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Capacitance and conductance of dots connected by quantum point contacts
ELSEVIER
Physica B 203 (1994) 432-439
Capacitance and conductance of dots connected by quantum point contacts Karsten Flensberg Mikroelektronik Centret, Danmarks Tekniske Universitet, DK-2800 Lyngby, Denmark
Abstract We study the transport properties of quantum dots and quantum point contacts in the Coulomb blockade regime and in the limit where the quantum point contact has nearly fully transmitting channels. We argue that the properties of the charge fluctuations through the contact can be dealt with using a one-dimensional representation and study the scaling behavior of the model. We find a cross-over between a low-energy regime with Coulomb blockade to a high-energy regime where quantum charge fluctuations are dominant. The cross-over energy defines an effective charging energy given by U* ~ U[1 - T] u/2, where U is the bare charging energy, T is the transmission coefficient, and N is the number of channels given by twice the number of quantum point contacts connected to the dot.
1. Introduction Transport measurements in devices such as quantum dots exhibit the so-called Coulomb blockade (CB) at low temperatures. This happens when the thermal energy is less than the charging energy associated with adding a single electron onto the dot. For the CB effect to occur it is necessary that the charge on the dot is well-defined, a condition which is satisfied if the tunneling resistance of the connecting junctions are much larger than the quantum resistance, h/e 2. However, it is interesting to study how the blockade effect is modified by quantum fluctuations. Much of the theoretical works have dealt with metallic systems with large tunnel junctions [ 1] and some works have focused on the case of one or few channels [2-7]. In this paper we consider the situation of a dot connected to lead through quantum point contacts (QPCs). The typical experimental setup [8, 9] in gated semiconductor structures is shown in Fig. 1. The quantum dot (QD) devices are well-suited for studying the influence of charge transfer fluctuations because the transmittance of the contacts can be controlled by the gates. In a nice experiment by Kouwenhouven et al. [9], Coulomb oscillation traces were measured for different values of the conductance of one of the QPCs leading into the dot. It was observed
0921-4526/94/$07.00(~)1994 ElsevierScienee B.V. Allrightsrese~ed
SSDI0921-4526(94)00327-0
QD
QPC2
Fig. I. Typical layout of the quantum dot experiments discussed in this paper. The dot is connected to leads by two quantum point contacts. Their widths can be controlled by the gates as can the potential of the dot. We study the case where the quantum point contacts have a conductance close to 2e2/h.
that the Coulomb blockade oscillations decreased in size as the conductance of the QPC approaches 2eZ/h, corresponding to a transmission coefficient of unity, T = 1. However, even when the transmission coefficient was very close to one (T ~ 0.9) CB could still be observed. In this paper we study how CB depends on the transmission coefficients of the QPCs in the case when the transmission coefficients are close to one. The main idea is to expand in the reflection coefficients [4,7,10] rather than in the transmission coefficient as is normally the case when the role of
1(. Flensberg / Physica B 203 (1994) 432~439 finite junction resistance is considered in metallic systems [1]. In Section 2 we formulate the model and there we argue that these systems may be described in terms of coupled one-dimensional chains, one for each channel through the QPCs. In Section 3, the model is transformed using the 1D bosonization techniques and an effective action is derived. The action is studied by means of a renormalization-group analysis, and in Section 4 it is used to calculate the conductance.
2. Model Hamiltonian
2.1. Interactions Here we discuss the role of the electron--electron interactions in the quantum dot geometry shown in Fig. 1. The electron-electron interactions in the tunnel barrier region are neglected. Furthermore we assume that a large number of modes are occupied in the leads, as well in the dot, such that the electron-electron interaction is well screened and we can view the QD as a capacitor. In terms of time scales this involves the approximation that the rearrangement of the charge distribution in the dot happens on a time scale (typically of the order of the inverse plasma frequency in the dot) much faster than the characteristic time of the tunnel event, rD ~- h/(e2/2CD), where CD is the total capacitance of the dot (dot-to-lead and dot-to-gates). With these approximations we can write the interaction energy as simply the electrostatic energy Hint = U ( Q I
+ Q 2 ) 2,
(1)
where U = e2/2Co, and Qi is the number of electrons that have passed through contact number i into the dot. We repeat the assumptions that lead to the electrostatic model for the electron-electron interactions: (i) a large density of states in the dot and in the leads such that the level spacing is small compared to the relevant energy scales for the experiment, (ii) there is no coherent transport of electrons between the QPCs, (iii) interactions in the contacts are neglected. When the width of the contact is small compared to the dot size, the traversal time for the passage through the QPC is much smaller than ZD and hence the transfer of charge from the leads to the dot is governed by the correlation energy in the dot.
2.2. The quantum points contacts The electrostatically confined quantum point contacts are well approximated as being effectively one dimensional in the neck region [11]. As the contact widens the adiabatic one-dimensional picture breaks down and the
433
one-dimensional (1D) states are scattered into the twodimensional states of the dot or the leads. However, here we will argue that this does not destroy the one-dimensional nature of the problem. This is equivalent to the X-ray edge treatment in terms of 1D Tomonaga model as was done by Schotte and Schotte [12] where it was assumed that the core-hole potential only couples to the s-wave component of the band electrons. In our case the states in the dot or the leads also couples to a single channel in neck and hence the one electron part of the Hamiltonian may be mapped to a 1D problem. This will be shown in some more detail in the following. We start our discussion by considering two semi-infinite one-dimensional wires which are connected by tunneling, see Fig. 2(a). The Hamiltonian for this system reads H :
~
~,,~ ( 4 c , + d ~ d , ) +
k----~
~
(toc,*d, + h . c . ) ,
k,k t --oo
(2) where to is the hopping matrix element from one wire to the next, and c, d are annihilation operators for electrons in the two separate chains. Now by introducing the transformations
ak = (ck + c - k ) / v ~ 4k = (ck - c _ , ) / v ~ ,
k > 0,
(3)
bk = (dk + d - k ) / x / 2 Bk = (dk -- d - k ) / x / 2 the Hamiltonian becomes
t k--O
+ ~
(toa~b, + h.c.).
