Part II Spin blockade with spin singlet electrons
9. Results and Discussions
9.2 The Two-Electron Excitation Spectrum
Similar to the measurement in Fig. 7.6 (c) where the single quantum dot with two electrons shows the singlet-triplet ground-state transition in magnetic fields [56], we measure the current strip by sweeping gate voltages, VgL/Vg , under various magnetic fields from 0 to 10 T with fixed |Vsd| = 4 mV (see Fig. 9.2 (a)), and the result is in Fig. 9.2 (b).
(a)
Fig. 9.2 (a) The Coulomb diamond diagram at 0 T. To measure the two-electron state energy spectrum, we sweep gate voltages along the yellow arrow to trace the variation of chemical potential of (0,2) states from 0 to 10 T, and the result is shown in Fig. 9.2 (b).
The yellow line indicated µ2(0,2) is the ground (0,2) state before the S-T transition.
In Fig. 9.2 (b), the 1s2 orbital S(0,2) state (indicated by the solid line) and the 1s2p+ orbital T(0,2) state (dashed line) are clearly resolved and show the ground-state transition at 5 T. Another excited state appears for H > 7 T (dotted line), and undergoes the second
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ground-state transition at 9 T. This state is suggested to be the spin singlet state with high angular momentum [82]. The energy difference between the triplet ground state and the singlet excited state reaches to be maximum at ~ 7.5 T. Zeeman splitting was not observed owing to the expected small g-factor of our device. All these behaviors are consistent with previous experimental results in vertical single-dot devices [56, 77].
(b)
Fig. 9.2 (b) The excitation energy spectrum for (0,2) states, measured at Vsd = − 4 mV.
The first current strip exhibits the (0,2) state evolution among the solid line of the 1s2 orbital (0,2) state with spin singlet, the dashed line of the 1s2p+ orbital (0,2) state with spin triplet, and the dotted line for another (0,2) state with spin singlet.
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9.3 Spin Blockade in the Vsd-Vg Diagram under Magnetic Fields Going through the Singlet-Triplet Transition
The Coulomb diamond in an intensity plot of differential conductance, dIsd/dVsd, is measured from 0 to 9 T, and Figs. 9.3 (a-f) show data taken at the condition before, during and after the S-T transition.
Begin with Fig. 9.3 (a) measured at 0 T, the SB appears on the positive Vsd side of the Coulomb diamond with the total electron number N = 2. The right corner of the SB region is partially cut by a current threshold that runs nearly parallel to the vertical axis and is indicated by an arrow. The current-carrying cycle for triplet states, T(1,1) → T(0,2) → (0,1), takes place under this condition as previous studies. We also see a “current peak line”
appears at two borders: between the SB region and the N = 1 Coulomb blockade region and between the SB region and the N = 3 Coulomb blockade region. SB is relieved on these two borders for T(1,1) aligned with the Fermi energies of the source and drain electrodes, respectively. In all measurements in the Vsd − Vg diagram, we tuned both VgL and Vg , so that the current peak lines touch Vsd = 0 in the dIsd/dVsd.
Corresponding to Sec. 9.1, increasing the magnetic field yet further before the S-T transition, the current threshold (due to T(1,1) → T(0,2) tunneling) indicated with arrows in Figs. 9.3 (a-c) shifts to a lower Vsd, and the current-suppressed area due to SB is decreased. At 5.0 T, near the S-T transition, the SB region completely disappears and leaves only the N = 2 Coulomb blockade region. Here at this time, both S(1,1) and T(1,1) can tunnel into (0,2) states. Further increasing the magnetic field to be at the condition after the S-T transition, Coulomb diamond data again show the current threshold, which implies tunneling into an (0,2) excited state (indicated by arrows in Figs. 9.3 (e-f)). We observed that the current threshold shifts to a higher Vsd with further increasing magnetic field. However, this threshold becomes blurred and difficult to trace for H > 8.4 T.
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Both results of the (0,2) excitation spectrum (Fig. 9.2 (b)) and Vsd values at the current threshold (Figs. 9.3 (a-f)) give the magnetic-field-dependent energy difference between the ground and excited (0,2) states, Δ𝐸𝑔𝑒 [56, 71]. We summarized them in Fig. 9.3 (g) using circles and triangles, respectively, and the results are nearly identical. This agreement confirms that SSB can take place in the range below the current threshold for H > 5 T.
