- Email: [email protected]
The effect of the interdot Coulomb interaction on Kondo resonances in series-coupled double quantum dots
Physica B 406 (2011) 749–755
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
The effect of the interdot Coulomb interaction on Kondo resonances in series-coupled double quantum dots Shao-quan Wu n, Jia-feng Chen, Guo-ping Zhao College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China
a r t i c l e in f o
abstract
Article history: Received 11 June 2010 Received in revised form 16 November 2010 Accepted 17 November 2010 Available online 30 November 2010
We theoretically investigate the effect of the interdot Coulomb repulsion on Kondo resonances in the series-coupled double quantum dot coupled to two ferromagnetic leads. The Hamiltonian of our system is solved by means of the slave-boson mean-field approximation, and the variation of the density of states, the transmission probability, the occupation number, and the Kondo temperature with the interdot Coulomb repulsion are discussed in the Kondo regime. The density of states is calculated for various interdot Coulomb repulsions with both parallel and antiparallel lead-polarization alignments. Our results reveal that the interdot Coulomb repulsion greatly influences the physical property of this system, and relevant underlying physics of this system is discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Interdot Coulomb interaction Kondo resonances Slave-boson mean-field Approximation Density of states
1. Introduction In recent years, double-quantum-dot (DQD) systems have attracted much attention because of their importance in physics and device applications [1–6]. In contrast to the single quantum dot (QD), DQD systems have richer physics. On the one hand, DQD systems provide an ideal model system for studying the two impurity effect, which has long been the highlight of condensedmatter physics. On the other hand, a series-coupled DQD can cross over from the weak tunnelling coupling regime to the strong tunnelling coupling regime by varying the dot–dot coupling strength. In the weak tunnelling coupling regime, when the temperature is below the Kondo temperature, the Kondo resonance is formed in each dot and lead, and the tunnelling is dominated by hopping between two Kondo states. On the contrary, in the strong tunnelling coupling regime, the electrons are delocalized over two dots, and two dots are coupled coherently and form an artificial molecule. Therefore, DQD systems have been suggested as possible candidates for building blocks of a quantum computer and other possible device applications. Recent theoretical and experiment studies have shown that coherent couplings between two dots can be fulfilled by adjusting relevant parameters properly when DQD systems are coupled to electrodes [6–9], where the Kondo resonance peak splits into two peaks as a result of the energy difference between bonding and antibonding states. Most of the previous studies on DQD systems have treated effects of the intradot
n
Corresponding author. E-mail address: [email protected] (S.-q. Wu).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.11.057
Coulomb repulsion but have ignored effects of the interdot Coulomb repulsion [6,7,9]. However, the interdot Coulomb repulsion is known to play a significant role in electronic transport through DQD systems.
2. Model and solution In this article, we investigate the effect of the interdot Coulomb repulsion on the Kondo resonance in the series-coupled DQD coupled to two ferromagnetic (FM) leads. The schematic diagram of our device is presented in Fig. 1. A DQD system can be described by a two-fold degenerate twoimpurity Anderson Hamiltonian with the interdot tunnelling. In the limit of the intradot Coulomb repulsion U0 -N, the Hamiltonian can be written in terms of auxiliary SB operators plus constraints [6–10] H¼
X
X
kas
as
ekas ckyas ckas þ
y eas fas fas þ
tc X y ðf bL byR fRs þfRys bR byL fLs Þ N s Ls
1 X y Vkas ðckyas bya fas þ fas ba ckas Þ þUð1byL bL Þð1byR bR Þ þ pffiffiffiffi N kas X X y þ la ð fas fas þ bya ba 1Þ a
ð1Þ
s
where ckyas ðckas Þ is the creation (annihilation) operator for an electron with energy ekas and spin s in the lead a(a ¼L, R), and dyas ðdas Þ is the creation (annihilation) operator of electrons in the left or right dot. According to the slave-boson representation,
750
S.-q. Wu et al. / Physica B 406 (2011) 749–755
y dyas ðdas Þ is replaced by dyas -fas ba ðdas -bya fas Þ, where ba(fas) is the slave-boson (pseudo-fermion) annihilation operator of an empty state (singly occupied state). Vkas(tc) is the tunnelling amplitude between dots and leads (between two dots), while the fifth term is used to describe effects of the interdot Coulomb repulsion and the Lagrange multiplier la is introduced to incorporate the constraint P y y s ¼ mk fas fas þ ba ba ¼ 1 imposed on slave particles in the sixth term. For simplicity, we only consider symmetric dots with P eLs ¼ eRs ¼ e0 and the line width function is Gas ðeÞ ¼ p kas Vkas 2 dðeekas Þ. We introduce the effective spin polarization
e~ a ¼ e0 þ la . The total Hamiltonian is now replaced by ~ ¼ H
tc
þ
2
ρ (ε)
0.18
ð3Þ
0
ε/ΤΚ = 0
0.16
0.75 1.5
0.14
3
0.12
0
ε/ΤΚ = 0.75 0
ε/ΤΚ = 1.5 0
ε/ΤΚ = 3
0.10
0.15
0.08 0.06
0.10
0.04
0.05
0.02 0.00
0.00 0.16
0
0.5
ε/ΤΚ = 0
0.14
0 ε/ΤΚ = 0.75 0 ε/ΤΚ = 1.5 0 ε/ΤΚ = 3
0.12 0.10 T (ε)
X Z de o ~2 ~2 ðee~ LðRÞ ÞGL,LðR,RÞ s ðeÞ þU b LðRÞ ð1N b RðLÞ Þ 4 p s
ð4Þ 2 a a ~ ~ Here G s ¼ b a Gs ða ¼ L,RÞ, and fa(e) is the Fermi distribution function. By using the renormalized parameters determined self-consistently
0
0.20
a
o of the Keldysh Green Here GL,LðR,RÞ E s ðeÞ is theD Fourier transform function Gao, aus ðttuÞ i fayus ðtuÞfas ðtÞ . The first equation represents the constraint imposed on the slave particles, while the second one is obtained from the stationary condition, namely, the boson field is time independent at the mean-field level. From the equation of motion of the operator fas [9–11], we have the explicit form of the Green function n h i o ~ LðRÞ ðee~ RðLÞ Þ2 þðG ~ RðLÞ Þ2 þ fRðLÞ ðeÞG ~ RðLÞ t~ 2 2i fLðRÞ ðeÞG s s s c o GL,LðR,RÞs ðeÞ ¼ 2 ~ L Þðee~ R þ iG ~ R Þt~ 2 ðee~ L þ iG s s c
ε/ΤΚ = 0
0.25
s
X 2 2 2 2 2 y ckas Þ þ Uð1Nb~ L N b~ R þ N 2 b~ L b~ R Þ þ la ðN b~ a 1Þ V~ kas ðckyas fas þ fas
lLðRÞ b~ LðRÞ ¼ i
Fig. 1. Schematic picture of serial double quantum dot system coupled to FM leads. U is interdot Coulomb repulsion in DQD, tc is interdot coupling and Vkas(a ¼ L, R, s ¼ m,k) represents dot-lead tunnelling for electron with spin s.