(4)
k, kt =O
It is clear that the states corresponding to the A and B separate out and we are left with the "standard" tunneling Hamiltonian approach to the Coulomb blockade problem. The charge that has passed through the tunnel junction is as usual given by
Q = l ~(a~ak - btkbk).
(5)
Similarly, for a two-dimensional geometry as for example the one depicted in Fig. 2(b), we can also formulate it in terms of a 1D problem, provided that the coupling between the two sides of the junction is through a single channel. The tunneling part is thus taken to be HT = to ~ULt(X= 0, y = 0)tPR(X = 0, y = 0) + h.c.,
(6)
K. Flensberg I Physica B 203 (1994) 432~139
434
where L, R means left and right sides, respectively. Expanding on the eigenstates in polar coordinates, ~L(R)(r, 0) = Ea(b)mKeim°jm(Kr),
(7)
mK where J,. is the usual Bessel function of the first kind, we obtain
(amKamK + b~xbmK) k (tOatm=O,Kbm=O,K' + h . c . ) . K,K t 0
Q = ~ x~=o(a~,xao,K - b~,xbo.K).
(10)
where ~
n-l k-
t Cnk ~nk Cnk
(11)
c~
(8)
=
The tunneling term only couples the m = 0 components, and consequently the m # 0 parts of the Hamiltonian are unimportant. Again, we get for the charge that passed through the contact, 1
H = H0 + Hbarr +Hint,
Ho = ~
m K=O
+
value at 2k F will matter. (2) The charging energy is assumed to be given by the charge that has passed through the contact, which is taken to be the charge to (say) the left of the barrier. The model Hamiltonian now reads
(9)
The Hamiltonian is now equivalent to the 1D model considered above. We can work backwards from the rn = 0 part of Eq. (8) to the Hamiltonian in Eq. (2) for two 1D wires connected by tunneling at the ends. Generally, when the electron enters the 2DEG from the single mode channel formed by the QPC, it is coupled to a particular linear combination of eigenstates modes in the dot or lead. Therefore, the mapping carried out above can be generalized so that the two sides of the junction always couple through a single state. Assuming a constant density of states in the dot and lead, all the above models map to each others. It is important to note that this mapping to a one-dimensional system only works because the interactions is assumed to be of the simplified electrostatic form as discussed in Section 2.1. If interactions between electrons in the different states are included correctly, the interactions would mix the different harmonics. Having established that the motion of the electrons from dot to lead through a quantum point contact can be viewed as a purely 1D problem, we continue by considering the case when the transmission through the QPC is close to unity, i.e., in the region just prior to the first conductance plateau. In the tunneling models discussed above it corresponds to the case where the tunneling matrix element is close the optimum value where T : 1 (here T is the transition coefficient), Itol = l/up, where p is the density of states [3]. In this case it is more appropriate to view the tunnel barrier as a scattering potential since we, eventually, want to do perturbation theory in the backscattering amplitude. The following model is therefore adopted: ( 1) the single particle part is a single ID channel (plus spin degeneracy) with a scattering barrier V(x). Furthermore, for simplicity we will use a delta-function barrier, since for the low-energy transport properties which we are interested in, only the
L k, kt=_oo
n=l
Hint = U
~
(CnkCnk -~-
,
(13)
where V, are the scattering potentials, L is the normalization length, and Q, the charge that has passed into the dot from channel n. The channel indexes n = 1. . . . . N denote the different point contacts leading into the dot and their spin degeneracy. The total number of channels N is therefore given by twice the number of point contact leading into the dot. In the case of an applied magnetic field the spin degeneracy is lifted and thus the number of channels may be reduced. However, this introduces the further complication that the edge states may be coherently transmitting through several contacts and we shall not discuss that situation here. The charges Q, in our model are given by
Q, =
#
dxp,(x)
(14)
CX3
where we have introduced the Fourier transforms: p(x) = f dk/(2rO eikXp(k). This concludes the formulation of the model. In the following sections it is transformed in terms of coupled Tomonaga-type models utilizing the standard bosonization procedure.
3. Bosonization of the model Hamiltonian
It tums out to be useful to use a bosonization scheme for the model Hamiltonian because by this mapping the interaction term as well as the unperturbed parts of the kinetic energy are bilinear, while the backscattering caused by the barrier is of a more complicated form. The Tomonaga bosonization starts with the basic approximation that the single electron energies are linear in the quantum number k (throughout we use h = 1): ~,k = v ~ , ( I k l - k F , ) .
(15)
In the bosonization procedure any dependence on the bandwidths are ignored. For reviews of the boson representation
K. Flensberg/ Physica B 203 (1994) 432-439 of the Tomonaga and Luttinger models see, e.g., Refs. [1315] The first step in the bosonization procedure is to separate the electron operators into left- and right-moving electron operators: ck =
{Ckl, ck2,
k > 0, k < 0.
be expressed as a bilinear form and in order to deal with this term, we must use the boson representation of the Fermi operators. We will use a formulation due to Schotte and Schotte [ 12] for the X-ray edge problem. Now define the fermion operators
(16)
1
Cni = ~
define corresponding density operators, p~(k) = ~,,ctk,gc(k+k,)~, where i = 1,2. When neglecting terms of order k/k F, p~ obey boson-type commutation relations, e.g., [p,o,2)(-k),p.,{L2}(k')] = ±(kL/2rt)6,,,6~,. Now, since the unperturbed Hamiltonian and the density operators commute as [Ho, p,{l.2)(k)] = ±kp,o2)(k), we may write H0 in the form
435
~-2c.ki,
(22)
V lvL k
We
where NL is the number of states in our normalization volume: NL = Ek 1. We have therefore introduced an upper cutoff kD, that is given by the bandwidth: kD = 2nNL/L. The new local operators obey the usual anticommutation relations. Following Schotte and Schotte we note that [p(k), C t ] = C t, which suggests that we can write
2~ N
Ho = T~-2vF, )-2 [p,,(k ) p , , ( - k ) + p , 2 ( - k )p,2(k )] l.a n=l
CJi.2 =
exp
+ --~--k);.~ n [p.,,2(k) - p.,.2(-k)]
(23)
N VFn
where the field operator 4} and its conjugate momentum P are given by m ( x ) + p2(x) =
~x4~,
p l ( x ) - p2(x) = - v / ~
.
