1. IntroductionSemiconductor quantum dot system is considered as a promising candidate for quantum computation due to its ability to confine single electron at discrete energy levels and fine manipulation of parameter space.[1,2] During the past two decades, tremendous progress has been made in understanding the charge and spin transport for single and double quantum dot structures both in experimental and theoretical aspects.[3–9] Recently, the research interest has focused on the triple quantum dot (TQD) systems. Experimentally, TQDs are used to realize multielectron hybrid qubit,[10–13] spectral absorbers,[14,15] charge detectors,[16] and current rectifiers.[17,18] While in the theoretical field, TQD structure offers the potential to study some new fascinating phenomena at low temperatures, including the Kondo effect,[19–22] the Fermi liquid theory,[23,24] the quantum interference,[25,26] and various kinds of quantum phase transitions (QPT).[27–31]
TQD structure could be organized in serial,[32,33] parallel,[34–36] or triangular[37–40] spatial configuration, which may naturally result in different quantum behaviors. Among these configurations, the parallel TQD (PTQD) system is a basic one, which has also inspired a wealth of research. Although many works have been done related to the PTQD model, the charge transport and phase transition are less studied beyond the Kondo regime, which, in all cases, are important for the dynamic and static properties, even for the potential application in nanodevices. In this paper, we consider a PTQD device connected to two conduction leads (see Fig. 1(a)) deviated from the Kondo regime. With the usage of Wilson’s numerical renormalization group (NRG) method, we find the inter-dot hopping t may lead to some interesting quantum behaviors in the QPT and charge transport at the strongly correlated limit, quite different from the Kondo regime. More importantly, these behaviors depend closely on the initial electron number on the dots, which is usually adjusted by an external gate voltage. For instance, when the initial charge number of the dots 
, the linear conductance G reveals a peak shape as t increases. An orbital spin-1/2 Kondo effect could be found between the conduction leads and the bonding orbital. When t is large enough, dot 1 (3) and dot 2 form a spin singlet, while dot 3 (1) is empty. In the regime 
, G could be considered as a part of the above case and the quantum state is similar to the empty case as t is large enough. When the initial charge number 
, we find the ground state is a magnetic-frustration phase and G is nearly zero when t is absent. This phase is sensitive to t, and is replaced by another mixture state even t is just 
. A first order QPT could be identified both in dynamic and static physical quantities.
We organize the remaining sections as follows. In section 2, we present the model Hamiltonian, and the calculation formulas. In section 3, we describe the numerical result and discussion. Finally, a conclusion is given in section 4.
2. Model, method, and formulasThe schematic view of the PTQD structure connected to two conduction leads (source (S) and drain (D)) is presented in Fig. 1(a), and the second quantized Hamiltonian for the whole system is written as
Here,
HS and
HD correspond to the noninteracting source and drain leads respectively with
(
α = S, D):
, where
is the creation (annihilation) operator for an electron with wave vector
k, spin
σ (
or
) in lead
α, and
denotes the single-electron energy.
Hdots describes the energy of the PTQD:
Here,
is the annihilation (creation) operator for the electrons on dot site
i (
i = 1, 2, 3).
is the number operator.
ϵ and
U are the energy level and the intra-orbital Coulomb repulsion respectively, while
describes the inter-dot hopping between dots
i and
j. To avoid too many parameters, we choose
. Finally, the last term in Eq. (
1) illustrates the tunneling coupling between conduction bands and dot levels:
. Here,
describe the couplings of the
electron states belonging to lead
α to the
i-th dot site. We assume
is symmetric with respect to the source and drain leads.
In the present work, we are mainly interested in the QPT and the charge transport of the PTQD device in the strong correlation regime, thus a sophisticated many-body technique must be used. Therefore, we use the NRG method,[41–43] to solve Eq. (1), which is an ideal technique to calculate both static and dynamical properties of the Anderson-type model. We first choose a constant density of states for the conduction leads 
, with 2D being the band width, and D is chosen as the energy unit in the following discussions. We also approximate a constant hybridization strength between the conduction band and quantum dots 
. In our NRG calculation, Λ, the renormalization parameter, is set to be 1.8 ∼2.4, and the number of the low-lying energy states kept at each iteration is about 3000.
