1

Enhancement of theTwo-channel Kondo Effectin Single-Electron Boxes

Frithjof B. AndersUniversity of Bremen

10. April 2003

Collaborators:A. Schiller, E. Lebanon

Hebrew University, Jerusalem, Israel

cond-mat/0303248

2

Contents

1. Introduction

• Reminder: one and two-channel Kondo effect• Coulomb Blockade in Quantum Dots• Model: Quantum Box coupled to a reservoir of electrons through a single

resonant level• Low temperature dynamics

2. Analytical Treatment

3. Selection of NRG results

• Effective capacitance• Low temperature energy scale• Finite charging energy on the resonant level

4. Conclusion and outlook

3

Single-Channel Kondo Effect

Local spin ~Sloc

band

Jlocal Spin

H =∑~kσ

�~kc†~kσc~kσ− JzszbS

zloc

−J⊥

2

(s

+b S−loc + s

−b S

+loc

)

J < 0: IR divergence∫ xnxn+1

d�

�= − ln(x)

3

Single-Channel Kondo Effect

Local spin ~Sloc

band

Jlocal Spin

H =∑~kσ

�~kc†~kσc~kσ− JzszbS

zloc

−J⊥

2

(s

+b S−loc + s

−b S

+loc

)

J < 0: IR divergence∫ xnxn+1

d�

�= − ln(x)

=⇒ Screeningcloud

☞ J < 0

−→ IR Problem☞ for T → 0:

global singlett

☞ ω, T < TK :universality

4

Two-Channel Kondo Effect

local spin coupled to two channels

−+s sSloc

H =∑~kσα

�~kc†~kσαc~kσα − J~Sloc

∑α

~sα

TK ∝ e− 1Jρband

☞ J < 0 −→ IR problem☞ at T = 0: no singlett

☞ universality:χ(T ), γ(T ) ∝ log(T/TK) =⇒non-Fermi liquid due to mag. scat-tering

0.1 1.0 10.0�

T/TK

0

0.5

1

1.5

2

TK *

χ(T

)

magnetic susceptibility

single channel modeltwo channel model

5

Single-Channel vs Two-Channel Kondo Effect

Quantity 1-Channel 2-Channel

Modell

band

Jlocal Spin

band

band

JIsospin (Charge)

Fixed point Fermi-liquid non-Fermi liquid

J →∞ J = 1

S(T = 0) 0 12 ln(2)

χ(T ) ∝ 1TK ∝ −1TK

ln(T/TK)

C(T ) ∝ TTK ∝ −TTK

ln(T/TK)

6

Charging of a Quantum Box: the Coulomb-Blockade

quantum

V

boxgaselectron

b

Q-Box:∆E � tb, T, TK, D

Ec = e2/C0

-1 0 1

N

Ebox

-1 0 1

N

Ebox

1/2 3/2 5/2

VC0/e

1

2

3

/e

Ĥc = Q̂2

2C0+ VbQ̂

6

Charging of a Quantum Box: the Coulomb-Blockade

quantum

V

boxgaselectron

b

Q-Box:∆E � tb, T, TK, D

Ec = e2/C0

-1 0 1

N

Ebox

-1 0 1

N

Ebox

1/2 3/2 5/2

VC0/e

1

2

3

/e

Ĥc = Q̂2

2C0+ VbQ̂

tb

gaselectron quantum

V

box

b

Question:effectivecapacitancefor tb 6= 0?Nature of thefixed point?

7

Quantum Dots und Quantum Boxes

Berman et. al. PRL 81, 161 (1999)

7

Quantum Dots und Quantum Boxes

Berman et. al. PRL 81, 161 (1999)

Hbox =∑kσ

�Bkσc†BkσcBkσ +

e2

2C0

(Q̂/e−Ng

)2; Ng = VbC0/e

7

Quantum Dots und Quantum Boxes

Berman et. al. PRL 81, 161 (1999)