0 ε/ΤΚ = 0 ε/ΤΚ = 0 ε/ΤΚ =
as
X y ðfLs fRs þ fRys fLs Þ
By means of the equation of motion method, we can obtain the following self-consistent equations: X Z de 2 1 o GL,LðR,RÞ b~ LðRÞ ¼ i s ðeÞ þ 2 4 p s
lead R
0.30
y e~ a fas fas þ t~ c
ð2Þ
U dot R
X
X
kas
VkR
dot L
lead L
ekas ckyas ckas þ
kas
strength p given by p ¼ ðGas Gas Þ=ðGas þ Gas Þ (0 rpr1); here GLm ¼ GRm ¼ ð1 þ pÞG0 and GLk ¼ GRk ¼ ð1pÞG0 are for the parallel (P) leadpolarization alignment configuration, while GLm ¼ GRk ¼ ð1 þpÞG0 and GLk ¼ GRm ¼ ð1pÞG0 are for the antiparallel (AP) lead-polarization alignment configuration. G0 describes the coupling between the QD and the electrode without internal magnetization and is used as the unit of energy. According to the slave-boson mean-field approximation [11], bosonpfields by their mean values, namely, ffiffiffiffi are approximated pffiffiffiffi ba ðtÞ= N- ba ðtÞ = N ¼ b~ a , V~ kas ¼ Vkas b~ a , t~ c ¼ tc b~ L b~ R , and
V
X
0
ε/ΤΚ = 0 0
ε/ΤΚ = 0.75
0.4
0
ε/ΤΚ = 1.5 0
ε/ΤΚ = 3
0.3
0.08
0.2
0.06 0.04
0.1
0.02
0.0
0.00 0
1
2
3
4 U
5
6
7
0
1
2
3
4
5
6
7
U
Fig. 2. Density and transmission probability of states of left and right dot with P configuration, calculated for various values of energy e (e ¼ 0,0:75 103 G0 , 1:5 103 G0 and 3 103 G0 ) in the equilibrium case. (a) tc=0.3, (b) tc=1.5, (c) tc=1.5, and (d) tc=1.5.
S.-q. Wu et al. / Physica B 406 (2011) 749–755
1.0
1.0
Tk
0.9 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0 1
2
3 U
4
5
Tk
0.9
0
751
6
0.0
0
1
2
3
4
5
6
U
Fig. 3. Occupation number and Kondo temperature of double-quantum-dot (DQD) system with P configuration, calculated for various values of interdot Coulomb repulsion U in equilibrium case. (a) tc=0.3, and (b) tc=1.5.
0.30
0.18
P = 0.0 U = 0.0 P = 0.3 U = 0.0 P = 0.3 U = 1.5
0.25
P = 0.3 U = 3.0
P = 0.3 U = 0.0
0.14
P = 0.3 U = 6.0
P = 0.3 U = 3.0
0.12 ρ↑ (ε)
0.20 ρ↑ (ε)
P = 0.0 U = 0.0
0.16
0.15
0.10 0.08 0.06
0.10
0.04 0.05
0.02 0.00
0.00 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0 ε/ΤΚ
1
2
3
4
5
6
0 ε/ΤΚ
0.40
P = 0.0 U = 0.0 P = 0.3 U = 0.0 P = 0.3 U = 1.5 P = 0.3 U = 3.0
0.35 0.30
0.24
P = 0.0 U = 0.0
0.22
P = 0.3 U = 0.0 P = 0.3 U = 1.5
0.20
P = 0.3 U = 3.0
0.18 0.16 ρ↓ (ε)
0.25 ρ↓ (ε)
0
0.20
0.14 0.12
0.15
0.10
0.10
0.06
0.08 0.04
0.05
0.02
0.00
0.00 -6
-5
-4
-3
-2
-1
0 0 ε/ΤΚ
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0
1
2
3
0 ε/ΤΚ
Fig. 4. DOS for up- and down-spin electrons of dots in case of (a and b) tc ¼ 0.3 and (c and d) tc ¼1.5 with P configuration in equilibrium case.
4
5
6
752
S.-q. Wu et al. / Physica B 406 (2011) 749–755
in Eq. (3), we obtain the current I through this system Z 2eG0 X I¼ deðfL ðeÞfR ðeÞÞTs ðeÞ h s
as e0 ¼ 3.5G0 (Kondo regime with the Kondo temperature TK0 ¼ 103 G0 ). We choose two typical values—tc ¼ 0.3 for the weak tunnelling coupling regime and tc ¼1.5 for the strong tunnelling coupling regime.
ð5Þ
Here the transmission probability is given by 2 ~L ~R 2t~ c G sGs Ts ðeÞ ¼ 2 L ~ Þðee~ R þiG ~ R Þt~ 2 ðee~ L þiG s s c
ð6Þ 3. Results and discussion
While the density of states (DOS) for the dot a(a ¼L or R) is obtained as 1
2
ra, s ðeÞ ¼ b~ a ImGra, as ðeÞ p Here Gra, as ðeÞ is the retarded Green function and is given by
At first, in Fig. 2(a) and (b), we display the variation of the DOS of dots and the transmission probability with U at p¼ 0. Obviously, the DOS of dots and the transmission probability is suppressed when increasing the value of U for various electron energies (e). With the increase in U, the DOS of dots and the transmission probability then decrease to a plateau, and so on. But, the DOS of dots is more strongly suppressed in strong tunnelling coupling regime than in the weak tunnelling coupling regime, while the transmission probability is more strongly suppressed in the weak tunnelling coupling regime than in the strong tunnelling coupling regime, and all line shapes of the DOS of dots and the transmission probability exhibit steps. In addition, the further the value of e from the Fermi
ð7Þ
RðLÞ
GrL,LðR,RÞs ðeÞ ¼
~ ðee~ RðLÞ þ iG s Þ L ~ ~ R Þt~ 2 ðee~ L þ iG s Þðee~ R þ iG s c
ð8Þ
In our calculation, the temperature is taken as zero for the equilibrium and non-equilibrium cases, while the bandwidth of the leads is taken as D ¼60G0 and the bare level for two dots is fixed
0.30
0.18
P = 0.0 U = 0.0
0.25
P = 0.3 U = 1.5
P = 0.3 U = 0.0 P = 0.3 U = 3.0
0.14
P = 0.3 U = 3.0
0.20
ρL↑ (ε), ρR↓ (ε)
ρL↑ (ε), ρR↓ (ε)
P = 0.0 U = 0.0
0.16
P = 0.3 U = 0.0
0.15 0.10
P = 0.3 U = 6.0
0.12 0.10 0.08 0.06 0.04
0.05
0.02 0.00
0.00 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-6
6
-5
-4
-3
-2
-1
0 ε/ΤΚ
1
2
3
4
5
6
0 ε/ΤΚ
0.40
0.20
P = 0.0 U = 0.0 P = 0.3 U = 1.5
P = 0.3 U = 0.0 P = 0.3 U = 3.0
0.16
P = 0.3 U = 3.0
0.30
P = 0.0 U = 0.0
0.18
P = 0.3 U = 0.0
0.35
P = 0.3 U = 6.0
0.14 ρR↑ (ε), ρL↓ (ε)
ρR↑ (ε), ρL↓ (ε)
0
0.25 0.20 0.15
0.12 0.10 0.08 0.06
0.10
0.04
0.05
0.02
0.00
0.00 -6
-5
-4
-3
-2
-1
0 0 ε/ΤΚ
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0
1
2
0 ε/ΤΚ
Fig. 5. DOS of left and right dots at various U: tc ¼ 0.3 (a and b) and tc ¼ 1.5 (c and d) with AP configuration in equilibrium case.