(18) These fields obey the commutation rule [c~(x),P(x')] = i ~ ( x - x ' ) . The momentum P is the difference between left and right movers and is thus equivalent to the current operator. In the boson language the charges Q. are given by Qn =
//
,
k>0
d x [pnl (X) q- pn2(X)]
= -~n ~b,(0).
where the prefactor is fixed so that the anticommutation relations are fullfilled for n = n' and i = i '. For different n's and i's anticommutations relations must also be maintained and in principle it is necessary to include an extra phase factor in the bosonized form of the C's [ 15]. However, we can show that this complication can be disregarded [6]. The backscattering term can now be written in terms of the boson operator by using Eq. (23), and we obtain [6] /-/b~ = ~ ~a~V"[exp (2iv/'~bln(0)) + h.c.] ,
(24)
t/=l
where a, --- 1/2~kD, defines the inverse upper cutoff. We have now finally arrived at fully bosonized xepresentation of the model Hamiltonian, Eqs. (17), (20) and (24). In the next section we further reduce the model to an effective action involving only the local fields q~,(0).
(19)
3.1. Effective action This expression is valid to leading order in k/k F. The charging energy thus becomes Hint = _U /'c
{b,(0)
.
(20)
The barrier term splits into two terms when the electrons are divided into the left- and right-mover sections: a forward and backward scattering part, v.
t
The model Hamiltonian has now been transformed into a series of coupled Luttinger models. Noting that Hi.t and Hbarr both only contain the operators ~b.(x = 0), we can reduce the problem further by integrating out all the q~.(x :fi 0) degrees of freedom. [4, 6, 10] The unperturbed part of the Hamiltonian then gives rise to an action [4,10], So =
,
~ ~1~o.114,.(io~.) I, n= 1
(21) The forward-scattering term can be absorbed as a phaseshift and is not important. The backscattering term cannot
(25)
co n
where we have used the notation ~b(x = 0, r) -- q~(z) and (on is the boson Matsubara frequencies (not to be confused with the channel index n). The co,-sum has a upper cutoff given by the bandwidth. The cutoff is given by o9c = roy F/a
K. Flensberg/ Physica B 203 (1994) 432~t39
436
(see Ref. [6], Appendix A). The total imaginary time action thus becomes U1
N
2
N Vn f 3
+~-]~a-7. [ dz cos (2v/~qS.(v)). n=l
(26)
n J0
is more appropriate and that the Coulomb blockade is restored. This happens when the renormalized reflection amplitude: R* ~ Ro(U*/U) -2IN is of the order of one, which gives a condition for the scale of the renormalized charging energy, U* ~ U0[l - T] m/2.
(31)
From this we conclude that the measured charging energy or capacitance is renormalized according to Eq. (31 ). Recent experiments [ 17] indeed support this result.
4. Renormalization-group analysis - effective capacitance
We now do a perturbative renormalization-group analysis in the backscattering matrix elements 1,I.. Keeping only one V. finite (n = n0), we can integrate out the N - 1 remaining ~b.40 fields. We then obtain
5[¢.0] = ~E[~°I [Ck.o(i~,)12h(l~°.l)
5. Conductance o f t h e Q D d e v i c e
To be more specific, we will now calculate the conductance of the quantum dot devices depicted in Fig. 1, which has two quantum point contacts connected to the dot. Two different configurations are considered, see Fig. 3. For the first configuration (denoted as case A) both contacts are
~ojt
+ [l~dz V"° cos (2x/x~b.o(Z)) ,
Jo
(27)
(a) to
ano
where
g=l-l/N,
(28)
(b)
and
U/~+ Io9.l/N h(Ic°"l)= U/rc + lo).i/(gg ) --*
{ 1 for I~o.l>U.
(29)
For low energies the action is equivalent to that of an interacting Luttinger liquid with a single impurity [4]. There 9 represents the interaction strength, being less than one for repulsive interaction. For large frequencies o9. >>U, where the charging energy is unimportant, the action reduces the case of free electrons. Next, we perform a perturbative renormalization-group transformation in the usual way by integrating out the high frequencies [16]. For large frequencies (o). >>U) the action is in a fixed point and we can scale the high frequency cutoff down to the charging energy U. For frequencies smaller than U we take the action to be given by that ofh(¢o.) = 1. Then we can continue the scaling, and following the standard method [16], we get that
dV"°=(1-g)V"°~
~
~ \Uo](U'~-' = (U*)~
,:m
Fig. 2. Schematical drawing of two one-dimensional systems connected by tunneling (a), and two 2D point contacts also connected by tunneling with to being the tunneling matrix element. In the text it is shown that these two systems are equivalent.
Contact IAV
V
Contact 1 A
~Contact 2
Case A
Contact 2
Case B
(30) Here dl = -dU/U. We see that the barrier grows under scaling, and perturbation theory breaks down at small energies. This means that at small energies the tunneling picture
Fig. 3. The configurations of the device in Fig. 1 that is considered in Section 5. Case A: both contacts have a small reflection coefficient. Case B: contact 2 is in the tunneling regime whereas contact 1 has channels with small reflection coefficient.
K. Flensber9 / Physica B 203 (1994) 432-439 nearly perfectly transmitting. For the second case (case B) one contact (contact 1) is biased in the strong tunneling regime whereas the second contact is in the weak tunneling regime.
437
a
=
.2
5.1. Confiouration A Configuration A has two contacts with Nt, N2 channels, respectively. Then the action is 0.8
s--~
I~.ll,~l., 12+ ~ 1~o.11~2.~1~
t " ¢on
Lnl=l
~--~ 0.2
n2= I
+ - - Iqbl(iO~n)+ q~2(ko.)l 2 7~
III
,I
00"
I 0.,5
,
/'"
I," l
,
I 1.5
,
kT/U
+ ~ ~ ~12=|
cos(2v~b.2(z)) ] .
