A Comparison with the Theory of Granular Metals

Let us discuss that our results can be satisfactorily, but not fully, interpreted in terms of the recent theory of granular metals. We have fitted our measured σ ∝ ln T results with the predicted σ(T) (see Sec. 2.4). To carry out the least-squares fits, we assume a value of the charging energy Ec ≈ 10 k_𝐵T^∗, where T^∗ is the temperature below which the argument of a logarithmic function [48]. Inspection of Table 1 indicates that in the junctions A–C, we obtained g ≫ 1. This result is in good consistency with the prerequisite for the predicted σ(T) to be applicable. On the other hand, our extracted value of g ≃ 1 in the junction D implies that this sample falls marginally inside the regime of validity of the predicted σ(T). We notice that the predicted σ(T) was formulated by considering a periodic cubic array of uniformly sized grains and neglecting dispersion of the intergrain tunneling conductance [25], while our samples contained random arrays of varying-sized, disk-shaped granules [6]. Therefore, a close quantitative comparison of our (and other groups’ [24, 49, 50]) experiment with theory is not possible at this stage.

Another important feature of the predictions of the theory of granular metals is that the predicted σ(T) should be valid at any magnetic field. This is indeed confirmed by our experiment. We have measured σ(T) of the Cr electrode in the junction A between 4 and 20 K in both zero magnetic field and in a perpendicular magnetic field of 4 T (see Fig. 4.4 (a)). The measured values are the same to within our experimental uncertainty. On contrary,

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the WL effect, if any exists, should be very sensitive to and suppressed by even a small magnetic field [47]. If the EEI effect were responsible, we should then have observed a √𝑇, but not a ln T, dependence in this sample in this temperature interval [51]. Therefore, both the WL and EEI effects are irrelevant to our observations in Figs. 4.1 (a-b).

The mean energy-level spacing δ in our Cr granules may be evaluated as follows. We have carried out AFM studies of a film deposited under conditions similar to those used for the fabrication of the junction B. We found that the Cr film formed a granular structure consisting of disk-shaped grains of ≈ 60 ± 20 nm in diameter and ≈ 1.5 ± 1 nm in height, along with a few larger aggregations. By taking an average diameter of ∼ 60 nm and an average height of ∼ 1.5 nm, we obtain an estimate of δ = 1/N_{𝑐𝑟,𝑑}(0) V̅ ≈ 2 µeV, where V̅ is the average granule volume. This δ value in turn suggests a characteristic temperature T_𝐵 ≈ gδ/k_𝐵 ≈ 1 K above which the predicted σ(T) equation is expected to apply.

Experimentally, the σ ∝ ln T law in our junction B is observed in the temperature interval 0.3– 7 K, see Fig. 4.1 (b). This degree of agreement is satisfactory, considering that the evaluations of parameters in a granular sample unavoidably involve large uncertainties.

Using our estimated values of Ec ≈ 6 meV (see Table 1) and δ ≈ 2 µeV, we obtained the ratio Ec/δ ≈ 3 × 10³. This ratio suggests the existence of a broad range for logarithmic corrections to conductivity. However, even under such circumstances, Feigel’man et al. have theoretically shown that a simple σ ∝ ln T law should still hold in a wide range of temperature [52]. This prediction is confirmed by the present experiment.

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(a)

Fig. 4.4 (a) Plots of temperature-dependent conductivities in Junction A under magnetic field H = 0 T (triangles) and H = 4 T (squares). The small difference between two curves is within our experimental uncertainty. The results confirm that the WL effect (which is sensitive to the magnetic fields) is irrelevant to our system, while the prediction on conductivity in the theory of granular metals should be valid at any magnetic field.

