF "k!k !m"3 ! eA0
$#%
!! = (!1, !2)t ! (Uk!q, Vk!q)t
!! = (!1, !2)t ! ("Vk+q, Uk+q)t
vF ! 106m/s ! c/300
field near K
field near K !
Spinor structure <ーー sublattices
band gap m! 0
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Bilayer graphene
chiral Schroedinger eq. positive- and negative-energy states w. quadratic dispersion gapless
bilayer graphite
BABA
BA BAB B
BA
B B
B
BA
BA
A
A
AAA
A