(32)
n2
Here we have defined ~; = E .N;i= j q~i~;, i = 1,2. The current is given by e
Fig. 4. The conductance of the device in configuration A (see Fig. 3) to second order in the backscattering matrix elements (Eq. (37)). The conduetances have been normalized with respect to their respective large temperature values, GN. The inset shows the cross-over temperature kT*. The result in Eq. (31) is shown with the dashed line. In the plot we have defined the cross-over temperature as the temperature where G(kT) crosses zero.
•
d = - 2x/~O,
O = (qh - ~2),
(33)
5.1.2. Second order in Vn and we calculate the conductance using the Kubo formula: G =
llm//(~ + i6),
The conductance to second order is calculated along the same lines as in Ref. [6]. We obtain after some algebra that
(34)
G-- e2hN, + N2NIN2[ I - f(kT/u)R'--~l ~+ R2Nl J
(37)
w h e r e / / i s the current--current correlation function. Here the function f is given by
5.1.1. Zeroth order in V, When V, = 0 we can express the action in terms of 6) alone, and obtain S[~9] = l ~ l o ( i t o , ) i 2 O o l ( i o 9 , ) , r
(38) where
(N1 - N 2 ) 2 [co.l(N, +N2)+rcco2/U"
B(t) = exp
Now using H(ko.) = e2c02/(4~)(10(i0~.)12), we obtain for the DC-conductance to zeroth order in the backscattering (restoring h),
( ×
(NI +N2)
f
'~
(e - i ~ ' - 11[1 +nB(a)] "~ dc ei~-~-(.~-~/777-N.~----~.~12,t. (39)
c~
The limiting behavior of the f-function is
G(O)= e__22 N1N2
h NI q-N2"
dt B( t ) f ' ~ d~ e -iEt[-~EEns (E)] ,
(35)
oJn
D ~ l ( k o , ) _ N1 + N2 ]~o.I
2F
f(kT/U) =
(36)
which is just the series conductance of the two contacts. It is independent of temperature. However, the next-order term does depend on temperature and in fact diverges at low temperatures in agreement with the results of the previous section.
f(kT/U) --~
1 for kT~>U, (kT/U)_2/(N,+N2) for kT~.U.
(40)
In the limit of large temperatures the conductance reduces to that of the contacts in series (adding G; = Nil( 1 + R~), i = 1,2 in series yields the conductance in Eq. (37) to first
K. Flensber01Physica B 203 (1994) 432-439
438
order in the R's). For low temperatures, we obtain a powerlaw behavior and in fact the correction term diverges at low temperatures. We have also calculated for the pe~urbative form of the conductance numerically. Fig. 4 shows the conductance given by Eq. (37) as a function of temperature. The conductance is shown for different values of the reflection coefficient (R~ = R2) and for the case with N1 = N2 = 2. The conductance crosses zero indicating the break-down of perturbation theory and that CB is restored. In the inset we show the temperature at which the conductance in Eq. (37) crosses zero as a function of(1 - T) 2. The linear dependence support the scaling result in Eq. (31 ). In recent experiments [17] on a double dot device the predicted scaling behavior was indeed observed.
5.2. Configuration B Now we discuss the case shown in Fig. 3(b), where the second contact has a small transmission coefficient T2 ~ 1, whereas contact number one has a small reflection coefficient. In this case, we can use the standard tunneling scheme to calculate the conductance of the device. We find
i = -~G2 f
where x is a constant of order one. Again we see that the correction increases at low energies.
6. Conclusions We have established an analogy between Coulomb blockaded system with quantum points contacts and a Luttingertype model. The formalism introduced here allows us to study the behavior for any number of connecting channels. In particular, we have studied the case when a QD is strongly coupled to the leads through QPCs with nearly perfectly transmitted modes. A perturbative calculation in the reflection coefficients was performed and we saw that the perturbation theory breaks down at a cross-over temperature and we interpreted this as the temperature below which Coulomb blockade is restored. The cross-over temperature defines an effective capacitance, T* = e2/2C *. It would be interesting to test experimentally the predicted dependence of the number of channels and the dependence of the cross-over temperature on the nominal conductance.
do:, ~-~x[(o9 - eV)ns(~o - eV) Acknowledgement
-(~o + eV)ns(o9 + eV)]P(co),
(41) This work was supported by the Carlsberg Foundation.
where the function P(E) is given by
/i[-,o,+ue~,l, 'J e--i[Hlel+U(Ql+l)2]t )
P(t) = \ e L
=
Texp
dt'Ql(t')
-2iU
References e-iU,,
/11
Q1 = - - ~ 1
•
(42)
(There is a misprint in Ref. [6] in Eq. (8).) Here Htel is the free electron part of contact one. The bosonization scheme is now carried out for contact one alone, since contact two is already taken care of. We then obtain that to zeroth order in the reflection coefficient of contact one, 2 o~ p(0)(t)=exp(~f_o d_e(e i2 -
e
1)[l+nB(e)].'~
1 + [rte/N1U] z
] .
(43)
It is interesting to note that this is identical to the result of a single tumael junction coupled to an environment with impedance Z = h/e2Nl [18]. To next leading order we find for the conductance,
G(eV, T) ..~ Go ( max(e~-'kT) ) 2IN' × (l-x(max(e-~'kT))-z/N'N,R,),
(44)
[1] G. Sch6n and A.D. Zaikin, Phys. Rep. 198 (1990) 237; A.A. Odintsov, Zh. Eksp. Teor. Fiz. 94 (1988) 312 [Soy. Phys. JETP 67 (1988) 1265]; S.V. Panyukov and A.D. Zaikin, J. Low Temp. Phys. 73 (1988) 1; W. Zwerger and M. Scharpf, Z. Phys. B 85 (1991) 421; K. Flensberg and M. Jonson, Phys. Rev. B 43 (1991) 7586; A.D. Zaikin and S.V. Panyukov, Phys. Lett. A 183 (1993) 115. [2] K.A. Matveev, Zh. Eksp. Teor. Fiz. 99 (1991) 1598 [Sov. Phys. JETP 72 (1991) 892]. [3] H.O. Frota and K. Flensberg, Phys. Rev. B 46 (1992) 15 207. [4] C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220; Phys. Rev. B 46 (1992) 15233. [5] K.A. Matveev and L.I. Glazman, Phys. Rev. Lett. 70 (1993) 990; Physica B 189 (1993) 266. [6] K. Flensberg, Phys. Rev. B 48 (1993) 11156. [7] K.A. Matveev, Physica 13 203 (1994) 404. [8] U. Meirav, M.A. Kastner and S.J. Wind, Phys. Rev. Lett. 58 (1990) 771; A.T. Johnson et al., Phys. Rev. Lett. 69 (1992) 1592; C. Pasquier et al., Phys. Rev. Lett. 70 (1993) 69. [9] L.P. Kouwenhoven et al., Z. Phys. B 85 (1991) 367. [10] A. Furusaki and N. Nagaosa, Phys. Rev. B 46 (1993) 3827. [11] L.I. Glazman, G.B. Lesovik, D.E. Khmel'nitskii and R.I. Shekhter, Pisma Zh. Eksp. Teor. Fiz. 48 (1988) 218 [JETP Lett. 48 (1988) 238. [12] K.D. Schotte and U. Schotte, Phys. Rev. 182 (1969) 479.