- 53 -

4.5 A Comparison with the Theory of Granular Metals --

Tuning to the differential conductance curves, we discuss the crossover behavior of prediction of ν₃(ϵ). However, a close comparison with theory cannot be made at this stage, because the numerical prefactor A in the predicted ν₃(ϵ) was calculated for the case when the logarithmic term is much smaller than 1 [26]. Quantitatively, the magnitudes of the G(V) dips we observed are much larger than that predicted by ν₃(ϵ) in Sec.2.4. On the other hand, our g values are larger than the critical intergrain tunneling conductance g_𝑐 = (2πd)⁻¹ ln(Ec/δ) (≈ 0.4, using the above Ec/δ value) [44]. Therefore, we do not expect to find a “hard” gap in our samples [27]. Our G(V) ∝√𝑉 results in the high bias voltage regime (V ≳ Vc) also have to await a future theoretical explanation [52].

Finally, in the junction D, we did not observe any G(V) ∝ ln V dependence even at relatively low bias voltages, which may be due to the fact that g ≃ 1 in this sample and thus the prediction of ν₃(ϵ). is marginally applicable. However, recall that we found the σ ∝ ln T behavior, as predicted by the predicted σ(T). In fact, G(V) ∝√𝑉 was observed in this sample in a wide range of |V | ≈ 1– 100 mV at 2.5 K, the insert of Fig. 4.2 (e).

Furthermore, we found that in this particular sample, the scaled differential conductance [G(V, T) − G(0, T)]/√T versus the combined parameter √e|V|/k_𝐵T for different

- 54 -

measurement temperatures between 2.5 and 32 K collapse closely onto a single curve, Fig. 4.5 (a). This result strongly suggests the existence of a universal scaling function in the g ≃ 1 regime. That is, there exists a function f such that G(V, T) − G(0, T) = √T × f(√eV/k𝐵T), where f should depend on the combined parameter eV/k𝐵T, instead of depending independently on eV or k_𝐵T. For comparison, Fig. 4.5 (b) shows the unscaled G(V) versus bias voltage V at five measurement temperatures. Previously, a universal scaling behavior of differential conductance has been theoretically predicted for 3D weakly disordered homogeneous conductors (Refs. 13 and 36). Our observation of Fig. 4.5 (b) demonstrates that a universal scaling phenomenon also exists in the granular case.

- 55 -

(a)

Fig. 4.5 (a) Normalized differential conductance, (G(V, T) − G(0, T) )/√T, as a function of the combined parameter √eV/k_𝐵T for the junction D at five measurement temperatures. Notice that the data points collapse closely.

- 56 -

(b)

Fig. 4.5 (b) Unscaled G(V) versus bias voltage V at (from bottom up) 2.5, 5.0, 9.0, 16 and 32 K.

- 57 -

Chapter 5

Summary

We have measured the conductivities σ(T) in granular Cr electrodes and the differential conductances G(V) in Al/AlOx/Cr tunnel junctions at liquid-helium temperatures.

In samples with dimensionless intergrain tunneling conductances g ≫ 1 , we found σ ∝ ln T and G(V) ∝ ln V at low bias voltages. These results are satisfactorily understood in light of the recent theory of granular metals. A crossover of G(V) from the ln V to √𝑉 dependence was observed at high bias voltages. In a sample with g ≃ 1, we found σ ∝ ln T and G(V) ∝√V in a wide bias voltage interval. Moreover, the normalized differential conductance [G(V, T) − G(0, T)]/√T reveals a universal scaling behavior with the combined parameter√e|V|/k_𝐵T in a wide range of temperature. This last observation requires a further theoretical explanation. Finally, we would like also to note that, while the theory of granular metals considers a periodic array of uniformly sized grains, in real samples one often has some distribution in granule size. The effect of such size distribution on our results in the present study has yet to be fully addressed.

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Part II

Spin blockade with spin singlet electrons

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Chapter 6

Introduction

6.1. Motivation

Quantum dot (QD) systems, developed with the growing of the semiconductor device fabrication technique, provide a possible candidate for the application on the quantum computer area [53-55] and a path to investigate the fundamental physic, especially atomic physics, in more easily accomplishable experimental conditions [56,57].

In classical computers, information is represented by a string of bits indicating as “0”

and “1”. However, in the quantum world, if a system can be controlled between two states, resembling as the mentioned “0” and “1”, the wave property enables the information to be computed in a combination or superposition of the two states. And, this so called “quantum parallelism” ability makes quantum computers more powerful to solve the problems which are considered intractable nowadays [58].