Hbox =∑kσ

�Bkσc†BkσcBkσ +

e2

2C0

(Q̂/e−Ng

)2; Ng = VbC0/eH =

α=L,B∑kσ

�αkσc†αkσcαkσ+Ec

(Q̂/e−Ng

)2+ tB

∑σ

{c†B0σcL0σ + c

†L0σcB0σ

}

8

Problem: Charging of the Box

t Ec(N̂B −Ng)2

8

Problem: Charging of the Box

t Ec(N̂B −Ng)2

N̂tot = N̂L + N̂B

8

Problem: Charging of the Box

t Ec(N̂B −Ng)2

N̂tot = N̂L + N̂B

= N̂L − N̂B + 2N̂B

8

Problem: Charging of the Box

t Ec(N̂B −Ng)2

N̂tot = N̂L + N̂B

= N̂L − N̂B + 2N̂B

T̂ z = Ntot/2 =12

N̂L − N̂B︸ ︷︷ ︸τzc

+ N̂B︸︷︷︸τzbox

T̂ z = τzc + τzbox

8

Problem: Charging of the Box

t Ec(N̂B −Ng)2

N̂tot = N̂L + N̂B

= N̂L − N̂B + 2N̂B

T̂ z = Ntot/2 =12

N̂L − N̂B︸ ︷︷ ︸τzc

+ N̂B︸︷︷︸τzbox

T̂ z = τzc + τzbox

t

View τzbox as new degree of freedom which keeps trace of how manyelectrons are on the box =⇒ see talk by E. Lebanon

9

Low Temperature Dynamics

H =∑α=L,B

∑k,σ

�kαc†αkσcαkσ +

∑σ

tb{c†0Bσc0Lσ + c

†0Lσc0Bσ

}+ Ec(n̂B −N)2

• if Ec � kBT : only < n|N̂B|n >= n0, n0 + 1 are possible☞ Mapping: τ zbox = 1/2 Isospin represents NB = n0, n0 + 1 (Matveev 91)

☞ replace Hc by H ′c = −τzboxeV , V = 2EC∆N/e =

CbC Vb

9

Low Temperature Dynamics

H =∑α=L,B

∑k,σ

�kαc†αkσcαkσ +

∑σ

tb{c†0Bσc0Lσ + c

†0Lσc0Bσ

}+ Ec(n̂B −N)2

• if Ec � kBT : only < n|N̂B|n >= n0, n0 + 1 are possible☞ Mapping: τ zbox = 1/2 Isospin represents NB = n0, n0 + 1 (Matveev 91)

☞ replace Hc by H ′c = −τzboxeV , V = 2EC∆N/e =

CbC Vb

H =∑k,σα

�kαc†αkσcαkσ +

∑σ

tb{c†0Bσc0Lστ

++ c

†0Lσc0Bστ

−}− τ zboxeV

=⇒ anisotropic two-channel Kondo model: (Spin, Isospin)→ (channel index α, Spin)

10

Problem:

tb

gaselectron quantum

V

box

b

• condition: ∆E � TK � Ec

• problem: TK = tB e−1/(tBρ)

☞ 2C Kondo effect might not be observable in semiconductors(Zarand et. al. 2000)

11

Low-Temperature Model

gaselectron quantum

t

V

tL

Ed

SET

b

box

b

Gramespacher & Matveev PRL 85, 4582 (2000)

Large U → local momentregime

SU(4) symmetry ?poster by K. Le Hur and

P. Simon

H =∑k,σα

�kαc†αkσcαkσ + �d

∑σ

d†σdσ + Und↑nd↓ +

∑σ

tL{c†0Lσdσ + h.c.

}+∑σ

tB{c†0Bσdστ

++ d

†σc0Bστ

−}− τ zboxeV

12

Analytical Treatment for U = 0

• Diagonalization of lead plus SET level

HD

=∑α=L,B

∑σ

∫�a†�ασa�ασd�− hSz

+∑σ

∫d�

∫d�′

2J(�, �

′)[a†�Bσa�′LσS

++ h.c.