Notice that, in order to have Δ𝐸𝑔𝑒 , we first convert the difference in the vertical Vg axis in Fig. 9.2 (b) to Vsd; then, eVsd values from these two results are multiplied by the voltage drop proportion on the barriers [83], estimated from the slopes of Coulomb diamonds.
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(a-b)
Figs. 9.3 (a-b) dIsd/dVsd plots under the magnetic fields of (a) 0.0 T, (b) 2.0 T. The arrows mark the threshold of SB.
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(c-d)
Figs. 9.3 (c-d) dIsd/dVsd plots under the magnetic fields of (c) 4.0 T, (d) 5.0 T. The arrow marks the threshold of SB. At 5.0 T, SB is completely relieved and only the Coulomb blockade region is left.
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(e-f)
Figs. 9.3 (e-f) dIsd/dVsd plots under the magnetic fields of (e) 6.2 T, (f) 7.4 T. The arrow s mark the threshold of SSB.
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(g)
Fig. 9.3 (g) Plots of the energy difference between the ground and 1st excited (0,2) states, ΔEge. Circles indicate data extracted from (0,2) excitation spectrum in Fig. 9.2 (b), and triangles are data from the series of Coulomb diamond measurements shown in Figs. 9.3 (a-f). ΔEge measured as VgL/Vg (Fig. 9.2 (b)) and Vsd (Figs. 9.3 (a-f)) are converted to energy using the voltage drop ratio of three tunneling barriers.
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9.4 Hyperfine Interaction Leads to a Short Lifetime of Spin Singlet State
Figure 9.4 (a) shows the Isd − Vsd curve measured along the dashed line in the insert.
The current step at Vsd ~ 3 mV is the tunneling threshold of the first excited (0,2) state.
The “leakage current” of ~ 10 p is seen between the Coulomb blockade region and the threshold. Although the order of the current level is the same as the one above the threshold (~ 15 p ), we consider this value of leakage current consistent with SSB.
In the SSB region, the blocked S(1,1) state is nearly degenerated with one of the unblocked triplets, T0(1,1), where T0 is the zero component of triplet states. In the presence of the hyperfine interaction, nuclear spins generate a randomly fluctuating effective magnetic field ΔBnuc . Whenever S(1,1) and T0(1,1) are close enough to be ≲ gµ𝐵ΔBnuc, the two states will mix with each other efficiently via the hyperfine interaction.
The energy difference between S(1,1) and T0(1,1) is introduced in Sec. 7.4. With the result of a double-dot device with similar barrier thicknesses, the S(1,1) − T(1,1) energy difference near zero magnetic field is estimated to be 0.42 to 0.83 𝜇eV [84]. For △ Bnuc inversely proportional to the square root of the nuclei number, our effective ΔBnuc is
~ 10 mT [63] (each electron resides in our system confronting 105 nuclei in the GaAs dot with an effective diameter ~ 30 nm and the lattice constant = 0.57 nm ). Thus, the hyperfine induced mixing is dominant in our devices.
The time required for the mixing due to hyperfine interaction was measured in the lateral double quantum dots to be ~ 10 ns [63]. Our leakage current of 10 p suggests that the average tunneling interval is e/(10 p ) ~ 10 ns, where e is the elementary charge.
This time interval is consistent with the expected lifetime of S(1,1)! Notice that the S(1,1) − T(1,1) energy difference is a function of tc, therefore diminishing the interdot tunnel barrier, less than the 8 nm in our system, will lift the degeneracy of the S(1,1) and
- 113 - T0(1,1) states and decrease the leakage current in SSB.
(a)
Fig. 9.4 (a) The Isd − Vsd curve along the dashed line in the insert Vsd − Vg diagram measured at 7.4 T. A leakage current of 10 p is observed in the SSB region.
