- Evolution of a Theoretical Model
8.6 Breathing of Compressible Strips
8.6.2 Compressible Strips Close to the Quantum Dots
The periodic conductance modulations are not caused by an electrostatic shift of the single-electron tunneling thresholds due to the presence of an additional charge in their vicinity. However, the effect leading to a rearrangement of charge in the surrounding of the antidot arises due to the spatial reduction of the closed wave function enclosing the antidot (see chapter 6). Thish/eperiodic breathing might also occur in the configuration here for the landscape in front of the quantum dots. A qualitative sketch of the process is displayed in figure 8.8a. A closed wave function is expected to incorporate both quantum dots as well as the compressible strips close to the quantum dots (white dotted line). In case of an increasing magnetic flux density δB the wave function will reduce in size to preserve the enclosed magnetic flux. It thereby acts against the electrostatic potential
4Estimation of an ellipsoidal quantum dot with a short half-axis of a= 50 nm and a long half-axis of a= 70 nm, respectively.
160
8.6 Breathing of Compressible Strips created by the etched groove with the side-gates attached to it. At a certain threshold it will be energetically favorable to add a magnetic flux quantum to the wave function and relax to a larger perimeter with a lower electrostatic energy. In consequence, a reiterative breathing (white arrows) of the wave function while tuning the magnetic flux with a peri-odicity of one flux quantum φ0 =h/e occurs. With the electrostatic potential landscape being shallow in the regions of the quantum dot tunneling barriers - compared with the electrostatic potential formed by the etched groove or metal top-gates - the spatial re-duction in the perimeter of the closed wave function is expected to occur mainly here. In contrast to the antidot configuration from chapter 6, here the spatial reduction is present on the inner perimeter of the compressible landscape facing towards the patterned area free of electrons. This is illustrated by the red area in the enlargement of the right hand figure panel. In consequence, the periodic breathing could tune the tunneling barrier of the quantum dots on a magnitude observable in the experiments. The tunnel coupling of the quantum dots to the leads is thereby expected to increase as the wave function reduces its extension. Hence, the linewidth of the single-electron tunneling peak broadens and its conductance value may increase, unless the peak conductance value already yields the maximum of I/VDS= 1e2/h. Both processes are sketched qualitatively in figure 8.8b as blue and red curve, respectively. Applying this qualitative hypothesis the expected conductance for tuning the magnetic flux density versus the quantum dot plunger gate-voltage in figure 8.9 can be drawn. In this situation both quantum dots are considered as coherent system with only a single transmission probability, as depicted in the insets.
Both the non-capped behavior as well as the capped conductance value atI/VDS= 1e2/h of the single-electron tunneling peak intersection are illustrated. The variation of the linewidth and height of the single-electron tunneling peak repeats at a rate of one flux quantum in the magnetic flux density. Interestingly, this simple model reproduces qual-itatively similar results as obtained experimentally, compare with figure 8.6. However, the finite tilt of the iso-phase line (connecting the maximum or minimum of the conduc-tance modulations) seen in the experimental data requires an additional mechanism5 to be present.
The reiterative change of the tunneling barriers was only discussed qualitatively so far, but the strength of the process has to be evaluated quantitatively. For this purpose, the two quantum dots are considered as a coherent system that is described by a total transmission probability, as depicted in figure 8.2b. The arrangement is considered as two sequential tunneling barriers in one dimension leading to resonant tunneling. The transmission coefficient through a resonant system as a function of the energy of an incident particle relative to the resonance energy Epk reads [24]
Ttot(E,Γ, Tpk) = Tpk
1 + E−Epk
Γ/2
!2
−1
. (8.3)
It takes the form of a Breit-Wigner resonance (centered around Epk), where Tpk is the transmission coefficient of the peak and Γ is the FWHM of the resonance. In case of small, but equal transmission coefficients for the tunneling barriers of both quantum dots a perfect transmission6 is obtained with Tpk = 1, independent of however opaque the
5A possible influence of the plunger gate-voltage is discussed in chapter C.2.11 of the appendix.
6Approximation for small transmission coefficients leads to Tpk ≈(4T1T2)/(T1+T2)2, where Ti with i∈ {1,2} is the transmission coefficients of a single barrier.
8 Two Parallel Quantum Dots - Evolution of a Theoretical Model
B
y x
D
X X
(a)
I/V
DSV
G1 e²/h
(b)
Figure 8.8 (a) A closed wave function (white dotted line) connected throughout the two quantum dots and the compressible strips nearby is expected to be present.