We can summarize five requirements [59] for establishing quantum computers. And, through the introduction on these requirements in the following paragraph, we could know how spins are treated as quantum bit (qubit) in double QD systems [53, 54, 59] and what kind of roles the spin blockade phenomenon can play [60-63]. These both motive us to study our topics in this thesis:

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1. A system with a few discrete quantum states

A two-level system is normally ideal for representing the qubit. For a single spin (a one-bit operation), the two levels can be spin up and spin down respectively; for a two-bit operation, the two-particle spin configuration as spin singlet and spin triplet can be are utilized as two discrete levels.

2. Able to accomplish logic gates

A XOR (or controlled-NOT) gate, known to be employable for any arbitrary quantum computation, can be produced by two square roots of the swap operation of a two-qubit gate with a set of a one-qubit gate.

3. High coherence time

Spin state coherence time should be longer enough to execute a logic gate accurately.

That is, it should be at least 1000 times larger than the logic gate operation time. As we know, spin is less sensitive to the environment as comparing with charge. However, according to materials composed of the QD systems, the spin orbital interaction and/or the hyperfine interaction will play important parts to influence the coherence.

4. Able to initialize the qubit state 5. Able to measure the qubit state

Before having a brief illumination on the Pauli spin blockade (SB) in a double QD system in Sec. 7.3, we can first consider it just as a spin dependent barrier that only allows electrons to flow through if they are spin singlet, whereas spin triplets will lead the system to a blockade. This property helps out both the requirement 4 & 5 [60-63], since we can simply prepare a two-electron system to stay in triplet spin states for experiments (i.e. to initialize the qubit state). Further, we can measure or identify which qubit state the system is by only

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observing the electron-flow signals or the tunneling current (i.e. to measure the qubit state).

In this work, we observe a novel spin blockade phenomenon in the double QD system -- the singlet spin blockade (SSB). In contrast to the Pauli spin blockade resulting from forming one of three triplet states randomly, whenever SSB takes place, it blocks with the only state -- electron spin singlet. We propose a mechanism to utilize the singlet spin blockade and the hyperfine interaction to entangle the nuclear spins in Sec. 6.2. For a two-qubit operation, we need to establish links between distant qubits; that is, to entangle states between separated quantum systems. In the past studies, the entanglement between electron spins or between electron spin and nuclear spin is possible [53, 65]. However, so far the entanglement between nuclear spins has not been achieved. In Table 2, the conditions or developments for electron spins and nuclear spins as quantum information carriers in GaAs, which composes of our QD system, are summarized.

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Table 2 Electron spins vs. nuclear spins as quantum information carriers

Electron Spin in GaAs Nuclear Spins in GaAs

Two-level system ○ ○(Four level)

Initialization ○ ○(Polarization ~ 40%) [66]

Coherence ○(~ µs) [63] ○(~ ms) [67]

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6.2 ( A Proposal )

To Entangle Nuclear Spins with Singlet Spin Blockade

The singlet spin blockade occurs when there is only one electron staying in each of the double QDs and the two-electron spin configuration is singlet. In this section, we propose a mechanism that while the singlet spin blockade phenomenon is relieved via the spin flip-flop process due to the hyperfine interaction between electron spin and nuclear spin, the nuclear spins residing in two QDs are entangled at the same time.

In Fig. 6.2 (a), each of the two QDs has a single electron interacting with one of cross-dot interactions are tiny, so that the α12 and α21 terms can be dropped. If we further assume that α11 and α22 have the same strength as α11 ≈ α22 = , then the hyperfine

- 64 - same dot are opposite, the spin flip-flop process is executable to exchange their spin states.

However, if we only consider the spin states before and after the interaction (such as the example above, | ⇑₁⇑₂↑₁↓₂> turns to be | ⇑1⇓₂↑₁↑₂>), it looks like that the interaction exchange electron and nuclear spin states with each other no matter whether their spins direct oppositely or not. Since if the spin states of the electron and the nuclei have the same orientation, the states after the interaction will be the same as those after the exchange (if it really happens).

