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close to one of the largest
5 eV for Mn in Zn. In the rest of this work,
we shall therefore use the Kondo Hamiltonian
[
95
]
as our staring point.
Our analysis begins with the Hamiltonian for free Dirac electrons in
graphene and consider interaction with a single magnetic impurity at the
origin. We first find that we need a critical Kondo coupling J
c
to get
Kondo screening for neutral graphene. This is because of linear vanishing
of density of states at the fermi level. By applying a gate voltage we can
change the chemical potential and make the density of states finite at the
fermi level. This is the tunability offered by graphene.
After a detailed analysis we obtain the expression for
J
2
.
J
c
(
q, T
), the crit-
ical value in the large
N
limit as,
J
c
(0)
1
ln
1
/q
2
ln (
Λ)
−
1
J
c
(
q, T
)=
−
2
q
k
B
T/
(31)
π
v
F
k
c
)
2
/
where the temperature
Λ=
π
2
Λ is the critical coupling in the absence of the gate voltage and q =
eV
Λ
k
B
T
is the infrared cutoff,
J
c
(0) = (
. We have omitted all subleading non-divergent terms which are not
important for our purpose. For
, we thus have, analogous to
the Kondo effect in flux phase systems
[
88
]
, a finite critical Kondo coupling
J
c
(0) =
V
=0=
q
π
2
Λ
20 eV which is a consequence of vanishing density of states
at the Fermi energy for Dirac electrons in graphene. Of course, the mean-
field theory overestimates
J
c
requires a more sophisticated analysis which we have not attempted here.
The presence of a gate voltage leads to a Fermi surface and consequently
J
c
.
A quantitatively accurate estimate of
J
c
(
q, T
→
T →
J<J
c
(0) and
)
0as
0. For a given experimental coupling
temperature
T
, one can tune the gate voltage to enter a Kondo phase. The
T
∗
(
temperature
) below which the system enters the Kondo phase for a
physical coupling
J
q
q, T
∗
)=
can be obtained using
J
c
(
J
which yields
k
B
T
∗
=Λexp
(1
/q
2
])
− J
c
(0)
/J
)
/
(2
q
ln[1
(32)
T
∗
35K
[
96
]
.We
For a typical
J
2 eV and voltage
eV
0
.
5eV,
stress that even with overestimated
J
c
, physically reasonable
J
leads to
T
∗
for a wide range of experimentally tunable
experimentally achievable
gate voltages.
We now discuss the possible ground state in the Kondo phase quali-
tatively.
J
c
implies that the
ground state will be non-Fermi liquid as also noted in Ref. [88] for flux
phase systems. In view of the large
In the absence of the gate voltage a finite
J
c
estimated above, it might be hard to
realize such a state in undoped graphene. However, in the presence of the
gate voltage, if the impurity atom generates a spin half moment and the
