renormalization
group study)
Electron (a
transport
in multilevel
quantum dots
David Logan
Oxford
With Martin Galpin, Chris Wright
[ cond-mat arXiv:0906.3169 ]
Warwick Workshop on Quantum Simulations,
24.8.09
NATO
ARW,
Yalta, 20.9.07.
1. Introduction.
What are quantum dots? ….
Vsource
drain
´ Vsd
...... measure current across dot, I ´ I(Vsd );
and differential conductance:
dI
Gc (Vsd ) =
dVsd
dot
the zero-bias conductance is
Gc (Vsd = 0)
(s.t. Gc = 1=R if ohmic)
Vgate ´ Vg
: applied gate voltage, shifts level
energies of dot relative to chemical
potential /fermi level of leads.
And the dot itself could be e.g….
Semiconductor quantum dot:
GaAs/n-doped GaAs
heterostructure (‘top’ view).
Vtl
source lead
Vr
VG
drain lead
Vbl
e.g. zero-bias conductance
vs gate voltage, and its
T-dependence:
van der Wiel et al Science,
289, 2105 (2000).
Molecular quantum dot:
V
N
C
Liang, Shores,Bockrath, Long, Park,
Nature, 417, 725 (2002).
Single-molecule transistor, based
on di-vanadium complex.
e.g. differential conductance vs
Vsd ; and its T-dependence:
Vsd
Nanotube quantum dot:
Vsource drain
´ Vsd
A
nanotube
SiO2
doped Si
e.g. differential conductance
‘maps’ in the (Vg, Vsd )-plane:
Makarovski, Liu, Finkelstein,
PRL, 99, 066801 (2007).
Vgate
´ Vg
1. Introduction.
Moreover, however obvious, remember:
The ‘system’ is dot + leads, so:
dot
– this is a ~1023 electron system.....
.... of interacting electrons ....
.... that is not in general in
equilibrium (save for zero
bias, Vsd  0).
1. Introduction.
Why study quantum dots? ....
– central to molecular electronics/device physics: single-electron
transistors, molecular rectifiers, transistor arrays and logic gates,
spintronics, quantum computation, entanglement etc.
– tunnel-coupled quantum dots provide controlled/tunable realisation of
strongly correlated ‘quantum impurity’ systems; which embody much
of many-body theory in a nutshell, and has led to strong resurgence of
interest in Kondo and related physics...
e.g. the spin-1/2 Kondo effect in a semiconductor QD, manifest in the
unitarity limit for the zero-bias conductance ......
Spin-1/2 Kondo effect in a semiconductor QD ….
Odd electron valleys:–
– spin-1/2 Kondo effect;
– unitarity limit in zero-bias
conductance for T ¿ TK:
– associated in effect with a
single dot level, singly occupied.
– Kondo progressively killed by
increasing T.
Vtl
source lead
Vr
Vgl
drain
Vbl
van der Wiel et al, Science (2000).
Even electron valleys:
– no Kondo; suppressed conductance
Odd electron valleys:
– spin-1/2 Kondo effect;
van der Wiel et al, Science (2000).
This odd/even (or ‘Kondo/non-Kondo’)
alternation is common in QDs ……
…. but not ubiquitous:
– if the dot level spacing is sufficiently
small, might expect instead to see S=1
Kondo physics in an even valley,
i.e.
S=1 (ferromag
Hund’s coupling).
odd
even
Such behaviour is indeed
observed ....... e.g......
van der Wiel et al, Science (2000).
Triplet Kondo:
Kogan et al, PRB 67, 113309 (2003)
Even valley: also strongly enhanced
zero-bias conductance, indicative of
(underscreened) spin-1 Kondo.
Talk about aspects of the S=1 and
S=1/2 Kondo regimes, and the
transitions between them ……
Odd valley: spin -½ Kondo, strongly
enhanced zero-bias conductance.