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Physica B 203 (1994) 432-439
Capacitance and conductance of dots connected by quantum point contacts Karsten Flensberg Mikroelektronik Centret, Danmarks Tekniske Universitet, DK-2800 Lyngby, Denmark
Abstract We study the transport properties of quantum dots and quantum point contacts in the Coulomb blockade regime and in the limit where the quantum point contact has nearly fully transmitting channels. We argue that the properties of the charge fluctuations through the contact can be dealt with using a one-dimensional representation and study the scaling behavior of the model. We find a cross-over between a low-energy regime with Coulomb blockade to a high-energy regime where quantum charge fluctuations are dominant. The cross-over energy defines an effective charging energy given by U* ~ U[1 - T] u/2, where U is the bare charging energy, T is the transmission coefficient, and N is the number of channels given by twice the number of quantum point contacts connected to the dot.
1. Introduction Transport measurements in devices such as quantum dots exhibit the so-called Coulomb blockade (CB) at low temperatures. This happens when the thermal energy is less than the charging energy associated with adding a single electron onto the dot. For the CB effect to occur it is necessary that the charge on the dot is well-defined, a condition which is satisfied if the tunneling resistance of the connecting junctions are much larger than the quantum resistance, h/e 2. However, it is interesting to study how the blockade effect is modified by quantum fluctuations. Much of the theoretical works have dealt with metallic systems with large tunnel junctions [ 1] and some works have focused on the case of one or few channels [2-7]. In this paper we consider the situation of a dot connected to lead through quantum point contacts (QPCs). The typical experimental setup [8, 9] in gated semiconductor structures is shown in Fig. 1. The quantum dot (QD) devices are well-suited for studying the influence of charge transfer fluctuations because the transmittance of the contacts can be controlled by the gates. In a nice experiment by Kouwenhouven et al. [9], Coulomb oscillation traces were measured for different values of the conductance of one of the QPCs leading into the dot. It was observed
0921-4526/94/$07.00(~)1994 ElsevierScienee B.V. Allrightsrese~ed
SSDI0921-4526(94)00327-0
QD
QPC2
Fig. I. Typical layout of the quantum dot experiments discussed in this paper. The dot is connected to leads by two quantum point contacts. Their widths can be controlled by the gates as can the potential of the dot. We study the case where the quantum point contacts have a conductance close to 2e2/h.
that the Coulomb blockade oscillations decreased in size as the conductance of the QPC approaches 2eZ/h, corresponding to a transmission coefficient of unity, T = 1. However, even when the transmission coefficient was very close to one (T ~ 0.9) CB could still be observed. In this paper we study how CB depends on the transmission coefficients of the QPCs in the case when the transmission coefficients are close to one. The main idea is to expand in the reflection coefficients [4,7,10] rather than in the transmission coefficient as is normally the case when the role of
1(. Flensberg / Physica B 203 (1994) 432~439 finite junction resistance is considered in metallic systems [1]. In Section 2 we formulate the model and there we argue that these systems may be described in terms of coupled one-dimensional chains, one for each channel through the QPCs. In Section 3, the model is transformed using the 1D bosonization techniques and an effective action is derived. The action is studied by means of a renormalization-group analysis, and in Section 4 it is used to calculate the conductance.
2. Model Hamiltonian
2.1. Interactions Here we discuss the role of the electron--electron interactions in the quantum dot geometry shown in Fig. 1. The electron-electron interactions in the tunnel barrier region are neglected. Furthermore we assume that a large number of modes are occupied in the leads, as well in the dot, such that the electron-electron interaction is well screened and we can view the QD as a capacitor. In terms of time scales this involves the approximation that the rearrangement of the charge distribution in the dot happens on a time scale (typically of the order of the inverse plasma frequency in the dot) much faster than the characteristic time of the tunnel event, rD ~- h/(e2/2CD), where CD is the total capacitance of the dot (dot-to-lead and dot-to-gates). With these approximations we can write the interaction energy as simply the electrostatic energy Hint = U ( Q I
+ Q 2 ) 2,
(1)
where U = e2/2Co, and Qi is the number of electrons that have passed through contact number i into the dot. We repeat the assumptions that lead to the electrostatic model for the electron-electron interactions: (i) a large density of states in the dot and in the leads such that the level spacing is small compared to the relevant energy scales for the experiment, (ii) there is no coherent transport of electrons between the QPCs, (iii) interactions in the contacts are neglected. When the width of the contact is small compared to the dot size, the traversal time for the passage through the QPC is much smaller than ZD and hence the transfer of charge from the leads to the dot is governed by the correlation energy in the dot.