]J(�, �′) = 2tB

√ρB(�)ρ

effL (�

′)

• Limit of large band width ρeffL (�) =1

π

ΓL

(�− �d)2 + Γ2Lwith ΓL = πt2LρL(0)

Ed

transistorsingle electronquantum box

ΓL

Per. RG in weak coupling (tB/tL � 1): TK = tB√π

2e−π

2

4tLtB

13

Analytical Treatment for PH-Asymmetry

Definition:

δQ(V ∗) = Q(V ∗)− 1/2 = 〈τzbox〉(V ∗) = 0

Poor man’s scaling

V ∗(�d) = −(ΓB/π) ∗ [1 + 2 ∗ ln(0.5 +D/2|�d|)]

13

Analytical Treatment for PH-Asymmetry

Definition:

δQ(V ∗) = Q(V ∗)− 1/2 = 〈τzbox〉(V ∗) = 0

Poor man’s scaling

V ∗(�d) = −(ΓB/π) ∗ [1 + 2 ∗ ln(0.5 +D/2|�d|)]

transistorsingle electronquantum box

EdΓL

14

Effective Capacitance of the Quantum Box

10−7

10−5

10−3

10−1

101

T/TK

0

0.2

0.4

0.6

0.8

C(V

B,T

)kBT

K/e

2

0

0.01

0.03

0.1

0.3

1

ΓL/D = 0.0157

ΓB/D = 0.0157

TK/D = 0.0063

→ TK/ΓB ≈ 0.4

SETLEAD Q−BOX 0

dε = 0

C(T ) = −1

20TKlog

(T

TK

)+ b

15

Kondo Temperature

0 2 4 6 8 10

(ΓL/ΓB)1/2

10−12

10−10

10−8

10−6

10−4

10−2

100

k BT

K

� /D

(2ΓB/D)1/2

exp[−π2(ΓL/ΓB)1/2

/4]

NRG: Ns=1400

NRG: Ns=1500

NRG: Ns=2300

ΓL/D = 0.098

Weak coupling RG

TK = tB

√π

2e−π

2

4tLtB

16

Kondo Temperature

tL constant

0 0.1 0.2 0.3 0.4 0.5

(2ΓB/πD)1/2

0

0.5

1

1.5

2

k BT

K/ Γ

L

tB constant

0 0.05 0.1 0.15 0.2

(2ΓL/πD)1/2

0

0.2

0.4

k BT

K/Γ

B

ΓB/D=0.0039

17

Single-Electron-Transistor: �d 6= 0

10−7

10−5

10−3

10−1

101

T/TK

0

0.25

0.5

0.75

1

C(V

2CK,T

)kBT

K/e

2

−1.75 −1.5 −1.25 −1

eVB/�ΓL

−0.3

−0.2

−0.1

0

0.1

−δQ

(VB,T

)/e

εd=ΓL=ΓB=0.01D

−0.8 −0.6 −0.4 −0.2 0

εd/D

0

0.005

0.01

0.015

0.02

0.025

eV*/

D

1600 states

1400 states

(2ΓB/πD)ln[1+D/|εd|]

−0.1 −0.05 0

ΓL/D=ΓB/D=0.01

0

0.01

0.02

εd

LEAD Q−BOX 0

SET Poor man’s scalingV∗(�d) = −(ΓB/π) ∗ [1 + 2 ∗ ln(0.5 +D/2|�d|)]

18

Coulomb-Blockade on the SET ( �d = −U/2)

ΓB/ΓL = 0.36

10−4

10−3

10−2

10−1

10� 0

10� 1

10� 2

T/TK

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TKC

eff

Ed=−U/2, ΓB/�ΓL=0.36

U/ΓL=1U/ΓL=2U/ΓL=4U/ΓL=6U/ΓL=8U/ΓL=16

0�

10 20 30 40U/ΓL

10−4

10−3

10−2

10−1

100

TK

10−4

10−3

10−2

10−1

100

101

T/TK

0.0

0.5

1.0

1.5

2.0

C(0

,T)k

BT

K/e

2

00.0250.050.10.150.2

U/D

ΓL/D=ΓB/D=0.0039

18

Coulomb-Blockade on the SET ( �d = −U/2)