Vtl
¢Vg
Vbl
Outline:-
2. Model
– 2-level dot coupled to two conducting leads.
– some issues/questions.
Numerical results from Wilson’s numerical renormalization group (NRG)* .
Method of choice – non-perturbative RG, providing essentially exact results
on low-temperature/energy scales central to the physics and to experiment.
+ Analytical methods where possible.
3. Results
– static/thermodynamic properties.
– evolution of Kondo scale.
– phase diagram(s).
– single-particle dynamics, and transport.
– experiment.
[* use both ‘standard’ NRG and full density matrix/complete Fock space NRG
(Peters, Pruschke, Anders; Weichselbaum, von Delft …)]
MODEL:
V2L
V2R
2
2-level dot, coupled
to two conducting
leads.
L
R
1
V1L
V1R
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1n
^ 2 ¡ JH ^s1 ¢ ^s2
(JH > 0 – ferromagnetic [Hund’s rule] coupling)
H0 =
X
X
X
X X
¸=L;R
HV =
²k cyk¸¾ ck¸¾
– two non-interacting, metallic leads
k;¾
¸=L;R i=1;2
Vi¸ (cyk¸¾ di¾ + dyi¾ ck¸¾ )
– dot/leads tunnel coupling
k;¾
Model rather general; and rich. Look first at the dot states arising in the ‘atomic limit’,
where dot decouples from leads .......
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
²2 = ²1
effectively a one-level problem;
‘filling level 1’.
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
²1
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
effectively a one-level problem;
‘filling level 1’.
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
effectively a one-level problem;
‘filling level 2’.
²1
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
effectively a one-level problem;
‘filling level 1’.
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
²1
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
All states shown are spin S=1/2 or 0.
Finally, there’s the S=1 state.
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
²1
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
Obvious question: what happens on
coupling to the leads? ....
– ‘deep’ in the S=1/2 regimes, low-energy
model is usual spin-1/2 Kondo.
1 2
1 2
1 2
1 2
So ultimate stable fixed point (FP) is
usual strong coupling FP: dot spin
quenched on scale TK ; strongly enhanced
T=0 zero-bias conductance, Gc (0) » 2e2 =h:
System a ‘normal’ Fermi liquid.
1 2
1 2
1 2
1 2
1 2
²1
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
Obvious question: what happens on
coupling to the leads? ...
– ‘deep’ in the S=1 regime, low-energy
model will be spin-1 Kondo.
1 2
1 2
Is it 1-channel or 2-channel spin-1 Kondo?
Depends on coupling to leads…..
1 2
1 2
1 2
If 2-channel, S=1 wholly quenched on
coupling to leads (strong coupling FP);
and strong suppressed dc, Gc (0) » 0
(Pustilnik/Glazman).
1 2
1 2
1 2
1 2
²1
If 1-channel, S=1 only partially quenched
on coupling to leads (‘underscreened’ S=1
FP, Nozieres/Blandin);in this case, strongly
enhanced Gc (0) » 2e2 =h:
System a ‘singular’ Fermi liquid
(Mehta, Andrei et al).
Questions contd:
– for S=1 regime, two-channel behaviour is generic. So apparent puzzle (Coleman
et al): why do exps see strongly enhanced Gc (0) » 2e2 =h?
Answer: low-energy model is generically 2-channel spin-1 Kondo with channel anisotropy:
J¡
L00
J+
R00
: J+ > J¡
2-stage quenching of spin-1:
00
– on ‘high’ scale TK+ » D exp(¡1=½J+ ), quench spin S=1 ! S=1/2 by coupling to R :
Here flow towards underscreened S=1 fixed point.
– then on ‘low’ scale TK¡ » D exp(¡1=½J¡ ); quench spin S=1/2 ! S=0 by coupling to L00 :
(flowing then to fully screened, strong coupling FP)
But the two scales may be vastly different in magnitude.
And if TK¡ ‘irrelevantly small’, in practice might as well be 0:
– experiment will then ‘see’ the underscreened spin-1 FP.