2.2. The quantum points contacts The electrostatically confined quantum point contacts are well approximated as being effectively one dimensional in the neck region [11]. As the contact widens the adiabatic one-dimensional picture breaks down and the
433
one-dimensional (1D) states are scattered into the twodimensional states of the dot or the leads. However, here we will argue that this does not destroy the one-dimensional nature of the problem. This is equivalent to the X-ray edge treatment in terms of 1D Tomonaga model as was done by Schotte and Schotte [12] where it was assumed that the core-hole potential only couples to the s-wave component of the band electrons. In our case the states in the dot or the leads also couples to a single channel in neck and hence the one electron part of the Hamiltonian may be mapped to a 1D problem. This will be shown in some more detail in the following. We start our discussion by considering two semi-infinite one-dimensional wires which are connected by tunneling, see Fig. 2(a). The Hamiltonian for this system reads H :
~
~,,~ ( 4 c , + d ~ d , ) +
k----~
~
(toc,*d, + h . c . ) ,
k,k t --oo
(2) where to is the hopping matrix element from one wire to the next, and c, d are annihilation operators for electrons in the two separate chains. Now by introducing the transformations
ak = (ck + c - k ) / v ~ 4k = (ck - c _ , ) / v ~ ,
k > 0,
(3)
bk = (dk + d - k ) / x / 2 Bk = (dk -- d - k ) / x / 2 the Hamiltonian becomes
t k--O
+ ~
(toa~b, + h.c.).
(4)
k, kt =O
It is clear that the states corresponding to the A and B separate out and we are left with the "standard" tunneling Hamiltonian approach to the Coulomb blockade problem. The charge that has passed through the tunnel junction is as usual given by
Q = l ~(a~ak - btkbk).
(5)
Similarly, for a two-dimensional geometry as for example the one depicted in Fig. 2(b), we can also formulate it in terms of a 1D problem, provided that the coupling between the two sides of the junction is through a single channel. The tunneling part is thus taken to be HT = to ~ULt(X= 0, y = 0)tPR(X = 0, y = 0) + h.c.,
(6)
K. Flensberg I Physica B 203 (1994) 432~139
434
where L, R means left and right sides, respectively. Expanding on the eigenstates in polar coordinates, ~L(R)(r, 0) = Ea(b)mKeim°jm(Kr),
(7)
mK where J,. is the usual Bessel function of the first kind, we obtain
(amKamK + b~xbmK) k (tOatm=O,Kbm=O,K' + h . c . ) . K,K t 0
Q = ~ x~=o(a~,xao,K - b~,xbo.K).
(10)
where ~
n-l k-
t Cnk ~nk Cnk
(11)
c~
(8)
=
The tunneling term only couples the m = 0 components, and consequently the m # 0 parts of the Hamiltonian are unimportant. Again, we get for the charge that passed through the contact, 1
H = H0 + Hbarr +Hint,
Ho = ~
m K=O
+
value at 2k F will matter. (2) The charging energy is assumed to be given by the charge that has passed through the contact, which is taken to be the charge to (say) the left of the barrier. The model Hamiltonian now reads
(9)
The Hamiltonian is now equivalent to the 1D model considered above. We can work backwards from the rn = 0 part of Eq. (8) to the Hamiltonian in Eq. (2) for two 1D wires connected by tunneling at the ends. Generally, when the electron enters the 2DEG from the single mode channel formed by the QPC, it is coupled to a particular linear combination of eigenstates modes in the dot or lead. Therefore, the mapping carried out above can be generalized so that the two sides of the junction always couple through a single state. Assuming a constant density of states in the dot and lead, all the above models map to each others. It is important to note that this mapping to a one-dimensional system only works because the interactions is assumed to be of the simplified electrostatic form as discussed in Section 2.1. If interactions between electrons in the different states are included correctly, the interactions would mix the different harmonics. Having established that the motion of the electrons from dot to lead through a quantum point contact can be viewed as a purely 1D problem, we continue by considering the case when the transmission through the QPC is close to unity, i.e., in the region just prior to the first conductance plateau. In the tunneling models discussed above it corresponds to the case where the tunneling matrix element is close the optimum value where T : 1 (here T is the transition coefficient), Itol = l/up, where p is the density of states [3]. In this case it is more appropriate to view the tunnel barrier as a scattering potential since we, eventually, want to do perturbation theory in the backscattering amplitude. The following model is therefore adopted: ( 1) the single particle part is a single ID channel (plus spin degeneracy) with a scattering barrier V(x). Furthermore, for simplicity we will use a delta-function barrier, since for the low-energy transport properties which we are interested in, only the
L k, kt=_oo
n=l
Hint = U
~
(CnkCnk -~-
,
(13)
where V, are the scattering potentials, L is the normalization length, and Q, the charge that has passed into the dot from channel n. The channel indexes n = 1. . . . . N denote the different point contacts leading into the dot and their spin degeneracy. The total number of channels N is therefore given by twice the number of point contact leading into the dot. In the case of an applied magnetic field the spin degeneracy is lifted and thus the number of channels may be reduced. However, this introduces the further complication that the edge states may be coherently transmitting through several contacts and we shall not discuss that situation here. The charges Q, in our model are given by
Q, =
#
dxp,(x)
(14)
CX3
where we have introduced the Fourier transforms: p(x) = f dk/(2rO eikXp(k). This concludes the formulation of the model. In the following sections it is transformed in terms of coupled Tomonaga-type models utilizing the standard bosonization procedure.
3. Bosonization of the model Hamiltonian
It tums out to be useful to use a bosonization scheme for the model Hamiltonian because by this mapping the interaction term as well as the unperturbed parts of the kinetic energy are bilinear, while the backscattering caused by the barrier is of a more complicated form. The Tomonaga bosonization starts with the basic approximation that the single electron energies are linear in the quantum number k (throughout we use h = 1): ~,k = v ~ , ( I k l - k F , ) .