The linear conductance through the PTQD G is calculated by the Meir–Wingreen’s formula[44]
where
is the Fermi function, and
T(
ω) is the transmission coefficient, which is defined as
Here,
is related to the retarded dot Green’s function
, and the Fermi function is chosen as
, hence the linear conductance at zero temperature in the limit of zero bias can be written as
The spin–spin correlation between dots i and j is labeled as 
, which could be calculated as follows in the NRG steps:
with
and
.
3. Results and discussionsIn the following discussion, we fix Γ = 0.01 and U = 0.1. Since 
, the electron–electron interaction on the PTQD becomes significantly important, and the system is in the strong correlation limit. When ϵ sweeps downwards, the total electron number on the dots 
increases from empty to full, i.e., 
to 6.0. In the regime of 
, three dots are singly occupied due to strong U, and a spin-3/2 Kondo effect could be found, resulting from the process of a local spin magnitude 1/2 is screened by the conduction leads. Thus this regime is considered as the Kondo regime,[45] which has inspired a wealth of research in the past decade. Since the effect of the inter-dot hopping t in this regime is referred by some previous works,[27] here we focus our attention on the QPT and charge transport affected by t deviate the Kondo regime.
3.1. Effect of t in the empty regimeLet us start our discussion when the quantum dot energy level ϵ is set to be in the unoccupied regime with 
. In Figs. 1(b)–1(c), we show the electron number 
on each dot, and the spin–spin correlation 
between dots i and j at zero temperature as functions of t with fixed ϵ = 0.1. Here, 
and 
when t sweeps, since dots 1 and 3 are identical. When t = 0, one may see 
and 
, since the PTQD is in the frozen-impurity fixed point with no residual spin. With increasing t, the charge number on each dot grows continuously, and 
decreases to negative. Because a negative t term in Eq. (2) reduces the total energy of the system, and favors an antiferromagnetic spin arrangement between neighboring dots with an effective strength 
. When t is large enough, 
, and 
, illustrating that dot 2 is occupied by a single electron, whereas only one of dots 1 and 3 is singly occupied, and the other one is empty. In this case, 
reaches to about −0.2, showing a spin singlet between dots 1 (3) and 2 is generated. Therefore, the many-body state of 
mainly includes configurations 
(or (
)) and (
) (or (
)), where dot 3 (1) is empty, while dots 1 (3) and 2 are singly occupied with antiparallel spin ordering. Here, it is notable that 
could not reach −0.75 of the ideal spin singlet, because the expected value of 
is unequal to 1.0. Besides, one also finds 
reaches to a larger negative value of −0.1, which can be attributed to the fact that in this regime, the Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions mediated by both the Kondo coupling between the leads and the dots and the inter-dot hopping t, which favor singly occupied states and parallel spin arrangement between different dots, become invalid. Therefore, in order to avoid the intra-orbital Coulomb repulsion and reduce the total energy, there are still some possibilities of dots 1 and 3 to form a negative spin–spin correlation. This behavior is similar to the case where the energy level goes through the intermediate valence regime in multi-quantum dot systems.[45–48]
Now we study the charge transport through the PTQD. In Fig. 2(a), we depict the transmission coefficient 
at zero temperature in terms of different t. When t = 0, 
, suggesting zero conductance (see Fig. 2(b)). The spectral weight of T(ω) locates in the regime 
, and a large peak is found at energy 
, which corresponds to the process of adding an electron on the bonding orbital as discussed in the following. With increasing t, the spectral weight moves to the direction 
. In this process, the conductance 
increases and reaches to its unitary limit of 1.0 at about t = 0.075. When t is large enough, the spectral weight locates in 
(e.g., t = 0.016 in Fig. 3(a)), and the conductance G decreases to 0.0 gradually. In this case, the PTQD is in a local moment formation, and the conductance is suppressed by the spin singlet between dots 1 (3) and 2, similar to the double quantum dot system with strong t in the Kondo regime.[49,50]
In order to get a clear understanding of the quantum behaviors above, we invite a canonical transformation of the three dot levels:
Under this transformation, the orbital levels are given by
,
, and
, respectively, thus
is the bonding orbital, while
and
are the anti-bonding orbitals. Besides, the
item turns out to be
, suggesting that only the bonding orbital hybridizes with the conduction leads. In Fig.