We’ll take this for granted here; in practice, consider effective
1-channel set-up from beginning.
Atomic limit: dot states
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
²2
Questions contd:
– where does exp fit in to figure?
1 2
1 2
– so does there exist a quantum phase
transition (QPT) between ‘regular’ spin1/2 Kondo and spin-1 Kondo (Pustilnik
/Borda)?
1 2
1 2
1 2
If so, what is the nature of the QPT?
1 2
– Can exp be explained?
1 2
1 2
1 2
²1
So consider explicitly:-
Vcosµ
Vsinµ
2
L
R
1
Vsinµ
[two leads of course, but of 1-channel form:
Vcosµ
2
1
V
R0
]
V
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1n
^ 2 ¡ JH ^s1 ¢ ^s2
H0 =
X
X
X
X X
¸=L;R
HV =
²k cyk¸¾ ck¸¾
– two non-interacting, metallic leads
k;¾
¸=L;R i=1;2
k;¾
Vi¸ (cyk¸¾ di¾ + dyi¾ ck¸¾ )
– dot/leads tunnel coupling
Recall discussion of ‘atomic limit’:-
What of the phase boundaries on coupling to
leads?
²2
– all phases that were S=1/2 or 0 in the atomic
limit are continuously connected (as we know..);
here the stable fixed point is the usual strong
coupling (or frozen impurity) FP.
1 2
1 2
– but the underscreened triplet state has a
distinct FP. So we expect a quantum phase
transition; and hence the qualitative
behaviour shown.
Indeed.
1 2
1 2
1 2
1 2
1 2
1 2
1 2
²1
For purposes of illustration here, will now fix
²1 , and progressively decrease ²2 from
‘high up’ in the spin-1/2 Kondo regime, down
into the spin-1 Kondo regime, to ²2 = ²1 :
²2
We’ll chose specifically ²1 = ¡U=2 ¡ U 0 (‘midpoint’);
the point ²2 = ²1 is then particle-hole symmetric.
1 2
Will consider sequentially:
– static/thermodynamic properties.
1 2
– evolution of the Kondo scale.
– phase diagram(s).
– single-particle dynamics.
– transport/differential conductance.
– and experiment.
²1
Entropy, Simp (T ) vs T =D:
Screened spin-1/2 Kondo phase.
²2 = +9 (and ²1 = ¡10)
– high ²2 À 0; so essentially
single-level S=1/2 Kondo for T . U:
TK
– intermediate ln(2) plateau,
characteristic of spin-1/2 local
moment FP.
ln(2)
– characteristic Kondo scale TK ;
below which dot spin quenched:
Simp (T = 0) = 0:
Usual spin-1/2 strong coupling FP.
²2
2
screened
0
Fixed ²1 = ¡U=2 , with U = 20; JH = 5; U = 0
2
(energies i.t.o. ‘hybridization strength’ ¡ = ¼V ½);
1
USC
– and progressively lowering ²2 :
²1
Entropy, Simp (T ) vs T =D:
Screened spin-1/2 Kondo phase.
Now decrease ²2 : ¡
TK
TK TK
²2 = 5; 4; 3:
– Kondo scale
progressively decreases.
ln(2)
²2
2
screened
USC
1
²1
Entropy, Simp (T ) vs T =D:
Screened spin-1/2 Kondo phase.
²2 = 2:8
TK vanishes at a critical ²2;c
(here ²2;c = 2:775)
– symptomatic of a QPT to the
underscreened triplet phase.
ln(2)
For ²2 < ²2;c have the
underscreened spin-1 phase ......
²2
2
screened
USC
1
²1
Entropy, Simp (T ) vs T =D:
Undercreened spin-1 Kondo phase.
²2 = 0
Throughout this phase,
Simp (T = 0) = ln 2
– characteristic of the USC
spin-1 fixed point.
ln(2)
But there is no low-energy scale
in this phase that vanishes as
transition approached.