3
4
5
6
S.-q. Wu et al. / Physica B 406 (2011) 749–755
energy (e ¼0), the stronger the DOS of dots, and the transmission probability are suppressed. The variation of the DOS of dots and the transmission probability with U can be explained as follows. At first, just as shown in Fig. 3, with the increase in U, the interdot Coulomb blockade effectively reduces the occupation number of dots, which leads to a weak Kondo resonance, and also the variation in the occupation number and the Kondo temperature of this system with U exhibits steps. Thus, the DOS of dots and the transmission probability are suppressed due to the interdot Coulomb blockade when increasing the value of U. Secondly, in the weak tunnelling coupling regime, electrons on an individual dot are quantized and the Kondo resonance is formed in each dot and lead, and so the tunnelling is dominated by hopping between two Kondo states, while in the strong tunnelling regime, the electrons are delocalized over two dots, and two dots are coupled coherently and form an artificial molecule. Therefore, the DOS of dots is larger in the weak tunnelling coupling regime than in the strong tunnelling coupling regime, and also it is easier for electrons to tunnel through the DQD system in the strong tunnelling coupling regime than in the weak tunnelling coupling regime.
753
To further describe the effect of the interdot Coulomb repulsion U on Kondo resonances in the series-coupled double quantum dot, we plot the variation in the DOS of dots with e in the equilibrium case (V¼0) and the P configuration for various values of U and p in Fig. 4. When U ¼0 and p¼0, in the weak tunnelling coupling regime (Fig. 4(a) and (b)), the Kondo resonance is formed in each dot and lead, and the tunnelling is dominated by hopping between the two Kondo states. But in the strong tunnelling coupling regime(Fig. 4(c) and (d)), two dots are coupled coherently and form an artificial molecule, and the bonding and antibonding orbits are formed by the coherent superposition of the Kondo states of each dot. So the Kondo resonance peak splits into two peaks. However, when p a0, the Kondo resonance width for up-spin electrons increases with the increase in p, while the width of the Kondo resonance for downspin electrons becomes smaller with increase in p. On the other hand, when increasing the value of the interdot Coulomb repulsion, the original Kondo peaks for up spins are compressed (Fig. 4(a) and (c); U¼1.5), and the width of the Kondo resonance for both up-spin electrons becomes smaller. Similarly, the Kondo resonances are sharpened for down spins with the increase in U (Fig. 4(b) and (d); U¼1.5), and the distance of two peaks diminish. When changing
0.16 0.25
0.12
0.20
0.10
0.15
ρL↓ (ε)
ρL↑ (ε)
0.14
0.08 0.06 0.04 0.02
0.10
V=0U=0
V=0U=0
0
V = 3TK U = 0 V=
3TK0
U=
0.05
50TK0
V = 3TK0 U = 50TK0
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0
V = 3TK U = 0
0.00 4
5
6
7
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
8
0 ε/ΤΚ
1
2
3
4
5
6
7
8
4
5
6
7
8
0 ε/ΤΚ
0.16 0.25
0.12
0.20
0.10
0.15
ρR↓ (ε)
ρR↑ (ε)
0.14
0.08 0.06
V=0U=0
0.05
0
0.04 0.02
V = 3TK U = 0
-8 -7 -6 -5 -4 -3 -2 -1 0 0 ε/ΤΚ
U=
100TK0
1
2
3
V=0U=0 0
V = 3TK U = 0 V = 3TK0 U = 50TK0
V = 3TK0 U = 50TK0 V=3TK0
0.10
0.00 4
5
6
7
8
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0 ε/ΤΚ
Fig. 6. DOS for left and right dots at p ¼0.3 with P configuration in non-equilibrium case: tc ¼1.5 for (a and b) up- and (c and d) down-spin electrons.