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Chapter 10
Summary and Further Works
In this thesis, we observe the a new form of the spin blockade -- the singlet spin state blockade (SSB) in two-electron charge diagrams of a vertical double QD system, where the (0,2) state takes the spin triplet as the ground state under high magnetic field. The source-drain voltage dependences of the current threshold from the SB and SSB regions are consistent with the measured magnetic field dependence of the excitation spectrum of the (0,2) states. The leakage current found in SSB gives the lifetime of S(1,1) ~ 10 ns restricted by the randomly fluctuating effective magnetic field owing to the hyperfine interaction.
Under SSB, as proposed in Sec. 6.2 or Appendix (A), two nuclear spins in each of the two dots respectively can be entangled whenever spin flip-flop process occurs due to the hyperfine interaction. In the further work, we would like to verify the the steady condition for the nuclear spins being spin singlet by measuring the nuclear magnetic fields with similar devices of a smaller interdot tunnel barrier. To emphasize that the S pumping in Ref. [85]
successively alters the gate voltage to let the system be at S(0,2) → S(1,1) → the degeneracy of S(1,1) − T(1,1) in lateral double QDs. The entanglement in S(1,1) can also pass to nuclei at the S(1,1) − T(1,1) degeneracy. However, to generate the nuclei
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entanglement by repeating the gate-control sequence (via capacitance coupling to the dots) takes much longer time than just staying SSB to wait for SSB lifting and taking places repeatedly.
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Appendix (A) The Expected Value of the Entangled Pairs of Nuclear Spins under the SSB condition
To calculate the steady condition that the system reaches as the singlet electron spin state repeatedly passes the entanglement to a nuclear spin pair from each of the two QDs via the spin flip-flop interaction, we first consider the simplest case N1 = N2 = 2, where the N1/N2 is the total number of the nuclear spins in QD1/QD2 as shown in Fig. A (a).
(a)
Fig. A (a) The schematic diagram of the electron/nuclear spins, represented as ⇑/↑, in a imaged system that there are only two nuclear spins in each dot. Under SSB, the system is blocked with electron spin singlet until it interacts with nuclear spins via the hyperfine interaction and release the SSB by spin flip-flop process which passes the entanglement to the nuclear pairs. The α indicates the hyperfine interaction strength (see Sec. 6.2).
In Fig. A (a), we assume N1 = N2 = 2. Since the nuclear spin direction can be either spin up or spin down, we express the mixed state of the four nuclear spins to be
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(α1↑1+ β1 ↓1)(α2 ↑2+ β2 ↓2)(α3 ↑3+ β3 ↓3)(α4 ↑4+ β4 ↓4) ≡ mix2
for |α𝑖|2+ |β𝑖|2 = 1, ↑, ↓ standing for the nuclear spin direction.
After the first time the singlet electron spin state interacting with the nuclear spins under the spin flip-flop process which can be regarded as an exchange states action (see Sec.
6.2), the nuclear spin state can be rewritten as
(↑1↓2−↓1↑2)(α3 ↑3+ β3 ↓3)(α4 ↑4+ β4 ↓4) ≡ S12𝑚𝑖𝑥34
this is considering the case that the nuclear spin 1 in QD1 and the nuclear spin 2 in QD2 are entangled to be spin singlet. However, we cannot actually distinguish or identify the nuclei, so the condition would be the combination of all the possibilities as
1 of the possible spin pairs has been already entangled, then we can ignore the condition the same pair entangled again at the next time. Therefore, the S12𝑚𝑖𝑥34 term become 0; for
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the S14𝑚𝑖𝑥32/S32𝑚𝑖𝑥14 terms, the entangled nuclear spin pair of 1 and 4/ 3 and 2 are broken to have the new entangled spin pair of 1 and 2. Further, we also may have two entangled pairs as S12𝑆34. Similarly, if we consider all the probabilities, the condition is
𝑆 ∙ 𝑚𝑖𝑥after 2 → 𝑛𝑑 time 1 flip-flop process happens again. Say if the nuclear spin 1 and 2 are entangled after one more interaction, then similarly
That is, as the spin flip-flop process exchanges the nuclear spin and electron spin states,
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In Fig. A (b), we express what and the possibilities the nuclear spin state changes to be as the spin flip-flop process repeatedly occurs under SSB due to the hyperfine interaction.