²2
2
screened
USC
1
²1
Entropy, Simp (T ) vs T =D:
Undercreened spin-1 Kondo phase.
²2 = ¡4
TK+
ln(3)
There is of course a
characteristic low-energy
scale in the USC phase – TK+ :
ln(2)
²2
2
screened
USC
1
²1
Entropy, Simp (T ) vs T =D:
Undercreened spin-1 Kondo phase.
²2 = ¡10 (= ²1 )
TK+
There is of course a
characteristic low-energy
scale in the USC phase – TK+ :
But it barely varies
throughout the phase.
²2
2
screened
USC
1
²1
Spin susceptibility T Âimp (T ); vs T =D:
For screened Kondo phase,
T Âimp (T ) ! 0
as T ! 0:
USC Kondo
For underseened spin-1 phase,
T Âimp (T ) !
1
4
screened Kondo
1
4
(i.e. Âimp (T ) »
with S=1/2).
as T ! 0:
S(S+1)
3T
n1 + n
^ 2 i): …i.e. the net dot charge
“Impurity” charge, nimp (' h^
²2;c
nimp
²2
– very boring, evolves smoothly through the transition.
But key re transport (see on)!
– note that transition occurs generally in a ‘mixed valent’
regime of non-integral charge. Implications?
²2
screened
USC
²1
Vanishing of Kondo scale as ²2 ! ²2;c + from screened Kondo phase: -
²2 ¡ ²2;c
screened
USC
Vanishing of Kondo scale as ²2 ! ²2;c + from screened Kondo phase: TK / exp
µ
¡a
²2 ¡ ~
~
²2;c
¶
²2 = ²2 =¡)
²2;c : (~
as ²~2 ! ~
– QPT of Kosterlitz-Thouless type
(consistent with no evidence for a
separate critical FP).
[²2 ¡ ²2;c ]¡1
[behaviour quite generic.........]
²2 ¡ ²2;c
screened
USC
Phase diagrams:Phase transition boundaries in
general continuous KT-transitions.
y = ²2 + 12 U + U 0
Exception: on the line ²1 = ²2 (y = x)
y=x
screened
0
USC
0
(U = 20; U 0 = 0; JH = 5)
– where a level crossing QPT arises
^
(as permitted by invariance of H
y
y
under ‘1-2’ transformation d1¾ $ d2¾ )
x = ²1 + 12 U + U 0
Single-particle dynamics and transport.
Vsinµ
Vcosµ
2
L
R
1
Vsinµ
Vcosµ
Can consider the single-particle spectra Dij (!) = ¡ ¼1 ImGij (!)
n
o
y
(with i,j referring to dot levels (1 or 2), and Gij (!) $ Gij (t) = ¡iµ(t)h di¾ (t); dj¾ i );
³
´
y
y
y
1
Equivalently, take even/odd combinations of dot levels, de¾ = p2 d1¾ § d2¾ ; and work with
o¾
Gee (!) =
oo
1
2
[ G11 (!) + G22 (!) § 2G12 (!)] ( and Geo (!) = 12 [G11 (!) ¡ G22 (!)] ):
Focus here on e-e spectrum Dee (!) = ¡ ¼1 ImGee (!)
– which determines the zero-bias conductance:
Gc (T ) = G0
Z
1
¡1
d! ¡
@f (!)
2¼¡Dee (!; T )
@!
(¡ = ¼V 2 ½)
4 ¡¡RL
2e2 2
2
2
2
¡
=
¡
cos
(µ) ;
¡
=
¡
sin
(µ);
[where: G0 =
with
sin 2µ and sin 2µ ´
R
L
h
(1 + ¡¡RL )2
s.t. G0 = 2e2 =h for equivalent coupling to leads, ¡L = ¡R .]
Single-particle dynamics.
(T = 0)
For screened Kondo phase.
2¼¡Dee (!) vs !=¡ (¡ = ¼V 2 ½) – ‘all scales’ overview.