(15)
In the bosonization procedure any dependence on the bandwidths are ignored. For reviews of the boson representation
K. Flensberg/ Physica B 203 (1994) 432-439 of the Tomonaga and Luttinger models see, e.g., Refs. [1315] The first step in the bosonization procedure is to separate the electron operators into left- and right-moving electron operators: ck =
{Ckl, ck2,
k > 0, k < 0.
be expressed as a bilinear form and in order to deal with this term, we must use the boson representation of the Fermi operators. We will use a formulation due to Schotte and Schotte [ 12] for the X-ray edge problem. Now define the fermion operators
(16)
1
Cni = ~
define corresponding density operators, p~(k) = ~,,ctk,gc(k+k,)~, where i = 1,2. When neglecting terms of order k/k F, p~ obey boson-type commutation relations, e.g., [p,o,2)(-k),p.,{L2}(k')] = ±(kL/2rt)6,,,6~,. Now, since the unperturbed Hamiltonian and the density operators commute as [Ho, p,{l.2)(k)] = ±kp,o2)(k), we may write H0 in the form
435
~-2c.ki,
(22)
V lvL k
We
where NL is the number of states in our normalization volume: NL = Ek 1. We have therefore introduced an upper cutoff kD, that is given by the bandwidth: kD = 2nNL/L. The new local operators obey the usual anticommutation relations. Following Schotte and Schotte we note that [p(k), C t ] = C t, which suggests that we can write
2~ N
Ho = T~-2vF, )-2 [p,,(k ) p , , ( - k ) + p , 2 ( - k )p,2(k )] l.a n=l
CJi.2 =
exp
+ --~--k);.~ n [p.,,2(k) - p.,.2(-k)]
(23)
N VFn
where the field operator 4} and its conjugate momentum P are given by m ( x ) + p2(x) =
~x4~,
p l ( x ) - p2(x) = - v / ~
.
(18) These fields obey the commutation rule [c~(x),P(x')] = i ~ ( x - x ' ) . The momentum P is the difference between left and right movers and is thus equivalent to the current operator. In the boson language the charges Q. are given by Qn =
//
,
k>0
d x [pnl (X) q- pn2(X)]
= -~n ~b,(0).
where the prefactor is fixed so that the anticommutation relations are fullfilled for n = n' and i = i '. For different n's and i's anticommutations relations must also be maintained and in principle it is necessary to include an extra phase factor in the bosonized form of the C's [ 15]. However, we can show that this complication can be disregarded [6]. The backscattering term can now be written in terms of the boson operator by using Eq. (23), and we obtain [6] /-/b~ = ~ ~a~V"[exp (2iv/'~bln(0)) + h.c.] ,
(24)
t/=l
where a, --- 1/2~kD, defines the inverse upper cutoff. We have now finally arrived at fully bosonized xepresentation of the model Hamiltonian, Eqs. (17), (20) and (24). In the next section we further reduce the model to an effective action involving only the local fields q~,(0).
(19)
3.1. Effective action This expression is valid to leading order in k/k F. The charging energy thus becomes Hint = _U /'c
{b,(0)
.
(20)
The barrier term splits into two terms when the electrons are divided into the left- and right-mover sections: a forward and backward scattering part, v.
t
The model Hamiltonian has now been transformed into a series of coupled Luttinger models. Noting that Hi.t and Hbarr both only contain the operators ~b.(x = 0), we can reduce the problem further by integrating out all the q~.(x :fi 0) degrees of freedom. [4, 6, 10] The unperturbed part of the Hamiltonian then gives rise to an action [4,10], So =
,
~ ~1~o.114,.(io~.) I, n= 1
(21) The forward-scattering term can be absorbed as a phaseshift and is not important. The backscattering term cannot
(25)
co n
where we have used the notation ~b(x = 0, r) -- q~(z) and (on is the boson Matsubara frequencies (not to be confused with the channel index n). The co,-sum has a upper cutoff given by the bandwidth. The cutoff is given by o9c = roy F/a
K. Flensberg/ Physica B 203 (1994) 432~t39
436
(see Ref. [6], Appendix A). The total imaginary time action thus becomes U1
N
2
N Vn f 3
+~-]~a-7. [ dz cos (2v/~qS.(v)). n=l
(26)
n J0
is more appropriate and that the Coulomb blockade is restored. This happens when the renormalized reflection amplitude: R* ~ Ro(U*/U) -2IN is of the order of one, which gives a condition for the scale of the renormalized charging energy, U* ~ U0[l - T] m/2.
(31)
From this we conclude that the measured charging energy or capacitance is renormalized according to Eq. (31 ). Recent experiments [ 17] indeed support this result.
4. Renormalization-group analysis - effective capacitance
We now do a perturbative renormalization-group analysis in the backscattering matrix elements 1,I.. Keeping only one V. finite (n = n0), we can integrate out the N - 1 remaining ~b.40 fields. We then obtain
5[¢.0] = ~E[~°I [Ck.o(i~,)12h(l~°.l)
5. Conductance o f t h e Q D d e v i c e
To be more specific, we will now calculate the conductance of the quantum dot devices depicted in Fig. 1, which has two quantum point contacts connected to the dot. Two different configurations are considered, see Fig. 3. For the first configuration (denoted as case A) both contacts are
~ojt
+ [l~dz V"° cos (2x/x~b.o(Z)) ,
Jo
(27)
(a) to
ano
where
g=l-l/N,
(28)
(b)
and
U/~+ Io9.l/N h(Ic°"l)= U/rc + lo).i/(gg ) --*
{ 1 for I~o.l
(29)
For low energies the action is equivalent to that of an interacting Luttinger liquid with a single impurity [4]. There 9 represents the interaction strength, being less than one for repulsive interaction. For large frequencies o9. >>U, where the charging energy is unimportant, the action reduces the case of free electrons. Next, we perform a perturbative renormalization-group transformation in the usual way by integrating out the high frequencies [16]. For large frequencies (o). >>U) the action is in a fixed point and we can scale the high frequency cutoff down to the charging energy U. For frequencies smaller than U we take the action to be given by that ofh(¢o.) = 1. Then we can continue the scaling, and following the standard method [16], we get that
dV"°=(1-g)V"°~
~
~ \Uo](U'~-' = (U*)~
,:m
Fig. 2. Schematical drawing of two one-dimensional systems connected by tunneling (a), and two 2D point contacts also connected by tunneling with to being the tunneling matrix element. In the text it is shown that these two systems are equivalent.
Contact IAV
V
Contact 1 A
~Contact 2
Case A
Contact 2
Case B
(30) Here dl = -dU/U. We see that the barrier grows under scaling, and perturbation theory breaks down at small energies. This means that at small energies the tunneling picture
Fig. 3. The configurations of the device in Fig. 1 that is considered in Section 5. Case A: both contacts have a small reflection coefficient. Case B: contact 2 is in the tunneling regime whereas contact 1 has channels with small reflection coefficient.