After the nth process or action, the probability of nuclear spins being mix2, 𝑆 ∙ 𝑚𝑖𝑥, and 𝑆2 is P0(𝑛), P1(𝑛) and P2(𝑛) respectively (note: P𝑖(n) for i indicating the number of nuclear spin pairs being as spin singlet). And, the matrix form is
[
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Under the steady condition, P𝑖(𝑛 + 1) should be equal to P𝑖(𝑛) for i = 0,1,2.
Therefore, we can solve the matrix and have P0 = 0, P1 =34 and P2 =14, and the expected value 𝑃̅ for nuclear spin pairs being singlet is
𝑃̅ = ∑ 𝑖 results be a half of this value, i.e.
𝑃 hyperfine interaction again passes the electron singlet state to nuclear spins, and the total possibility equals to 1:
(1) S𝑛+1mix𝑚−1: The interaction pairs one of the random nuclear spins in QD1 and that in QD2, so that there will be m × m = 𝑚2 possibilities, and we have one more extra entangled pair.
(2) S𝑛mix𝑚: One nuclear spin from the singlet nuclear spin pair entangles one of the random nuclear spins. Since one of the original pairs is taken apart to form a new pair, the number of the nuclear spin singlet keeps the same.
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(3) S𝑛−1mix𝑚+1: Two singlet spin pairs are broken and each provides one nuclear spin to form a new pair of spin singlet. The possibility will be n × (n − 1) = 𝑛2− 𝑛. Like in Fig. A (b), we also draw a diagram of the change of the nuclear spins among all possible conditions in Fig. A (d) for clear.
(c)
Fig. A (c) Assuming there are N nuclear spins in both QD1 and QD2, then after nth hyperfine interaction to exchange the spin state with electron spin singlet under SSB, the nuclear spins have mix𝑁, Smix𝑁−1, S2mix𝑁−2, … or S𝑁possible conditions.
(d)
Fig. A (d) The diagram of the nuclear spins changing among the mix𝑁, Smix𝑁−1, S2mix𝑁−2, … or S𝑁 possible conditions. Whenever the interaction acts at the condition of S𝑛mix𝑚 (n + m = N), it may change to be S𝑛+1mix𝑚−1, S𝑛mix𝑚 or S𝑛−1mix𝑚+1. The arrow show the change of the direction and the number above is the probability.
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Similarly, we can also calculate the expected value 𝑃̅ for nuclear spin pair being singlet with the following conditions:
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By setting the initial conditions, and repeatedly N times of the exchange state action to have the 𝑃0, 𝑃1…𝑃𝑁. Then, the expected value for nuclear spin pair being singlet is
𝑃̅
𝑁 =∑𝑁𝑖=0𝑛𝑃𝑖 𝑁
In the double QD system, if all the nuclear spins residing in each QDs with equal nuclei number are paired to be spin singlet, then we can image that the whenever we measured a nuclear magnetic field in one dot (for example, the fluctuating effective nuclear magnetic field ∆𝐵𝑛𝑢𝑐 is a function of √𝑁, and ∆𝐵𝑛𝑢𝑐 ~10 mT for N = 105 in GaAs dots), there should be the same amplitude of a nuclear magnetic field in the other dot. Furthermore, these two nuclear magnetic fields direct oppositely due to the spin singlet property.
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Appendix (B) Constant Interaction Model for Single QD
To understand the electronic transport properties through the quantum dot systems, we introduce the constant-interaction (CI) model. And, there are two assumptions under the CI model, and Fig. B (a) shows an equivalent circuit of a single dot system:
1. The Coulomb interaction of an electron on the dot with the environment and with other electrons on the dot can be parameterized with a constant capacitance C.
2. The influence of interaction on the discrete single-particle energy spectrum is ignored.
(a)
Fig. B (a) An equivalent circuit for the single-quantum-dot system. The dot connected to the leads via tunnel barriers is characterized by parallel series of the resistor Rs/Rd and the capacitor Cs/Cd, while it capacitively couples to the gate through a capacitor Cg.
Under the above assumption, we can write down the electrostatic energy U(N) of the
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Cd + Cg, and E𝑖 represents the i-th single-particle energy level. |e|N0 compensates the positive background charge due to the donors in the heterostructure, and Cg Vg, Cs Vs , Cd Vd terms represent effective induced charges that changes the electrostatic potential in the dot continuously.