²1 = ¡10 and
²2 = 4.5, 3.7, 3.5.
(²2;c = 2:775)
²2
screened Kondo
USC
0
Fix ²1 = ¡U=2 , with U = 20; JH = 5; U = 0
– and progressively lower ²2 :
²1
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5. (²2;c = 2:775)
For screened Kondo phase.
High energy features .........
2
2
1
1
2
2
1
1
2
2
1
1
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5. (²2;c = 2:775)
Kondo resonance
For screened Kondo phase.
Key low-energy feature is of course
the Kondo resonance;
with characteristic scale TK :
Kondo resonance progressively
narrows as transition approached,
and ........
²2
screened Kondo
USC
²1
Single-particle dynamics.
For underscreened Kondo phase.
²1 = ¡10 and ²2 = 2.7 (²2;c = 2:775)
..... vanishes as the transition to the
underscreened spin-1 Kondo phase is
crossed.
[Again, there is no low-energy scale in
the USC phase that vanishes as
transition approached from that side.]
²2
screened Kondo
USC
²1
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5, 2.7 (²2;c = 2:775)
How does the Kondo resonance
‘collapse’ as transition approached
from screened Kondo phase? ......
²2
screened Kondo
USC
²1
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5, 2.7 (²2;c = 2:775)
How does the Kondo resonance
‘collapse’ as transition approached
from screened Kondo phase? ......
– resonance narrows progressively,
and vanishes ‘on the spot’.
– and as it does so exhibits scaling
in terms of the Kondo scale TK :
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5. (²2;c = 2:775)
Scaling spectrum in
screened Kondo phase.
!=TK
Single-particle dynamics.
²1 = ¡10 and ²2 = 4.5, 3.7, 3.5, 2.7 (²2;c = 2:775)
– resonance narrows progressively,
and vanishes ‘on the spot’;
leaving an incoherent background in
underscreened phase (dashed line).
Dee (! = 0), and hence the (T=0)
zero bias conductance / Dee (0);
drops discontinuously across the
transition ........
2¼¡Dee (! = 0) vs ²2
(for ²1 = ¡U=2 = ¡10)
Gc (T = 0)
= 2¼¡Dee (! = 0)
G0
– discontinuity in zero-bias
conductance as cross transition
2¼¡Dee (0)
– discontinuity a ‘drop’ from
screened
underscreened;
as expect from collapsing Kondo
resonance.
underscreened phase
screened phase
²2
²2;c
²2
screened Kondo
USC
(U = 20; U 0 = 0; JH = 5)
²1
Single-particle dynamics.
For underscreened Kondo phase.
Dee (!)
!!0
» Dee (0) ¡
²2 = ¡2
²2 = ¡10
ln
2
³
b
j!j
S =1
TK
´
²2
screened Kondo
USC
0
Fix ²1 = ¡U=2 , with U = 20; JH = 5; U = 0
– and progressively lowering ²2 = ¡2; ¡4; ¡6; ¡8; ¡10:
²1
2¼¡Dee (! = 0) vs ²2
(for ²1 = ¡U=2 = ¡10)
Gc (T = 0)
= 2¼¡Dee (! = 0)
G0
– discontinuity in zero-bias
conductance as cross transition
2¼¡Dee (0)
– discontinuity a ‘drop’ from
screened
underscreened;
as expect from collapsing Kondo
resonance.
So how do we understand
this in general terms?
underscreened phase
screened phase
²2;c
(U = 20; U 0 = 0; JH = 5)
²2
It’s subtle ……
“Friedel-Luttinger sum rule”
Recall
Gc (T = 0)=G0
= 2¼¡Dee (! = 0)
= sin2 ±
(1)
with ± = arg fdetG(! = 0)g the (Fermi level) scattering phase shift.