K. Flensber9 / Physica B 203 (1994) 432-439 nearly perfectly transmitting. For the second case (case B) one contact (contact 1) is biased in the strong tunneling regime whereas the second contact is in the weak tunneling regime.
437
a
=
.2
5.1. Confiouration A Configuration A has two contacts with Nt, N2 channels, respectively. Then the action is 0.8
s--~
I~.ll,~l., 12+ ~ 1~o.11~2.~1~
t " ¢on
Lnl=l
~--~ 0.2
n2= I
+ - - Iqbl(iO~n)+ q~2(ko.)l 2 7~
III
,I
00"
I 0.,5
,
/'"
I," l
,
I 1.5
,
kT/U
+ ~ ~ ~12=|
cos(2v~b.2(z)) ] .
(32)
n2
Here we have defined ~; = E .N;i= j q~i~;, i = 1,2. The current is given by e
Fig. 4. The conductance of the device in configuration A (see Fig. 3) to second order in the backscattering matrix elements (Eq. (37)). The conduetances have been normalized with respect to their respective large temperature values, GN. The inset shows the cross-over temperature kT*. The result in Eq. (31) is shown with the dashed line. In the plot we have defined the cross-over temperature as the temperature where G(kT) crosses zero.
•
d = - 2x/~O,
O = (qh - ~2),
(33)
5.1.2. Second order in Vn and we calculate the conductance using the Kubo formula: G =
llm//(~ + i6),
The conductance to second order is calculated along the same lines as in Ref. [6]. We obtain after some algebra that
(34)
G-- e2hN, + N2NIN2[ I - f(kT/u)R'--~l ~+ R2Nl J
(37)
w h e r e / / i s the current--current correlation function. Here the function f is given by
5.1.1. Zeroth order in V, When V, = 0 we can express the action in terms of 6) alone, and obtain S[~9] = l ~ l o ( i t o , ) i 2 O o l ( i o 9 , ) , r
(38) where
(N1 - N 2 ) 2 [co.l(N, +N2)+rcco2/U"
B(t) = exp
Now using H(ko.) = e2c02/(4~)(10(i0~.)12), we obtain for the DC-conductance to zeroth order in the backscattering (restoring h),
( ×
(NI +N2)
f
'~
(e - i ~ ' - 11[1 +nB(a)] "~ dc ei~-~-(.~-~/777-N.~----~.~12,t. (39)
c~
The limiting behavior of the f-function is
G(O)= e__22 N1N2
h NI q-N2"
dt B( t ) f ' ~ d~ e -iEt[-~EEns (E)] ,
(35)
oJn
D ~ l ( k o , ) _ N1 + N2 ]~o.I
2F
f(kT/U) =
(36)
which is just the series conductance of the two contacts. It is independent of temperature. However, the next-order term does depend on temperature and in fact diverges at low temperatures in agreement with the results of the previous section.
f(kT/U) --~
1 for kT~>U, (kT/U)_2/(N,+N2) for kT~.U.
(40)
In the limit of large temperatures the conductance reduces to that of the contacts in series (adding G; = Nil( 1 + R~), i = 1,2 in series yields the conductance in Eq. (37) to first
K. Flensber01Physica B 203 (1994) 432-439
438
order in the R's). For low temperatures, we obtain a powerlaw behavior and in fact the correction term diverges at low temperatures. We have also calculated for the pe~urbative form of the conductance numerically. Fig. 4 shows the conductance given by Eq. (37) as a function of temperature. The conductance is shown for different values of the reflection coefficient (R~ = R2) and for the case with N1 = N2 = 2. The conductance crosses zero indicating the break-down of perturbation theory and that CB is restored. In the inset we show the temperature at which the conductance in Eq. (37) crosses zero as a function of(1 - T) 2. The linear dependence support the scaling result in Eq. (31 ). In recent experiments [17] on a double dot device the predicted scaling behavior was indeed observed.
5.2. Configuration B Now we discuss the case shown in Fig. 3(b), where the second contact has a small transmission coefficient T2 ~ 1, whereas contact number one has a small reflection coefficient. In this case, we can use the standard tunneling scheme to calculate the conductance of the device. We find
i = -~G2 f
where x is a constant of order one. Again we see that the correction increases at low energies.
6. Conclusions We have established an analogy between Coulomb blockaded system with quantum points contacts and a Luttingertype model. The formalism introduced here allows us to study the behavior for any number of connecting channels. In particular, we have studied the case when a QD is strongly coupled to the leads through QPCs with nearly perfectly transmitted modes. A perturbative calculation in the reflection coefficients was performed and we saw that the perturbation theory breaks down at a cross-over temperature and we interpreted this as the temperature below which Coulomb blockade is restored. The cross-over temperature defines an effective capacitance, T* = e2/2C *. It would be interesting to test experimentally the predicted dependence of the number of channels and the dependence of the cross-over temperature on the nominal conductance.
do:, ~-~x[(o9 - eV)ns(~o - eV) Acknowledgement
-(~o + eV)ns(o9 + eV)]P(co),
(41) This work was supported by the Carlsberg Foundation.
where the function P(E) is given by
/i[-,o,+ue~,l, 'J e--i[Hlel+U(Ql+l)2]t )
P(t) = \ e L
=
Texp
dt'Ql(t')
-2iU
References e-iU,,
/11
Q1 = - - ~ 1
•
(42)
(There is a misprint in Ref. [6] in Eq. (8).) Here Htel is the free electron part of contact one. The bosonization scheme is now carried out for contact one alone, since contact two is already taken care of. We then obtain that to zeroth order in the reflection coefficient of contact one, 2 o~ p(0)(t)=exp(~f_o d_e(e i2 -
e
1)[l+nB(e)].'~
1 + [rte/N1U] z
] .
(43)
It is interesting to note that this is identical to the result of a single tumael junction coupled to an environment with impedance Z = h/e2Nl [18]. To next leading order we find for the conductance,
G(eV, T) ..~ Go ( max(e~-'kT) ) 2IN' × (l-x(max(e-~'kT))-z/N'N,R,),
(44)
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