(a)

Fig. 6.2 (a) The schematic diagram of the hyperfine interaction in a double QD system, where ⇑/↑ shows the electron/nuclear spin direction and α is the coupling strength.

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Similarly, we can also see what would happen if the hyperfine interaction works on two electrons forming the spin singlet state |(⇑₁⇓₂−⇓₁⇑₂) ↑₁↑₂> the spin flip-flop process, no matter which direction the electron spin and the nuclear spin point to, can be retreated as an exchange spin state action within the same dot as just mentioned. And further, for the spin singlet state, even though we do not know the electron spin direction in one dot but we do know the electron spin in the other dot should direct oppositely. Therefore, when two electron spins respectively exchange the state with the nuclear spin in its own dots, the connection between nuclei in two separated dots is established. The probability of being either | ⇑₁⇓₂> | ⇓₁⇑₂> in electrons now passes toward an arbitrary nuclear spin pair which is entangled as forming to be the singlet state at the same time.

Compared with the Pauli spin blockade keeping the system in triplet spin states that we cannot determine the precious one, the singlet spin blockade with the aim of the spin flip-flop process via the hyperfine interaction provides a way to entangle the nuclear spins staying in two dots to be just as one spin state—the singlet spin state. We can also calculate how many nuclear spin pair will be entangled under the steady condition if this mechanism works repetitively. And, the calculation on the expected value of the entangled nuclear spin pairs under the steady condition and a possible verified measurement as our further work are introduced in Appendix (A).

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Chapter 7

Background and Theory

Basics of the semiconductor QDs and the relative spin-dependent electron transport behavior will be introduced in this chapter. We start with the general transport properties of Sec. 7.1 single quantum dot systems, and then extend to Sec. 7.2 the vertical double quantum dots which we chose as our system in this thesis. In such a system, we control and limit the electron number to be within one or two, so that we can see the occurrence of Sec.

7.3 (Pauli) spin blockade. Further, a more clear vision of this spin blockade behavior will be shown with an understanding on the two-electron energy diagram in Sec. 7.4. We introduce the singlet spin blockade (SSB) in Sec. 7.5, while the two-electron singlet-triplet ground state transition, which we demonstrate in this thesis to achieve SSB, is in Sec. 7.6.

7.1 Single Quantum Dot Systems

As the development in the semiconductor device fabrication technique grows, it is possible to restrict electrons in a nano- to micro-meter-sized device composed of 10³ to 10⁹ atoms. We call such a small device as Quantum dot (QD) where electrons are confined to a dimension close to de Broglie wavelength (~ 100 nm). Therefore, it has two distinct well-known features -- the discrete energy spectrum and manifest quantum effects, and is often regarded as an artificial atom [68].

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Since every semiconductor material has different or its own bandgap, the conduction band of multilayers of semiconductors can form a potential well with discrete energy levels sandwiched between barriers as in Fig. 7.1 (a). In such semiconductor heterostructures, the confinement is one dimensional, or along the z-direction in this thesis. All the parameters, including electrode reservoirs for electron transport between the external environment and QDs, are well-considered so that all the excited states in the z-direction will be empty; that is, only the ground state is occupied with electrons. For building a quantum dot, however, it is still not enough, and we need to further confine the x-y plane. This can be done just with the evaporated metal gates or combining with the etching technique. And, in this way, we will have disk-like shaped quantum dots. The metal gates surround the semiconductor dot, so that we can control the depletion region by applying the gate voltage, Vg, and the confinement in x-y plane or the size of the dot will be changed. That is to say, energy levels of the dot with respect to the reservoirs are able to be tuned with Vg. The equivalent circuit diagrams of the vertical and lateral QD systems are shown in Figs. 7.1 (b-c).

(a)

Fig. 7.1 (a) Schematic diagrams of semiconductor heterostructures such as AlGaAs/GaAs/AlGaAs that confine electrons to a plane.