The electron transport condition can be easily exhibited by the electrochemical potential, µ(N), defined as the energy needed to add the Nth electron into the dot. µs/µd are electrochemical potential of the source/drain, and only when µ(N) is within the bias window, i.e. µs ≥ µ(N) ≥ µd (see Fig. B (b)), electrons can flow through the system; beyond this condition, the system is in the so-called Coulomb blockade (CB) regions. Since the energy needed to add the Nth electron into the dot should be the difference of the total energy of the system with N and N-1 electrons, or 𝜇(𝑁) ≡ 𝑈(𝑁) − 𝑈(𝑁 − 1), we write down 𝜇(𝑁) as
𝜇(𝑁) = (𝑁 − 𝑁0−12) 𝐸𝑐 −|𝑒|𝐸𝑐(Cg Vg+ Cs Vs + Cd Vd) + 𝐸𝑁, for Ec = e2/C is the charging energy.
(b)
Fig. B (b) A schematic diagram of the levels in single quantum-dot device, the electron can flow via the system only when levels falls within the bias window determined by µs and µd [79].
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With the value of 𝜇(𝑁), we can have the boundary conditions of CB regions as a function of Vsd and Vg, i.e. the boundary of white regions in the Vsd − Vg charge diagram shown in Figs. B (c-d): First, considering the situation of Vsd ~ 0. The current only flows when µ(N, Vg) = µlead. In another word, along the Vg axis, the current peak can be predicted to occur with an interval of ∆Vg = (C/eCg)∆µ(N) given by µ(N, Vg) = µ(N + 1, Vg+ ∆Vg) = µlead. Note that the addition energy, 𝐸𝑎𝑑𝑑, is defined to be ∆µ(N) equal to ∆E + Ec. Second, if we applied ∆Vsd > 0 and assume that µs = e∆Vsd/2 and µd = −e∆Vsd/2, then ∆µ(N) can be expressed when µs = µ(N + 1) and µd = µ(N) or ∆µ(N) = e∆Vsd. This is the condition at the yellow crossing point of a pair of two lines in Fig. B (c). With the relations of ∆Vg = (C/eCg)∆µ(N) and ∆µ(N) = e∆Vsd, we can analysis the spectrum in Fig. B (d).
(c)
Fig. B (c) The schematic diagram for a Coulomb diamond. The alignment between 𝜇(𝑁) and at least one of µlead determines the boundaries.
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(d)
Fig. B (d) The energy spectrum of a single dot represented in dIsd/dVsd on the Vsd − Vg plane. The white area is the Coulomb Blockade region where no electron can flow through the dot until −|e|Vsd (= µs − µd ) ≥ Eadd) [56].
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Appendix (C) Constant Interaction Model for double QDs
(a)
Fig. C (a) As single dot in Fig. B (a), a double quantum dot system connected to the leads and gates can be also characterized by resistors and capacitors. Besides, the tunnel barrier separating two series-connected dots is represented by a resistor Rm and a capacitor Cm, while the tiny cross-capacitances (such as between Vg1 and dot 2) are ignored.
An equivalent circuit of a double quantum dot system under CI model is shown in Fig. C (a), and the coupling between two dots is assumed to be Cm. In single quantum dot system, the chemical potential includes the charge energy and the single-part energy parts:
µ(N) ≡ U(N) − U(N − 1) = (N − N0−12) Ec−Ec
|e|(Cg Vg+ Cs Vs+ Cd Vd) + EN
as well as the additional energy, 𝐸𝑎𝑑𝑑 ≡ ∆µ(N) = Ec+ ∆E. And, we have similar forms for double QD systems. We define µ1(2)(N1, N2) as the energy needed for the N1(2)-th electron entering the dot1(2) to occupy discrete level N1/N2, while there are already N2(N1) electrons in dot 2(1), i.e.
µ1(N1, N2)≡ 𝑈(N1, N2) − 𝑈(𝑁1− 1, N2) &
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µ2(N1, N2)≡ 𝑈(N1, N2) − 𝑈(𝑁1, N2− 1)
With respect to the single QD system, we have to further consider the interdot charge energy. Therefore, energy change of one dot as an electron entering into another dot.