Can show quite generally – independently of phase (screened/usc) – that
± =
¼
2 nimp
+ IL
with IL the Luttinger integral: IL = Im
Z
0
d! Tr
¡1
½
‘Friedel-Luttinger
sum rule’
¾
@§(!)
G(!)
@!
[and §(!) the 2x2 self-energy matrix, likewise for G(!); ]
For a normal Fermi liquid – the screened phase – Luttinger (integral) theorem holds
throughout the phase: IL = 0
So (2) gives ± = ¼2 nimp – i.e. the Friedel sum rule – and hence from (1):
2
2¼¡ Dee (! = 0) = sin
¡¼
2 nimp
¢
Well known (+ can check directly from NRG that IL = 0):
: screened Kondo
(2)
“Friedel-Luttinger sum rule”
Recall
Gc (T = 0)=G0
= 2¼¡Dee (! = 0)
= sin2 ±
(1)
with ± = arg fdetG(! = 0)g the (Fermi level) scattering phase shift.
Can show quite generally – independently of phase (screened/usc) – that
± =
¼
2 nimp
+ IL
with IL the Luttinger integral: IL = Im
Z
0
d! Tr
¡1
½
‘Friedel-Luttinger
sum rule’
(2)
¾
@§(!)
G(!)
@!
[and §(!) the 2x2 self-energy matrix, likewise for G(!); ]
But the underscreened phase is a ‘singular Fermi liquid’. There is no reason to suppose
Luttinger’s theorem (IL = 0) is satisfied in it. And it isn’t.
So a question arises: in the same way that IL has a characteristic value (of
zero) for a Fermi liquid phase, regardless of ‘bare’ parameters, might IL for
the USC phase likewise be an intrinsic characteristic of that phase?
Yes: throughout the USC phase (i.e. regardless of bare parameters U, U’, J etc), find:
jIL j =
¼
2
: underscreened phase
“Friedel-Luttinger sum rule”
Recall
Gc (T = 0)=G0
= sin2 ±
= 2¼¡Dee (! = 0)
(1)
with ± = arg fdetG(! = 0)g the (Fermi level) scattering phase shift.
Can show quite generally – independently of phase (screened/usc) – that
± =
¼
2 nimp
+ IL
with IL the Luttinger integral: IL = Im
Z
0
d! Tr
¡1
½
‘Friedel-Luttinger
sum rule’
(2)
¾
@§(!)
G(!)
@!
[and §(!) the 2x2 self-energy matrix, likewise for G(!); ]
But the underscreened phase is a ‘singular Fermi liquid’. There is no reason to suppose
Luttinger’s theorem (IL = 0) is satisfied in it. And it isn’t.
Throughout the USC phase (i.e. regardless of bare parameters U, J etc), we find: jIL j =
So (1),(2) give directly that:
2¼¡ Dee (! = 0) = sin2
So overall we have:
¡¼
2 [nimp ¡ 1]
¢
: underscreened phase
¼
2
“Friedel-Luttinger sum rule”
sin2
2¼¡ Dee (! = 0) =
sin2
¡¼
2 nimp
¢
¡¼
¢
[n
¡
1]
2 imp
: screened phase
: underscreened phase
(= cos2 ( ¼2 nimp ))
with:
Gc (T = 0)=G0 = 2¼¡Dee (! = 0)
– and since n im p in general evolves smoothly across the transition, the zero-bias
conductance jumps discontinuously.
Can test independently, since can determine Dee (!) and
nimp separately via NRG ……
2¼¡Dee (0)
crosses: from NRG calcn of Dee (! = 0):
dotted: results from direct NRG
calcn of nimp :
²2;c
2
sin
2¼¡ Dee (! = 0) =
¡¼
2 nimp
²2
¢
cos2 ( ¼2 nimp )
: screened phase
: underscreened phase
2
sin
But wait:
2¼¡ Dee (! = 0) =
¡¼
2 nimp
¢
cos2 ( ¼2 nimp )
: screened
: underscreened
– so if nimp 2 [ 32 ; 2] or [0; 12 ] then the conductance should increase
(rather than decrease), on passing from the screened to the USC phase.