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(b) (c)

Figs. 7.1 (b-c) The equivalent circuits for the (b) vertical and (c) lateral quantum dot system. The gate couples to the dot capacitively, and barriers between electrode reservoirs and the dot can be represented as a parallel connection of resistance and capacitor.

In a disk-like shaped QD system, the energy spectrum can be solved with a 2D harmonic parabolic confinement, and En, l = (2n + |l| + 1)ℏω₀ , for ℏω₀ : the electrostatic confinement energy, n : the radial quantum number, and l : the angular momentum quantum number. When we consider the electron transport conditions through the QD system, besides the energy level spectrum, the Coulomb repulsion among electrons also has to be taken into account when the total electron number, N > 1. The phenomenon that an electron needs to overcome this Coulomb repulsion energy to tunnel into the QD is called Coulomb blockade (CB) [57]. The constant interaction (CI) model generalizes both concepts of discrete levels and the Coulomb repulsion, which parameterizes the Coulomb interaction of an electron on the QD with a constant capacitance C as facing the environment and other electrons on dot. The details are in Appendix (B), and we calculate

1. The total energy of the dot, U(N)

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2. The electrochemical potential, µ(N), which is defined as the energy of adding the N-th electron into the dot, and is the difference in the total energy of the dot containing N and (N-1) electrons, i.e.

We can express the allowing condition for the current flow via the QD as µs ≥ µ(N) ≥ µd, where µs/ µd stands for the electrochemical potential of the source/ drain electrode.

There are two parameters affecting the electron transport: µ(N) for the dot, and µs/ µd for the electrode, and both of them are controllable. As in Fig. 7.1 (f), the transport window opened by µs, µd is determined by eVsd, or µs − µd = eVsd; meanwhile, µ(N) can be tuned by Vg as mentioned in the second paragraph.

With the basic understanding of the above, we analyze or measure the current behaviors in the QD from Figs. 7.1 (d-h). First, in the simplest case Vsd ≈ 0, a series of separated current peaks, or Coulomb oscillation peaks, shows up as applying continuously Vg (see Fig. 7.1 (d)). Figure 7.1 (e) is the corresponding situation of these peaks: only when µ(N) aligns with µs = µd, electrons are able to tunnel through the dot; the distance between the peaks is actually the addition energy, 𝐸_𝑎𝑑𝑑. Second, if we apply a finite Vsd so that eVsd ≥ E_𝑎𝑑𝑑 such as the condition in Fig. 7.1 (f), then the blockade will be

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completely released. In Fig. 7.1 (g), we sweep Vsd at continuously different Vg to have the Vsd − Vg diagram. Along the green line is the condition in Fig. 7.1 (e) and the blue spots are the condition in Fig. 7.1 (f). Since Vsd bias will partially cross on barriers and relatively tune µ(N), the blockade conditions are diamond-shape current-suppressed areas centered to the zero bias, and called to be Coulomb diamond. Alignments between µ(N) and µ_𝑠 or µ(N) and µ_𝑑 construct the edges or the boundary conditions of Coulomb diamonds (see Fig. 7.1 (h)). Notice that the size of every Coulomb diamond in Fig. 7.1 (g) [56] is actually different and especially large at N = 2,6,12 …. This arises from the atom-like property of the QD. Or specifically, the shell structure of the first few levels in a 2D atom is revealed -- 1s orbit allowing for 2 electrons, 2p^ (for 4 e^-), 3s and 3d^ (for 6 e^-)…. Therefore, we can see that the addition energy is especially larger when the electron is going to occupy the next shell.

(d)

Fig. 7.1 (d) Coulomb Oscillations, the current peaks measured at Vsd ≈ 0. The peaks only happens when µ(N) = µs = µd, and we can tune µ(N) with Vg . The difference

Fig. 7.1 (d) Coulomb Oscillations, the current peaks measured at Vsd ≈ 0. The peaks only happens when µ(N) = µs = µd, and we can tune µ(N) with Vg . The difference