While detuning the enclosed magnetic flux, the wave function will breathe periodically (white arrows) at the rate of one magnetic flux quantum. The spatial reduction is expected to occur mainly in the region of the tunneling barriers of the quantum dots, as illustrated in red on the right hand figure panel. The tunneling barriers are expected to be detuned by the breathing.
(b) In consequence, the linewidth of the single-electron tunneling peak broad-ens (blue). In addition, the peak conductance value can increase (red) unless it already yields the maximum conductance value ofI/VDS= 1e2/h.
162
8.6 Breathing of Compressible Strips
gatevoltage[a.u.]
magnetic field [a.u.]
1-Ttot
Ttot
0 5 10 15 20
0 5 10 15 20
0 0.5 1
I/VDS[a.u.]
(a)
gatevoltage[a.u.]
magnetic field [a.u.]
1-Ttot
Ttot
0 5 10 15 20
0 5 10 15 20
0 0.5 1
I/VDS[a.u.]
(b)
Figure 8.9 Qualitative estimation of the conductance taking a magnetic flux dependent breathing of the linewidth of the single-electron tunneling peaks into account.
The periodic conductance modulations emerge as the tunneling barriers of the quantum dots alter with the magnetic flux density change. Here, both quan-tum dots are considered as a coherent system with only a single transmission probability, as depicted in the insets. Conductance modulations of a nomi-nally conductance value ofI/VDS = 1e2/hfor (a) non-capped and (b) capped intersection.
8 Two Parallel Quantum Dots - Evolution of a Theoretical Model
individual barriers are [24]. A quantitative estimation on the change of the transmission coefficient ∆Ttot(E,Γ, Tpk) of equation (8.3) can be made using calculus of variations
∆Ttot(E,Γ, Tpk) =
"
dT(E,Γ) dΓ
#
∆Γ +
"
dT(E,Γ) dTpk
#
∆Tpk
=
8Tpk(E−Epk)2 Γ
hΓ2+ 4 (E−Epk)2i2
∆Γ +
"
Γ2
Γ2+ (E−Epk)2
#
∆Tpk, (8.4) with the change ∆Γ and ∆Tpk obtained from the experimental data. In the variational calculus two cases can be considered:
1. Unity Peak Transmission:
In case of a transmission coefficient Tpk = 1, the variation of the transmission coefficient in equation (8.4) turns to zero directly on resonance (E−Epk) = 0. This matches the behavior found experimentally for the case of a capped intersection of single-electron tunneling peaks with at least one quantum dot contributing a conductance value of I/VDS = 1e2/h to the common working point (compare with figure 7.16b).
2. Non-Unity Peak Transmission:
In case of a non-unity peak transmission coefficient Tpk < 1 the change ∆Tpk 6= 0 results in a variance of the transmission through the system even directly on resonance (E−Epk) = 0. This matches the behavior observed experimentally in case of each single-electron tunneling peak contributing a conductance value I/VDS <
1e2/h to the common working point (compare with figure 7.16a).
Case 1 shall be considered in the following quantitative estimation, as it poses a more convenient way to extract the variance of ∆Γ from the measured data. The conductance of the two parallel quantum dots as a function of the magnetic flux and the plunger gate voltage of a single quantum dot is displayed in figure 8.10. With the conduc-tance value of I/VDS = 1e2/h the peak transmission coefficient refers to Tpk = 1. The projection of the single-electron tunneling peak on the gate voltage axis allows to ex-tract Γs = α×9.82 mV and Γl = α×12.57 mV as the small and large linewidth7, re-spectively and thus determines ∆Γ = Γl −Γs. For the detuning from the peak of the resonance (E−Epk) =α×5.2 mV is chosen (blue line in figure 8.10a). Inserting these values and Γ = Γs in equation (8.4) yields ∆T(E, Tpk,Γ) = 0.14 for the trans-mission coefficient. This value agrees reasonably well to the peak-to-peak value of the conductance modulations of ∆I/VDS= (0.11±0.02)e2/h, displayed in figure 8.10c.
In the next step the link between a single tunneling barrier and its transmission prob-ability is made using the WKB-approximation8. The transmission probability through a potential barrier of arbitrary shape can thereby be approximated via
TB ≈e−
2
~ x2
R
x1
√
2m∗[V(x)−E] dx,
(8.5)
7The energy scale of the gate voltage axis has to be related to the charging energyEC of the quantum dot, resulting in a correction factor ofα= 0.0097 (see figure C.20 in chapter C.2 of the appendix).
8Acronym for Wentzel-Kramers-Brillouin.