Since a decrease in the conductance at the transition is naturally associated with a
‘collapsing’ Kondo resonance in the spectrum, this might suggest that an increase in
conductance is associated with a collapsing Kondo antiresonance.
Indeed it is: to illustrate, consider line y = -x; along
which nimp = 2 by symmetry.
y = ²2 + 12 U + U 0
nimp = 2
FL
0
USC
y = ¡x
0
x = ²1 + 12 U + U 0
Line y = -x; along which nimp = 2 :
2¼¡Dee (!)
Clear Kondo antiresonance in
screened Fermi liquid phase;
Again ‘collapses on spot’ as
approach transition …..
… jumping discontinuously as
enter USC phase.
… and again scaling in terms of
(vanishing) Kondo TK as it
does so.
!=¡
2
sin
2¼¡ Dee (! = 0) =
¡¼
2 nimp
¢
cos2 ( ¼2 nimp )
= 0
: screened
= 1
: underscreened
EXPERIMENT:
Kogan et al, PRB 67, 113309 (2003)
Even valley: also strongly enhanced
differential conductance, indicative of
spin-1 Kondo effect.
Odd valley: spin -½ Kondo,
strongly enhanced diff con.
EXPERIMENT:
Kogan et al, PRB 67, 113309 (2003)
For zero-bias conductance
take cut along Vds = 0
Exptl ‘trajectory’ on varying ¢Vg :
y = ²2 + 12 U + U 0
y = x + ±²
EXPT
y=x
nimp = 2
i.e.
²2 = ²1 + ±²
with level separation ±² fixed,
and ²1 / ¢Vg
And can show zero-bias conductance is
symmetric f’n of ¢Vg about ‘midpoint’ ( )
of trajectory in USC phase; which lies on
y=-x line with nimp = 2 .
0
y = ¡x
0
x = ²1 + 12 U + U 0
EXPERIMENT:
Kogan et al, PRB 67, 113309 (2003)
For zero-bias conductance
take cut along Vds = 0
EXPERIMENT:
USC
Screened (S)
Gc =(2e2 =h)
S
¢Vg (mV)
Experiment
Experimental T ' 30 mK:
EXPERIMENT:
Gc =(2e2 =h)
Experiment
¢Vg (mV)
Experimental T ' 30 mK:
EXPERIMENT:
USC
Screened (S)
Gc =(2e2 =h)
S
¢Vg (mV)
Experiment
Experimental T ' 30 mK:
EXPERIMENT / THEORY
Comparison to theory (T=0):
Not entirely trivial: need …..
U
¡
U0
¡
JH
¡
±~
²=
[²2 ¡ ²1 ]
¡
plus proportionality (‘c’) between
gate voltage and level energy:
¢Vg ¡ ¢Vg;mid
= c[~
²1 ¡ ²~1;mid ]
mV
and hybridization strength ¡ itself.
EXPERIMENT / THEORY
Comparison to theory (T=0):
Screened (S)
USC
Gc =(2e2 =h)
S
¢Vg (mV)
sin2
Gc (0)=G0 =
2
sin
¡¼
2 nimp
¡¼
¢
2 [nimp
: screened
¡ 1]
¢
(Here, nimp ' 1:9 and 2:1
: underscreened
at transitions.)
EXPERIMENT / THEORY
Comparison to theory (T=0):
USC
Screened (S)
Gc =(2e2 =h)
S
¢Vg (mV)
EXPERIMENT / THEORY
Comparison to theory (T>0):
USC
Screened (S)
T =¡ = 0:005
i.e. T ' 30mK with
Gc =(2e2 =h)
S
¡ ' 0:5meV
¢Vg (mV)
EXPERIMENT / THEORY: non-zero bias.
In zero-bias limit,
Gc (T; Vsd = 0)
=
G0
Z
1
¡1
d!