754
S.-q. Wu et al. / Physica B 406 (2011) 749–755
anti-parallel configuration as shown in Fig. 5. In this case, the DOS for up- and down-spins in leads are identical rLm + rRm ¼ rLk + rRk, and the spin current of electrons cannot be polarized, and up-spin electrons make the same contribution to the transmission probability as down-spin electrons do. Thus when increasing the value of p and U, the original up-spin (down-spin) peaks becomes broad and compressed for the left (right) dot (Fig. 5(a) and (c)), which are the same as those for upspin peaks in the parallel configuration (Fig. 4(a) and (c) ). Similarly, the original down-spin (up-spin) peaks become smaller and sharper for the left (right) dot (Fig. 5(b) and(d)), which are the same as those for down-spin (up-spin) peaks in the parallel configuration (Fig. 4(b) and (d)). In Fig. 6, the DOS of two dots with the P configuration in the nonequilibrium case are presented. When V¼0, the DOS for up-spin electrons shows a broad hump, while the DOS for down-spin electrons has sharp peaks with small splitting. With the increase in V, the Kondo resonance of the left (right) dot shifts, following the change in the chemical potential of the left (right) lead. Obviously, we can see that the width of the original symmetrical double peaks becomes smaller and dissymmetrical. In the left dot, the value of
the value of U, the resonance width becomes smaller and double peaks become more obscured (U¼3.0 and U¼ 6.0). These differences in the DOS of dots can be interpreted as follows. When the interdot Coulomb repulsion is ignored, the double occupancy of the DQD is allowable. But when U is large enough so that it cannot be ignored, the effect of Coulomb repulsion unfold slowly and the DQD cannot accommodate more than one electron. In this way, the original tunnelling of electrons through the DQD is suppressed, and electrons can go through this system only one by one [9–12]. In the strong interdot Coulomb repulsion regime, not only each of the dots has only one electron, but also there are no states of simultaneous occupation of two dots just as shown in Fig. 3. When p a0, in the P configuration, there is an overall asymmetry for up- and down-spins, namely, rLm + rRm 4 rLk + rRk, and the hybridization for up-spins is larger than for down-spins (Gam 4 Gak) because of the spin-dependent DOS in the leads. Thus, the average occupation of the QD with up-spins is larger than these with down-spins (ndm 4 ndk). All these enhance the transmission of up-spin electrons through the system, while down-spin electron transport through this system is suppressed. This changes for the
0.22
0.18
0.20
0.16
0.18 0.16
0.12
0.14
0.10
0.12
ρL↓ (ε)
ρL↑ (ε)
0.14
0.08
0.08
0.06
V=0U=0
0.04
3TK0
0.02 0.00
0.10
V=
0.06
U=0
0.04
V = 3TK0 U = 50TK0
0.02
V = 3T0K U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0.00 4
5
6
7
V=
3TK0
0
U = 100TK0
1
2
3
4
5
6
7
8
5
6
7
8
0.18
0.20
0.16
0.18
0.14
0.16
0.12
0.14 0.12
ρR↓ (ε)
ρR↑ (ε)
0
V = 3TK U = 50TK
0 ε/ΤΚ
0.22
0.10 0.08 V=0U=0
0.04
V = 3TK U = 0 0
0.04
0
V = 3TK U = 50TK
0.02
V = 3TK0 U = 100TK0
0 ε/ΤΚ
1
0.08
2
V=0U=0 0
0
-8 -7 -6 -5 -4 -3 -2 -1 0
0.10
0.06
0.06
0.00
0
V = 3TK U = 0
-8 -7 -6 -5 -4 -3 -2 -1 0
8
0 ε/ΤΚ
0.02
V=0U=0
3
4
5
6
7
8
0.00
V = 3TK U = 0 V=
0 3TK
0
U = 50TK
0
0
V = 3TK U = 100TK
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
4
0 ε/ΤΚ
Fig. 7. DOS for left and right dots at p ¼ 0.3 with AP configuration in non-equilibrium case: tc ¼1.5 for (a and b) up- and (c and d) down-spin electrons.
S.-q. Wu et al. / Physica B 406 (2011) 749–755
the left peak is enhanced and the peak becomes sharp, while the right peak is suppressed and the peaks become obscured (spin- up and spin-down) (Fig. 6(a) and (c)). On the contrary, in the right dot, the value of the right peak is enhanced and the peak becomes sharp, while the left peak is suppressed and the peak becomes obscured (Fig. 6(b) and (d)). Furthermore, when U ¼ 50TK0 , we surprisingly find that the width of the original dissymmetrical double peaks becomes broader in each dot (Fig. 6; U a0). By increasing the value of U, the above phenomena become more obvious. These phenomena are opposite to those in the equilibrium case. This happens because the DQD is regarded as a single quantum dot with two levels when the interdot coupling is strong and the effect of the interdot Coulomb repulsion in the DQD is sufficiently large [9,10]. At this time, the DQD cannot accommodate more than one electron and the ordinary spin Kondo effect is suppressed. However, the interdot Coulomb repulsion will lead to enhanced interdot charge fluctuation between two quantum dots, which will induce the ‘‘interdot Kondo effect’’ [12–14], which is important in determining the transport properties of this system. Namely, in the low bias voltage regime, the tunnelling rate of electrons with spin into (out from) the DQ is enhanced due to the effect of the interdot Coulomb repulsion. Finally, the DOS of two dots with the AP configuration in the non-equilibrium case are presented in Fig. 7. The peak positions of rLs(e) and rRs(e) follow the change of the left or right chemical potential. The value of the left and right peaks are enhanced or suppressed in the original symmetrical double peaks of the left (right) dot (Fig. 7; V ¼ 3 TK0 ); the dissymmetrical double peaks become broader in each dot (Fig. 7; Ua0), which are the same as the P configuration in the non-equilibrium case.
results are consistent with other author’s work [6,7]. However, when Ua0, we find some new phenomena (see Figs. 2–6). In both P and AP configurations, the original Kondo resonance is compressed in the equilibrium case, but becomes enhanced in the nonequilibrium case; the interdot Coulomb repulsion in the DQD is one of the important parameters to control transport phenomena via the modified Kondo resonances and the rich physical behaviour of this system can be attributed to the interdot Coulomb repulsion. We hope that this work will encourage further efforts, both theoretically and experimentally, to probe the Coulomb interaction effects on quantum dot systems.