ΓB/ΓL = 0.36

10−4

10−3

10−2

10−1

10� 0

10� 1

10� 2

T/TK

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TKC

eff

Ed=−U/2, ΓB/�ΓL=0.36

U/ΓL=1U/ΓL=2U/ΓL=4U/ΓL=6U/ΓL=8U/ΓL=16

0�

10 20 30 40U/ΓL

10−4

10−3

10−2

10−1

100

TK

10−4

10−3

10−2

10−1

100

101

T/TK

0.0

0.5

1.0

1.5

2.0

C(0

,T)k

BT

K/e

2

00.0250.050.10.150.2

U/D

ΓL/D=ΓB/D=0.0039

0 0.05 0.1 0.15 0.2 0.25U/D

10−6

10−5

10−4

10−3

10−2

TK/D

1000 2000 3000N

0

2

4

6

103 T

K/D

ΓL/D=ΓB/D=0.0039

particle-hole symmetry

19

Energy Flow: Symmetric case

0 10 20 30 400

0.5

1

1.5

2

U=0U=20Γ

20

Coulomb-Blockade on the SET ( �d = −U/2)

0.5 0.5

1 1

1.5 1.5

ΓL/ΓB=1

10−11

10−9

10−7

10−5

10−3

10−1

kBT/D

0 0

0.5 0.5

1 1

1.5 1.5ΓL/ΓB=16

0.5

1

1.5

C(0

,T)k

BT

K/e

2

ΓL/ΓB=4

0.5

1

1.5

χ(T

)/χ(

0)

ΓL/D = 0.0157, U/D = 0.3, λ = 2.3, N = 2300

particle-holesymmetry:�d = −U/2

andsymmetric bands

Two Kondo-effectsSET: single

channelQuantum Box:two-channel

21

Coulomb-Blockade on the SET ( �d = 0, U = 6)

1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10T

0

0.5

1

1.5

2

Cef

f(T

)/e2

VB=-1.22

VB=-1.23

VB=-1.23555

VB=-1.25

VB=-1.26

tL=t

B=0.1779, U=6, D=20, ε

d=0 ph asymmetry:

�d = 0, U/D = 0.3

22

Conclusion and Outlook

• The charging of a quantum box, coupled to a lead by tunneling through a single reso-nant level can be solve exactly by Wilson’s NRG for arbitrary parameters

• The degeneracy point is governed by a two-channel Kondo fixed point• The associated Kondo scale is enhanced up to resonant level width close to unitary

transmission• The Coulomb-blockade on the SET generates a second Kondo scale, the magnetic

Kondo scale• We find that the two-channel fixed point is stable upon increasing U for �d = 0 and�d = −U/2

Outlook• The problem can be solved with NRG away from the degeneracy point for arbitrary

charging energy Ec• The model is under investigation in the parameter regime suggested by Y. Oreg and

D. Goldhaber-Gordon

23

Numerical Renormalization Group Method

• Logarithmic discretisation of the energy mesh [D 1λn+1

, D 1λn ], n = 0, 1 · · ·☞ logarithmically divergent terms constant on each intervall

Λ0Λ−1Λ−20−Λ −Λ−1 −Λ−2 0• exact mapping of the Hamiltonian onto a semi-infinite chain:

(Wilson 1975, Krishnamurty et. .al 80)

HN−1

24

Numerical Renormalization Group Method

☞ Recursion relation:

HN−1

HN =√λHN−1 +

∑ασ

ξN−1[c†N−1ασcNασ + c

†NασcN−1ασ]

☞ If all eigenstates of HN−1 are known, i. e. HN−1|L〉 = EL|L〉, the matrix-elementsof HN

〈γ′L′|HN |γL〉 = ELδLL′δγγ′ +∑ασ

ξN−1〈γ′L′|c†N−1ασcNασ + c†NασcN−1ασ|γL〉

can be calculated, and HN diagonalized

TitelContentsSingle Channel Kondo EffectTwo-Channel Kondo Effecttable Two channel Kondo EffektCharging of a Quantum Box: the Coulomb-BlockadeExperiment und HamiltonianProblem: Charging of the BoxLow temperature dynamicsProblem with original modellLow-Temperature ModelAnalytical Treatment U=0Analytical Treatment PH ass. U=0Effektive Kapazität der QuantenboxKondotemperaturKondotemperatur IISingle-Elektron Transistor Ed >0Coulomb-Blockade: finite UCoulomb-Blockade: finite U two-kondo-effectsCoulomb-Blockade: finite U two-kondo-effectsConclusionSymmertic Energy FlowNRGNRG II: Recursion