2(c), we show the electron number on the bonding and anti-bonding orbitals
at zero temperature as functions of
t. Here,
. One may see as
t increases,
keep at 0.0, while
increases gradually, since
decreases with increasing
t, while
grows larger (retains at
ϵ). Correspondingly, the peak in
moves toward the left, for
is merely decided by the local density states of the bonding orbital. When
t = 0.075,
, and the present model maps to a single impurity Anderson model. An orbital spin-1/2 Kondo effect could be found between the conduction leads and the bonding orbital, and the linear conductance
G reaches to its unitary limit (Fig.
2(b)). Here, the value of
t could be estimated as follows: when the bonding orbital is singly occupied, annihilating (creating) an electron on this orbital will cost energy
, which gives
t = 0.075, and is consistent with our NRG result. When
t is large enough, the bonding orbital is almost fully occupied, thus the electrons could not transmit the PTQD.
3.2. Effect of t in the singly occupied regimeNow we turn to the case when the PTQD is singly occupied with 
. Figure 3(a)–3(b) show 
, and 
as functions of t at zero temperature for fixed ϵ = −0.004. It is seen 
and 
when t = 0, indicating the many-body state of 
mainly includes configurations such as (1 0 0), (0 1 0), and (0 0 1), where one of the dots is singly occupied while the other two are empty. With increasing t, 
grows, while 
decreases continuously. When t is large enough, 
, while 
, similar to the case of ϵ = 0.1 with large t. Thus dot 1 (3) is empty, while dots 3 (1) and 2 are singly occupied with antiparallel spin ordering.
To understand the charge transport through the PTQD, we illustrate 
for various t, G, and 
as functions of t in Figs. 4(a)–4(c), respectively. One finds 
reaches to almost 1.0 of the unitary limit when t is absent. Because in this case, the charge number on the bonding orbital 
(Fig. 4(c)), and the orbital spin-1/2 Kondo effect could be found between the conduction leads and the bonding orbital. As t increases, the peak in 
moves toward the left and the conductance decreases to 0.0 gradually due to decreasing 
. In this process, 
grows correspondingly, then stabilizes at 2.0 when t is large enough. In Fig. 3(c), we also give 
as functions of t for ϵ = −0.002. It is seen the behaviors are similar to the case of ϵ = −0.004, illustrating the discussions above are robust for the regime 
.
3.3. Effect of t in the doubly occupied regimeIn this subsection, we focus on the situation where the PTQD is doubly occupied with initial charge numer 
. In Figs. 5(a)–5(b), we show 
and 
as functions of t with ϵ = −0.01. When t = 0, our NRG results show 
and 
, thus two of the dots are singly occupied, while the remaining one is nearly empty. The ground state of the system is considered as a magnetic-frustration phase,[45,48] and the antiferromagnetic spin–spin correlation is resulted from the following fact: comparing with the Kondo regime where 
, the possibilities of each dot being singly occupied is weakened in the regime 
. This tendency may result in a reduction of the RKKY interaction, which is mediated by the Kondo coupling between conduction leads and quantum dots and favors parallel spin alignment between different dots. Therefore, to reduce the total energy, the local spins are more likely to form an anti-parallel spin ordering on two singly occupied dots. However, this magnetic-frustration phase is sensitive to the inter-dot coupling t, which breaks the symmetry of the three dots. As shown in Figs. 5(a) and 5(b), even t is just 
, 
, and 
become different with 
and 
, accompanied with 
. Hence the electrons on dots 1 (3) and 2 are arranged anti-ferromagnetically. Besides, one notices 
, thus the local spins on dots 1 and 3 are likely to be ferromagnetic due to the RKKY interaction mediated by t. Here, 
and 
could not reach the ideal value of spin singlet and triplet respectively, for the ground state here is also frustrated, and the many-body states of 
may include configurations such as (
) (or (
)), (
) and (
). As t increases to about 
, one finds an abrupt change in 
and 
, indicating a first order QPT, e.g., 
drops to 0.42, and 
drops to −0.1. When t is large enough, e.g., t = 0.2, 
, 
, 
, and 
, thus dot 1 (3) is empty, while dots 3 (1) and 2 are singly occupied with antiparallel spin configuration, similar to the cases above.