Acknowledgments This project was supported by the Scientific Research Funds of Education Department of Sichuan Province under Grant no. 09ZA090 and the Major Basic Research Project of Sichuan Province under Grant no. 2006J13-155. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
4. Conclusions In conclusion, when the effect of the interdot Coulomb repulsion in the DQD coupled to the FM leads in series is ignored, the above
755
[12] [13] [14]
K. Ono, D.G. Austing, Y. Tokura, S. Tarucha, Science 297 (2002) 1313. W.G. Van der Wiel, et al., Rev. Mod. Phys. 75 (2003) 1. R. Hanson, et al., Rev. Mod. Phys. 79 (2007) 1217. M.R. Galpin, et al., J. Phys.: Condens. Matter 18 (2006) 6571. ¨ W. Hofstetter, J. Konig, H. Scheller, Phys. Rev. Lett. 87 (2001) 156803. Y. Tanaka, N. Kawakami, J. Phys. Soc. Jpn. 73 (2004) 2795; Y. Tanaka, N. Kawakami, Phys. Rev. B 72 (2005) 085304. A. Georges, Y. Meir, Phys. Rev. Lett. 82 (1999) 3508. Y. Takazawa, Y. Imai, N. Kawakami, J. Phys. Soc. Jpn. 71 (2002) 2234. T. Aono, M. Eto, Phys. Rev. B 63 (1999) 125327. T. Pohjola, et al., Europhys. Lett. 40 (1997) 189. A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University, Cambridge, 1993. R. Sakano, N. Kawakami, Phys. Rev. B 72 (2005) 085303. L. Borda, et al., Phys. Rev. Lett. 90 (2003) 026602. U. Wilhelm, J. Weis, Physica E 6 (2000) 668; Q.F. Sun, H. Guo, Phys. Rev. B 66 (2002) 155308.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
The effect of the interdot Coulomb interaction on Kondo resonances in series-coupled double quantum dots Shao-quan Wu n, Jia-feng Chen, Guo-ping Zhao College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China
a r t i c l e in f o
abstract
Article history: Received 11 June 2010 Received in revised form 16 November 2010 Accepted 17 November 2010 Available online 30 November 2010
We theoretically investigate the effect of the interdot Coulomb repulsion on Kondo resonances in the series-coupled double quantum dot coupled to two ferromagnetic leads. The Hamiltonian of our system is solved by means of the slave-boson mean-field approximation, and the variation of the density of states, the transmission probability, the occupation number, and the Kondo temperature with the interdot Coulomb repulsion are discussed in the Kondo regime. The density of states is calculated for various interdot Coulomb repulsions with both parallel and antiparallel lead-polarization alignments. Our results reveal that the interdot Coulomb repulsion greatly influences the physical property of this system, and relevant underlying physics of this system is discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Interdot Coulomb interaction Kondo resonances Slave-boson mean-field Approximation Density of states
1. Introduction In recent years, double-quantum-dot (DQD) systems have attracted much attention because of their importance in physics and device applications [1–6]. In contrast to the single quantum dot (QD), DQD systems have richer physics. On the one hand, DQD systems provide an ideal model system for studying the two impurity effect, which has long been the highlight of condensedmatter physics. On the other hand, a series-coupled DQD can cross over from the weak tunnelling coupling regime to the strong tunnelling coupling regime by varying the dot–dot coupling strength. In the weak tunnelling coupling regime, when the temperature is below the Kondo temperature, the Kondo resonance is formed in each dot and lead, and the tunnelling is dominated by hopping between two Kondo states. On the contrary, in the strong tunnelling coupling regime, the electrons are delocalized over two dots, and two dots are coupled coherently and form an artificial molecule. Therefore, DQD systems have been suggested as possible candidates for building blocks of a quantum computer and other possible device applications. Recent theoretical and experiment studies have shown that coherent couplings between two dots can be fulfilled by adjusting relevant parameters properly when DQD systems are coupled to electrodes [6–9], where the Kondo resonance peak splits into two peaks as a result of the energy difference between bonding and antibonding states. Most of the previous studies on DQD systems have treated effects of the intradot
n
Corresponding author. E-mail address: [email protected] (S.-q. Wu).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.11.057
Coulomb repulsion but have ignored effects of the interdot Coulomb repulsion [6,7,9]. However, the interdot Coulomb repulsion is known to play a significant role in electronic transport through DQD systems.
2. Model and solution In this article, we investigate the effect of the interdot Coulomb repulsion on the Kondo resonance in the series-coupled DQD coupled to two ferromagnetic (FM) leads. The schematic diagram of our device is presented in Fig. 1. A DQD system can be described by a two-fold degenerate twoimpurity Anderson Hamiltonian with the interdot tunnelling. In the limit of the intradot Coulomb repulsion U0 -N, the Hamiltonian can be written in terms of auxiliary SB operators plus constraints [6–10] H¼
X
X
kas
as
ekas ckyas ckas þ
y eas fas fas þ
tc X y ðf bL byR fRs þfRys bR byL fLs Þ N s Ls
1 X y Vkas ðckyas bya fas þ fas ba ckas Þ þUð1byL bL Þð1byR bR Þ þ pffiffiffiffi N kas X X y þ la ð fas fas þ bya ba 1Þ a
ð1Þ
s
where ckyas ðckas Þ is the creation (annihilation) operator for an electron with energy ekas and spin s in the lead a(a ¼L, R), and dyas ðdas Þ is the creation (annihilation) operator of electrons in the left or right dot. According to the slave-boson representation,
750
S.-q. Wu et al. / Physica B 406 (2011) 749–755
y dyas ðdas Þ is replaced by dyas -fas ba ðdas -bya fas Þ, where ba(fas) is the slave-boson (pseudo-fermion) annihilation operator of an empty state (singly occupied state). Vkas(tc) is the tunnelling amplitude between dots and leads (between two dots), while the fifth term is used to describe effects of the interdot Coulomb repulsion and the Lagrange multiplier la is introduced to incorporate the constraint P y y s ¼ mk fas fas þ ba ba ¼ 1 imposed on slave particles in the sixth term. For simplicity, we only consider symmetric dots with P eLs ¼ eRs ¼ e0 and the line width function is Gas ðeÞ ¼ p kas Vkas 2 dðeekas Þ. We introduce the effective spin polarization
e~ a ¼ e0 þ la . The total Hamiltonian is now replaced by ~ ¼ H
tc
þ
2
ρ (ε)
0.18
ð3Þ
0
ε/ΤΚ = 0
0.16
0.75 1.5
0.14
3
0.12
0
ε/ΤΚ = 0.75 0
ε/ΤΚ = 1.5 0
ε/ΤΚ = 3
0.10
0.15
0.08 0.06
0.10
0.04
0.05
0.02 0.00
0.00 0.16
0
0.5
ε/ΤΚ = 0
0.14
0 ε/ΤΚ = 0.75 0 ε/ΤΚ = 1.5 0 ε/ΤΚ = 3
0.12 0.10 T (ε)
X Z de o ~2 ~2 ðee~ LðRÞ ÞGL,LðR,RÞ s ðeÞ þU b LðRÞ ð1N b RðLÞ Þ 4 p s
ð4Þ 2 a a ~ ~ Here G s ¼ b a Gs ða ¼ L,RÞ, and fa(e) is the Fermi distribution function. By using the renormalized parameters determined self-consistently
0
0.20
a
o of the Keldysh Green Here GL,LðR,RÞ E s ðeÞ is theD Fourier transform function Gao, aus ðttuÞ i fayus ðtuÞfas ðtÞ . The first equation represents the constraint imposed on the slave particles, while the second one is obtained from the stationary condition, namely, the boson field is time independent at the mean-field level. From the equation of motion of the operator fas [9–11], we have the explicit form of the Green function n h i o ~ LðRÞ ðee~ RðLÞ Þ2 þðG ~ RðLÞ Þ2 þ fRðLÞ ðeÞG ~ RðLÞ t~ 2 2i fLðRÞ ðeÞG s s s c o GL,LðR,RÞs ðeÞ ¼ 2 ~ L Þðee~ R þ iG ~ R Þt~ 2 ðee~ L þ iG s s c
ε/ΤΚ = 0
0.25
s
X 2 2 2 2 2 y ckas Þ þ Uð1Nb~ L N b~ R þ N 2 b~ L b~ R Þ þ la ðN b~ a 1Þ V~ kas ðckyas fas þ fas
lLðRÞ b~ LðRÞ ¼ i
Fig. 1. Schematic picture of serial double quantum dot system coupled to FM leads. U is interdot Coulomb repulsion in DQD, tc is interdot coupling and Vkas(a ¼ L, R, s ¼ m,k) represents dot-lead tunnelling for electron with spin s.