As mentioned in Chap. 2.2, electrons can transport through the system via two sequences of electron states as though one is going through the electron transport, while the other is through the hole transport:
(N1, N2) → (N1+ 1, N2) → (N1, N2+ 1) → (N1, N2) &
(N1,N2 + 1) → (N1+ 1,N2+ 1) → (N1+ 1,N2) → (N1,N2+ 1)
thus, these two situations construct the boundaries of the CB diamond edges in the Vsd − Vg plane. And whenever either of the conditions below is achieved, the current can flow through the dots, and the corresponding diagrams of chemical potentials for transport are in Figs. C (b-c) :
µs≥ µ1(N1+ 1, N2) ≥ µ2(N1, N2+ 1) ≥ µd
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or
µs≥ µ1(N1+ 1, N2+ 1);
µ1(N1+ 1, N2) ≥ µ2(N1, N2 + 1);
µ2(N1, N2+ 1) ≥ µd
(b)
(c)
Figs. C (b-c) Electrochemical potential levels in a double-dot system. Electrons can tunnel through the system via the sequence of
(b) (N1, N2) → (N1+ 1, N2) → (N1, N2+ 1) → (N1, N2) or (c) (N1,N2+ 1) → (N1+ 1,N2+ 1) → (N1+ 1,N2) → (N1,N2+ 1).
- 131 - samples) for the following fabrication processes (b) The simplified schematic diagram of the water structure composed of multilayers as Table 3.
1. Prepare wafer in suitable size:
Scribe and cut the wafer in a size of 𝟗 × 𝟏𝟎 𝐦𝐦.
2. Clean the surface & Etch the oxide on the surface
A. Immerse the wafer in a plastic beaker with IPA (Isopropyl alcohol).
B. Clean the wafer in the beaker with an ultrasonic cleaner for 5 mins (Twice).
C. Dry the wafer by blowing N2, and check the surface under microscopic.
D. Put the wafer in a plastic beaker with alkali for 30 secs to remove the oxide on the surface.
E. Rinse the wafer with flowing DI water over than 1 min, and make it dry by blowing N2.
F. Post-bake for 10 mins under 110℃.
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B. Back Contact the back contact and the number are fabricated with photolithography.
(c)
Fig. D (c) Schematic diagrams of the top views (left) and the side views (right) of the wafer during the back contact fabrication.
1. Photolithography: Spin coating and Exposure
A. Drop the photoresist on the center of the surface.
B. Spin coating in 3 stages:
500 rpm for 3 secs slop for 7 secs 4000 rpm for 40 secs.
C. Pre-bake (soft-bake) for 20mins under 80℃.
D. Expose with a mercury lamp for 12 secs.
2. Developing: Remove the photoresist
A. Develop with S351 developer, and sway for 40~60 secs.
B. Rinse with flowing DI water over than 1 min, and make it dry by blowing N2.
3. Evaporation and Lift off
A. Deposit 20 nm-thick Ti and then 200 nm-thick Au.
B. Immerse in Acetone for 5 mins, and spray Acetone to lift the photoresist off.
C. Immerse in clean Acetone for 1 min and rinse with flowing DI water.
D. Bake for 10 mins under 110℃.
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C. Top Contact In this step, the top contact is fabricated with E-beam lithography.
(d)
Fig. D (d) Schematic diagrams of the top views (left) and the side views (right) of the wafer during the top contact fabrication.
1. E-beam lithography: Spin coating and Exposure A. Drop PMMA on the center of the surface.
A. Develop with both IPA and OEBR-1000 under 13.6℃:
IPA for 1 min OEBR-1000 for 30 secs IPA for 1 min.
B. dry by blowing N2 (note : no DI water)
3. Evaporation and Lift off
A. Deposit 20 nm-thick Ti and then 100 nm-thick Au.
B. Lift off:
Acetone for 5 mins spray Acetone immerse in clean Acetone for 1 min.
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C. Rinse with flowing DI water and make it dry by blowing N2
D. Bake for 10 mins under 110℃.
E. UV Ozone stripper: Purge Ozone for 10 mins under 200℃.
E. UV Ozone stripper: Purge Ozone for 10 mins under 200℃.