¡@f(!)
2¼¡Dee (!)
@!
(f (!) = [e!=T + 1]¡1 )
…. is exact. At finite bias, cannot say anything exact. But if neglect explicit Vsd
dependence of self-energies,
·
¸
Z 1
¡@f
¡@f
Gc (T; Vsd )
(!)
(!)
L
R
' 12
d!
+
2¼¡Dee (!)
G0
@!
@!
¡1
1
with fº (!) = f(! § 2 eVsd ) for º = L=R lead. Calculable; and
Gc (T = 0; Vsd )
' ¼¡[Dee (! = + 12 eVsd ) + Dee (! = ¡ 12 eVsd )]
G0
– ‘symmetrised spectral sum’.
EXPERIMENT / THEORY: non-zero bias.
Differential conductance maps in (VG ; Vsd ) -plane:
Theory
Experiment
T = 30 mK
Not bad at all – what can we learn from it? …..
EXPERIMENT / THEORY: non-zero bias.
Gc =(2e2 =h)
Take cuts through maps at different
fixed values of gate voltage; to show
conductance as function of Vsd.
For ¢Vg = ¡10mV
– see single maximum at zero-bias
(ie Kondo resonance for USC phase).
Vsd (mV)
EXPERIMENT / THEORY: non-zero bias.
Now increase gate voltage to
Gc =(2e2 =h)
¢Vg = ¡6mV
– just entering screened FL phase.
Resonance beginning to split.
Vsd (mV)
EXPERIMENT / THEORY: non-zero bias.
Increase further to
Gc =(2e2 =h)
¢Vg = ¡5mV
moving further into screened phase.
Vsd (mV)
EXPERIMENT / THEORY: non-zero bias.
And further ….
Gc =(2e2 =h)
¢Vg = ¡3mV
– seeing clear development of
antiresonance (arising from the
Kondo antires in Dee (!)).
.
Vsd (mV)
EXPERIMENT / THEORY: non-zero bias.
Gc =(2e2 =h)
¢Vg = +2mV
Vsd (mV)
EXPERIMENT / THEORY: non-zero bias.
T = 30 mK
Gc =(2e2 =h)
So ‘vanishing Kondo antiresonance’ case
seems in this case to explain experiment
rather decently.
Vsd (mV)
...... and what you’ll see if you just ‘miss’ the transition:
Theory
Vsd =mV
¢Vg =mV
Experiment
...... and what you’ll see if you just ‘miss’ the transition:
Theory
Vsd =mV
Experiment
Vsd =mV
¢Vg =mV
¢Vg =mV
Concluding remarks.
Vcosµ
Vsinµ
Considered a ‘simple’
but rather general twolevel quantum dot:
2
L
R
1
Vsinµ
Vcosµ
HD = ²1 n
^ 1 + ²2 n
^ 2 + U (^
n1" n
^ 1# + n
^ 2" n
^ 2# ) + U 0 n
^1 n
^ 2 ¡ JH ^s1 ¢ ^s2
Continuous line of QPTs in (²1 ; ²2 ) -plane separating an underscreened spin-1 phase from
the usual screened phases:
– QPT usually of KT type, with a vanishing Kondo
scale as approach transition from screened side.
²2
screened
– rich range of behaviour in static & dynamic properties.
USC
²1
– zero-bias conductance explicable i.t.o. “LuttingerFriedel sum rules” for two phases; including anomalous
transport as transition crossed (+ generalisation of
‘Luttinger integral theorem’ to USC phase).
– both phases seen experimentally; and appear
explicable by theory.
…. so models aren’t purely ‘paradigmatic’.
Acknowledgements
Fithjof Anders, Fabian Essler, H R Krishnamurthy, Andrew Mitchell,
Michael Pustilnik, David Goldhaber-Gordon, Andrei Kogan.
Funding: EPSRC Programme Grant, Condensed Matter Theory Group, Oxford.