0 ε/ΤΚ = 0 ε/ΤΚ = 0 ε/ΤΚ =
as
X y ðfLs fRs þ fRys fLs Þ
By means of the equation of motion method, we can obtain the following self-consistent equations: X Z de 2 1 o GL,LðR,RÞ b~ LðRÞ ¼ i s ðeÞ þ 2 4 p s
lead R
0.30
y e~ a fas fas þ t~ c
ð2Þ
U dot R
X
X
kas
VkR
dot L
lead L
ekas ckyas ckas þ
kas
strength p given by p ¼ ðGas Gas Þ=ðGas þ Gas Þ (0 rpr1); here GLm ¼ GRm ¼ ð1 þ pÞG0 and GLk ¼ GRk ¼ ð1pÞG0 are for the parallel (P) leadpolarization alignment configuration, while GLm ¼ GRk ¼ ð1 þpÞG0 and GLk ¼ GRm ¼ ð1pÞG0 are for the antiparallel (AP) lead-polarization alignment configuration. G0 describes the coupling between the QD and the electrode without internal magnetization and is used as the unit of energy. According to the slave-boson mean-field approximation [11], bosonpfields by their mean values, namely, ffiffiffiffi are approximated pffiffiffiffi ba ðtÞ= N- ba ðtÞ = N ¼ b~ a , V~ kas ¼ Vkas b~ a , t~ c ¼ tc b~ L b~ R , and
V
X
0
ε/ΤΚ = 0 0
ε/ΤΚ = 0.75
0.4
0
ε/ΤΚ = 1.5 0
ε/ΤΚ = 3
0.3
0.08
0.2
0.06 0.04
0.1
0.02
0.0
0.00 0
1
2
3
4 U
5
6
7
0
1
2
3
4
5
6
7
U
Fig. 2. Density and transmission probability of states of left and right dot with P configuration, calculated for various values of energy e (e ¼ 0,0:75 103 G0 , 1:5 103 G0 and 3 103 G0 ) in the equilibrium case. (a) tc=0.3, (b) tc=1.5, (c) tc=1.5, and (d) tc=1.5.
S.-q. Wu et al. / Physica B 406 (2011) 749–755
1.0
1.0
Tk
0.9 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0 1
2
3 U
4
5
Tk
0.9
0
751
6
0.0
0
1
2
3
4
5
6
U
Fig. 3. Occupation number and Kondo temperature of double-quantum-dot (DQD) system with P configuration, calculated for various values of interdot Coulomb repulsion U in equilibrium case. (a) tc=0.3, and (b) tc=1.5.
0.30
0.18
P = 0.0 U = 0.0 P = 0.3 U = 0.0 P = 0.3 U = 1.5
0.25
P = 0.3 U = 3.0
P = 0.3 U = 0.0
0.14
P = 0.3 U = 6.0
P = 0.3 U = 3.0
0.12 ρ↑ (ε)
0.20 ρ↑ (ε)
P = 0.0 U = 0.0
0.16
0.15
0.10 0.08 0.06
0.10
0.04 0.05
0.02 0.00
0.00 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0 ε/ΤΚ
1
2
3
4
5
6
0 ε/ΤΚ
0.40
P = 0.0 U = 0.0 P = 0.3 U = 0.0 P = 0.3 U = 1.5 P = 0.3 U = 3.0
0.35 0.30
0.24
P = 0.0 U = 0.0
0.22
P = 0.3 U = 0.0 P = 0.3 U = 1.5
0.20
P = 0.3 U = 3.0
0.18 0.16 ρ↓ (ε)
0.25 ρ↓ (ε)
0
0.20
0.14 0.12
0.15
0.10
0.10
0.06
0.08 0.04
0.05
0.02
0.00
0.00 -6
-5
-4
-3
-2
-1
0 0 ε/ΤΚ
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0
1
2
3
0 ε/ΤΚ
Fig. 4. DOS for up- and down-spin electrons of dots in case of (a and b) tc ¼ 0.3 and (c and d) tc ¼1.5 with P configuration in equilibrium case.
4
5
6
752
S.-q. Wu et al. / Physica B 406 (2011) 749–755
in Eq. (3), we obtain the current I through this system Z 2eG0 X I¼ deðfL ðeÞfR ðeÞÞTs ðeÞ h s
as e0 ¼ 3.5G0 (Kondo regime with the Kondo temperature TK0 ¼ 103 G0 ). We choose two typical values—tc ¼ 0.3 for the weak tunnelling coupling regime and tc ¼1.5 for the strong tunnelling coupling regime.
ð5Þ
Here the transmission probability is given by 2 ~L ~R 2t~ c G sGs Ts ðeÞ ¼ 2 L ~ Þðee~ R þiG ~ R Þt~ 2 ðee~ L þiG s s c
ð6Þ 3. Results and discussion
While the density of states (DOS) for the dot a(a ¼L or R) is obtained as 1
2
ra, s ðeÞ ¼ b~ a ImGra, as ðeÞ p Here Gra, as ðeÞ is the retarded Green function and is given by
At first, in Fig. 2(a) and (b), we display the variation of the DOS of dots and the transmission probability with U at p¼ 0. Obviously, the DOS of dots and the transmission probability is suppressed when increasing the value of U for various electron energies (e). With the increase in U, the DOS of dots and the transmission probability then decrease to a plateau, and so on. But, the DOS of dots is more strongly suppressed in strong tunnelling coupling regime than in the weak tunnelling coupling regime, while the transmission probability is more strongly suppressed in the weak tunnelling coupling regime than in the strong tunnelling coupling regime, and all line shapes of the DOS of dots and the transmission probability exhibit steps. In addition, the further the value of e from the Fermi
ð7Þ
RðLÞ
GrL,LðR,RÞs ðeÞ ¼
~ ðee~ RðLÞ þ iG s Þ L ~ ~ R Þt~ 2 ðee~ L þ iG s Þðee~ R þ iG s c
ð8Þ
In our calculation, the temperature is taken as zero for the equilibrium and non-equilibrium cases, while the bandwidth of the leads is taken as D ¼60G0 and the bare level for two dots is fixed
0.30
0.18
P = 0.0 U = 0.0
0.25
P = 0.3 U = 1.5
P = 0.3 U = 0.0 P = 0.3 U = 3.0
0.14
P = 0.3 U = 3.0
0.20
ρL↑ (ε), ρR↓ (ε)
ρL↑ (ε), ρR↓ (ε)
P = 0.0 U = 0.0
0.16
P = 0.3 U = 0.0
0.15 0.10
P = 0.3 U = 6.0
0.12 0.10 0.08 0.06 0.04
0.05
0.02 0.00
0.00 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-6
6
-5
-4
-3
-2
-1
0 ε/ΤΚ
1
2
3
4
5
6
0 ε/ΤΚ
0.40
0.20
P = 0.0 U = 0.0 P = 0.3 U = 1.5
P = 0.3 U = 0.0 P = 0.3 U = 3.0
0.16
P = 0.3 U = 3.0
0.30
P = 0.0 U = 0.0
0.18
P = 0.3 U = 0.0
0.35
P = 0.3 U = 6.0
0.14 ρR↑ (ε), ρL↓ (ε)
ρR↑ (ε), ρL↓ (ε)
0
0.25 0.20 0.15
0.12 0.10 0.08 0.06
0.10
0.04
0.05
0.02
0.00
0.00 -6
-5
-4
-3
-2
-1
0 0 ε/ΤΚ
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0
1
2
0 ε/ΤΚ
Fig. 5. DOS of left and right dots at various U: tc ¼ 0.3 (a and b) and tc ¼ 1.5 (c and d) with AP configuration in equilibrium case.