Figure 6(a) and 6(b) describe 
for different t. It is seen when t is absent that there is a sharp dip at ω = 0, corresponding to zero conductance (see Fig. 6(c)), which could be attributed to the fact that in this case two singly occupied dots are arranged anti-parallelly, and the present PTQD model maps to a two impurity Anderson model with large inter-dot hopping and the Kondo peak is suppressed.[51] When t is turned on, e.g., t = 0.011, the dip is filled up continuously, and the charge transport is enhanced (Figs. 6(a) and 6(c)). In this process, the proportion of configuration (
) grows with increasing t, thus dot 1 (3) and dot 2 are more likely to form a spin singlet, and the remaining one contributes to the enhanced conductance. Correspondingly, the electrons on orbital 
increase to about 1.0. However, in this regime, G could not reach the unitary limit since orbital 
connected with 
through a hopping term 
, which arises during the canonical transformation above, i.e., Eq. (7). Since orbital 
is isolated, the present model could be considered as a side-coupled double dot structure, and the conductance decreases with growing inter-dot hopping.[52] When t exceeds the critical point 
, a broadened peak develops at the left side of 
, which moves toward the left when t grows continuously. In this process, 
and 
drop to zero, thus the PTQD model maps to a single impurity Anderson model. On the other hand, 
increases from 1.27 to 2.0 gradually, hence G decreases to 0.0 successively according to the Freidel sum rule.[53,54] It is worth noting that it is convenient for the critical point 
to be evaluated by the level of 
, i.e., with increasing t, the charge number 
drops from 1.0 to 0.0, the critical point is just the point where the singly-occupied energy level of 
equals to the empty level, which gives 
. In Fig. 5(c), we also show 
as functions of t for fixed ϵ = −0.008. One finds the behaviors are similar to ϵ = −0.01, thus our above discussions are robust in the regime 
.
For initial charge number 
, 5.0, and 6.0, the numerical results are similar to the cases 
, 1.0, and 0.0 respectively, due to particle-hole symmetry. Therefore, we neglect them in the present work.
4. ConclusionIn conclusion, we have studied the quantum phase transition and charge transport for a parallel triple dot device in the strongly correlated limit. We focus our attention on the effect of inter-dot hopping t beyond the Kondo regime, and we find the quantum behaviors depend closely on the initial electron number on the dots, which could be adjusted by the dot level through external gate voltage. For instance, when the initial charge number of the dots 
, it grows continuously with increasing t. When t is large enough, dot 1 (3) and dot 2 form a spin singlet, while dot 3 (1) is empty. The linear conductance G reveals a peak shape and reaches to almost its unitary limit at the point 
, where the bonding orbital is singly occupied and an orbital spin-1/2 Kondo effect could be found between the conduction leads and the bonding orbital. In the regime 
, G could be considered as a part of the above case and the quantum state is similar to the case 
as t is large enough. When the initial charge number 
, we find the ground state is a magnetic-frustration phase and G is nearly zero when t is absent. This phase is sensitive to t, and is replaced by another mixture state even though t is just 
. G increases due to the charge number of the bonding orbital grows to 1.0. When t increases to 
, a first order QPT could be found, and a large peak develops in the transmission coefficient 
. As t is large enough, one of dots 1 and 3 forms a spin singlet with dot 2, while the remaining one is empty, similar to 
and 0.0 with large t.
Our results could be used to describe the strongly correlated effects for triple level semiconductor or unimolecular electronic devices connected to the conduction band. We firmly believe that our work not only settled the underlying physical mechanism for the charge transport and phase transition for a PTQD system affected by the inter-dot hopping beyond the Kondo regime, but it may also provide appropriate guidelines for related experiments.