164
8.6 Breathing of Compressible Strips
gatevoltage[V]
magnetic field[T]
5.2 mV
-0.84 -0.83 -0.82 -0.81 -0.8
6.585 6.6 6.615 0
0.2 0.4 0.6 0.8 1
conductanceI/VDS
h e2 h
i
(a)
conductanceI/VDS
h e2 h
i
gate voltage[V]
FWHM
dataΓs
Γl
0 0.2 0.4 0.6 0.8 1
-0.85 -0.84 -0.83 -0.82 -0.81 -0.8 -0.79
(b)
conductanceI/VDS
h e2 h
i
magnetic field[T]
0.35 0.4 0.45 0.5 0.55 0.6 0.65
6.585 6.6 6.615
(c)
Figure 8.10 (a) Combined conductance of the two quantum dots as a function of the magnetic field and plunger gate voltage (top-view representation of the data from figure 7.16b). (b) Projection of the data from (a) on the gate voltage axis in order to extract the linewidths Γs=α×9.82 mV and Γl =α×12.57mV withα = 0.0097, respectively (lines as guide to the eyes). (c) Conductance as a function of the magnetic field taken along the cross-cut marked in blue in (a). The average peak-to-peak value yields ∆I/V = (0.11±0.02)e2/h.
8 Two Parallel Quantum Dots - Evolution of a Theoretical Model
x
1QD
x
2E
V0
F
x
0
Figure 8.11 Qualitative electrostatic potential cross cut through the quantum dot (black solid line). The effective potential for an electron can be approximated by an inverse parabola (red dotted line) with a height ofV0.
whereV(x) denotes the potential barrier andE is the energy of the electron in the 2DES with the effective massm∗ = 0.067mepassing the barrier. The potential cross cut through a quantum dot is qualitatively sketched in figure 8.11. The effective potential V(x) in the one-dimensional approximation can be modeled by an inverse parabola (red dotted line). It shall be defined as V(x) = −ax2 +V0, where a > 0 denotes the curvature of the parabola and V0 the maximum. The transmission probability yields after integration using the curvature dependent boundaries x1 =qV0−Ea and x2 =−qV0−Ea
TB = e−
2
~
√2m∗
x2
R
x1
√
[−ax2+V0−E]
dx
= e
−2
~
√ 2m∗
x2
√
V0−E−ax2+V20√−Ea arctan
√
√ ax
V0−E−ax2
x2=
√V0−E a
x1=−
√V0−E a
= e−π~√2m∗(V0−E)/√a. (8.6)
In the considerations above the periodic conductance modulations are presumed to arise from a variation of the width of the tunneling barrier d= 2q(V0−E)/a. Replacing the curvature a by solving equation (8.6) one obtains
d(TB, V0−E) = h π2√
2m∗
ln (TB)
√V0−E. (8.7)
By variational calculus the change of the width of the tunneling barrier is evaluated as a function of the varying tunneling probability,
∆d(TB, V0−E) = h π2√
2m∗
√ 1
V0−E
∆TB
TB . (8.8)
A value for the effective potential barrier height shall be approximated with (V0−E) = 5 meV. Using the relation between ∆Γ/Γ = ∆TB/TB one obtains a change in the tunneling barrier width of ∆d = 1.9 nm. This value has to be compared to a change in the circumference9 of the closed wave function of about∆U = 1.3 nm.
9Calculated from ∆U = 2πδrwith the change in the radius ofδr= 0.2 nm, deduced from the magnetic
166
8.6 Breathing of Compressible Strips Even if one accounts for the presence of four tunneling barriers, the change in circumfer-ence of the closed wave function and the change in the width(s) of the tunneling barriers are of the same order of magnitude for the chosen tunnel barrier height.
In summary, a breathing of a closed wave function enclosing both quantum dots and the compressible strips close to the dots yields both qualitatively and quantitatively a picture matching the observations made experimentally.
The breathing affects the tunneling barriers of the quantum dots and thus al-ters both the linewidth and the peak conductance value of the single-electron tunneling peaks. Here, both quantum dots were considered to form a coher-ent system described by a single transmission coefficicoher-ent. The peak conduc-tance value will only be modulated, if it deviates from its maximum value of I/VDS = 1e2/h. This mechanism can assumed to be the origin of the h/e periodic conductance modulations found in this operating regime with two parallel quantum dots considered as a single coherent system. In the presence of decoherence the breathing mechanism of the closed wave function breaks down and the two quantum dots have to be considered separately. In consequence, one obtains a coherent and hence modulated contribution to the conductance as well as an incoherent flux independent contribution. This might explain the reduced combined conductance of both quantum dots to a value below the sum of both contributing conductance values of either quantum dot.
flux density period.
8 Two Parallel Quantum Dots - Evolution of a Theoretical Model