3
4
5
6
S.-q. Wu et al. / Physica B 406 (2011) 749–755
energy (e ¼0), the stronger the DOS of dots, and the transmission probability are suppressed. The variation of the DOS of dots and the transmission probability with U can be explained as follows. At first, just as shown in Fig. 3, with the increase in U, the interdot Coulomb blockade effectively reduces the occupation number of dots, which leads to a weak Kondo resonance, and also the variation in the occupation number and the Kondo temperature of this system with U exhibits steps. Thus, the DOS of dots and the transmission probability are suppressed due to the interdot Coulomb blockade when increasing the value of U. Secondly, in the weak tunnelling coupling regime, electrons on an individual dot are quantized and the Kondo resonance is formed in each dot and lead, and so the tunnelling is dominated by hopping between two Kondo states, while in the strong tunnelling regime, the electrons are delocalized over two dots, and two dots are coupled coherently and form an artificial molecule. Therefore, the DOS of dots is larger in the weak tunnelling coupling regime than in the strong tunnelling coupling regime, and also it is easier for electrons to tunnel through the DQD system in the strong tunnelling coupling regime than in the weak tunnelling coupling regime.
753
To further describe the effect of the interdot Coulomb repulsion U on Kondo resonances in the series-coupled double quantum dot, we plot the variation in the DOS of dots with e in the equilibrium case (V¼0) and the P configuration for various values of U and p in Fig. 4. When U ¼0 and p¼0, in the weak tunnelling coupling regime (Fig. 4(a) and (b)), the Kondo resonance is formed in each dot and lead, and the tunnelling is dominated by hopping between the two Kondo states. But in the strong tunnelling coupling regime(Fig. 4(c) and (d)), two dots are coupled coherently and form an artificial molecule, and the bonding and antibonding orbits are formed by the coherent superposition of the Kondo states of each dot. So the Kondo resonance peak splits into two peaks. However, when p a0, the Kondo resonance width for up-spin electrons increases with the increase in p, while the width of the Kondo resonance for downspin electrons becomes smaller with increase in p. On the other hand, when increasing the value of the interdot Coulomb repulsion, the original Kondo peaks for up spins are compressed (Fig. 4(a) and (c); U¼1.5), and the width of the Kondo resonance for both up-spin electrons becomes smaller. Similarly, the Kondo resonances are sharpened for down spins with the increase in U (Fig. 4(b) and (d); U¼1.5), and the distance of two peaks diminish. When changing
0.16 0.25
0.12
0.20
0.10
0.15
ρL↓ (ε)
ρL↑ (ε)
0.14
0.08 0.06 0.04 0.02
0.10
V=0U=0
V=0U=0
0
V = 3TK U = 0 V=
3TK0
U=
0.05
50TK0
V = 3TK0 U = 50TK0
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0
V = 3TK U = 0
0.00 4
5
6
7
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
8
0 ε/ΤΚ
1
2
3
4
5
6
7
8
4
5
6
7
8
0 ε/ΤΚ
0.16 0.25
0.12
0.20
0.10
0.15
ρR↓ (ε)
ρR↑ (ε)
0.14
0.08 0.06
V=0U=0
0.05
0
0.04 0.02
V = 3TK U = 0
-8 -7 -6 -5 -4 -3 -2 -1 0 0 ε/ΤΚ
U=
100TK0
1
2
3
V=0U=0 0
V = 3TK U = 0 V = 3TK0 U = 50TK0
V = 3TK0 U = 50TK0 V=3TK0
0.10
0.00 4
5
6
7
8
V=3TK0 U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0 ε/ΤΚ
Fig. 6. DOS for left and right dots at p ¼0.3 with P configuration in non-equilibrium case: tc ¼1.5 for (a and b) up- and (c and d) down-spin electrons.
754
S.-q. Wu et al. / Physica B 406 (2011) 749–755
anti-parallel configuration as shown in Fig. 5. In this case, the DOS for up- and down-spins in leads are identical rLm + rRm ¼ rLk + rRk, and the spin current of electrons cannot be polarized, and up-spin electrons make the same contribution to the transmission probability as down-spin electrons do. Thus when increasing the value of p and U, the original up-spin (down-spin) peaks becomes broad and compressed for the left (right) dot (Fig. 5(a) and (c)), which are the same as those for upspin peaks in the parallel configuration (Fig. 4(a) and (c) ). Similarly, the original down-spin (up-spin) peaks become smaller and sharper for the left (right) dot (Fig. 5(b) and(d)), which are the same as those for down-spin (up-spin) peaks in the parallel configuration (Fig. 4(b) and (d)). In Fig. 6, the DOS of two dots with the P configuration in the nonequilibrium case are presented. When V¼0, the DOS for up-spin electrons shows a broad hump, while the DOS for down-spin electrons has sharp peaks with small splitting. With the increase in V, the Kondo resonance of the left (right) dot shifts, following the change in the chemical potential of the left (right) lead. Obviously, we can see that the width of the original symmetrical double peaks becomes smaller and dissymmetrical. In the left dot, the value of
the value of U, the resonance width becomes smaller and double peaks become more obscured (U¼3.0 and U¼ 6.0). These differences in the DOS of dots can be interpreted as follows. When the interdot Coulomb repulsion is ignored, the double occupancy of the DQD is allowable. But when U is large enough so that it cannot be ignored, the effect of Coulomb repulsion unfold slowly and the DQD cannot accommodate more than one electron. In this way, the original tunnelling of electrons through the DQD is suppressed, and electrons can go through this system only one by one [9–12]. In the strong interdot Coulomb repulsion regime, not only each of the dots has only one electron, but also there are no states of simultaneous occupation of two dots just as shown in Fig. 3. When p a0, in the P configuration, there is an overall asymmetry for up- and down-spins, namely, rLm + rRm 4 rLk + rRk, and the hybridization for up-spins is larger than for down-spins (Gam 4 Gak) because of the spin-dependent DOS in the leads. Thus, the average occupation of the QD with up-spins is larger than these with down-spins (ndm 4 ndk). All these enhance the transmission of up-spin electrons through the system, while down-spin electron transport through this system is suppressed. This changes for the
0.22
0.18
0.20
0.16
0.18 0.16
0.12
0.14
0.10
0.12
ρL↓ (ε)
ρL↑ (ε)
0.14
0.08
0.08
0.06
V=0U=0
0.04
3TK0
0.02 0.00
0.10
V=
0.06
U=0
0.04
V = 3TK0 U = 50TK0
0.02
V = 3T0K U = 100TK0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
0.00 4
5
6
7
V=
3TK0
0
U = 100TK0
1
2
3
4
5
6
7
8
5
6
7
8
0.18
0.20
0.16
0.18
0.14
0.16
0.12
0.14 0.12
ρR↓ (ε)
ρR↑ (ε)
0
V = 3TK U = 50TK
0 ε/ΤΚ
0.22
0.10 0.08 V=0U=0
0.04
V = 3TK U = 0 0
0.04
0
V = 3TK U = 50TK
0.02
V = 3TK0 U = 100TK0
0 ε/ΤΚ
1
0.08
2
V=0U=0 0
0
-8 -7 -6 -5 -4 -3 -2 -1 0
0.10
0.06
0.06
0.00
0
V = 3TK U = 0
-8 -7 -6 -5 -4 -3 -2 -1 0
8
0 ε/ΤΚ
0.02
V=0U=0
3
4
5
6
7
8
0.00
V = 3TK U = 0 V=
0 3TK
0
U = 50TK
0
0
V = 3TK U = 100TK
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
4
0 ε/ΤΚ
Fig. 7. DOS for left and right dots at p ¼ 0.3 with AP configuration in non-equilibrium case: tc ¼1.5 for (a and b) up- and (c and d) down-spin electrons.
S.-q. Wu et al. / Physica B 406 (2011) 749–755
the left peak is enhanced and the peak becomes sharp, while the right peak is suppressed and the peaks become obscured (spin- up and spin-down) (Fig. 6(a) and (c)). On the contrary, in the right dot, the value of the right peak is enhanced and the peak becomes sharp, while the left peak is suppressed and the peak becomes obscured (Fig. 6(b) and (d)). Furthermore, when U ¼ 50TK0 , we surprisingly find that the width of the original dissymmetrical double peaks becomes broader in each dot (Fig. 6; U a0). By increasing the value of U, the above phenomena become more obvious. These phenomena are opposite to those in the equilibrium case. This happens because the DQD is regarded as a single quantum dot with two levels when the interdot coupling is strong and the effect of the interdot Coulomb repulsion in the DQD is sufficiently large [9,10]. At this time, the DQD cannot accommodate more than one electron and the ordinary spin Kondo effect is suppressed. However, the interdot Coulomb repulsion will lead to enhanced interdot charge fluctuation between two quantum dots, which will induce the ‘‘interdot Kondo effect’’ [12–14], which is important in determining the transport properties of this system. Namely, in the low bias voltage regime, the tunnelling rate of electrons with spin into (out from) the DQ is enhanced due to the effect of the interdot Coulomb repulsion. Finally, the DOS of two dots with the AP configuration in the non-equilibrium case are presented in Fig. 7. The peak positions of rLs(e) and rRs(e) follow the change of the left or right chemical potential. The value of the left and right peaks are enhanced or suppressed in the original symmetrical double peaks of the left (right) dot (Fig. 7; V ¼ 3 TK0 ); the dissymmetrical double peaks become broader in each dot (Fig. 7; Ua0), which are the same as the P configuration in the non-equilibrium case.
results are consistent with other author’s work [6,7]. However, when Ua0, we find some new phenomena (see Figs. 2–6). In both P and AP configurations, the original Kondo resonance is compressed in the equilibrium case, but becomes enhanced in the nonequilibrium case; the interdot Coulomb repulsion in the DQD is one of the important parameters to control transport phenomena via the modified Kondo resonances and the rich physical behaviour of this system can be attributed to the interdot Coulomb repulsion. We hope that this work will encourage further efforts, both theoretically and experimentally, to probe the Coulomb interaction effects on quantum dot systems.
Acknowledgments This project was supported by the Scientific Research Funds of Education Department of Sichuan Province under Grant no. 09ZA090 and the Major Basic Research Project of Sichuan Province under Grant no. 2006J13-155. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
4. Conclusions In conclusion, when the effect of the interdot Coulomb repulsion in the DQD coupled to the FM leads in series is ignored, the above
755
[12] [13] [14]
K. Ono, D.G. Austing, Y. Tokura, S. Tarucha, Science 297 (2002) 1313. W.G. Van der Wiel, et al., Rev. Mod. Phys. 75 (2003) 1. R. Hanson, et al., Rev. Mod. Phys. 79 (2007) 1217. M.R. Galpin, et al., J. Phys.: Condens. Matter 18 (2006) 6571. ¨ W. Hofstetter, J. Konig, H. Scheller, Phys. Rev. Lett. 87 (2001) 156803. Y. Tanaka, N. Kawakami, J. Phys. Soc. Jpn. 73 (2004) 2795; Y. Tanaka, N. Kawakami, Phys. Rev. B 72 (2005) 085304. A. Georges, Y. Meir, Phys. Rev. Lett. 82 (1999) 3508. Y. Takazawa, Y. Imai, N. Kawakami, J. Phys. Soc. Jpn. 71 (2002) 2234. T. Aono, M. Eto, Phys. Rev. B 63 (1999) 125327. T. Pohjola, et al., Europhys. Lett. 40 (1997) 189. A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University, Cambridge, 1993. R. Sakano, N. Kawakami, Phys. Rev. B 72 (2005) 085303. L. Borda, et al., Phys. Rev. Lett. 90 (2003) 026602. U. Wilhelm, J. Weis, Physica E 6 (2000) 668; Q.F. Sun, H. Guo, Phys. Rev. B 66 (2002) 155308.