Introduction

Single electron tunnel (SET)1 is an inspiring topic in quantum box transistors (QBTs)2,3 owing to its diverse applications in single-electron devices such as digital memory cells, high precision electro-mechanical sensors and standardized meters4,5 that consume one-tenth of pico-watt energy to upgrade the science and technology of semiconductors. With the application of a fixed, small bias on the drain-source electrodes and adjusting the bias on the gate electrode in a QBT, single electron tunnel peaks in current-voltage (I–V) characteristics can be manifested at a broad temperature range from milli Kelvin to 300 K. These peaks have been used to disclosure striking electro-quantum-mechanical effects which are known as Coulomb blockade6, Coulomb-blockade oscillations7,8, Coulomb staircases9,10, single electron charging11,12,13, zero-dimension states14,15,16, competing channels17, and two-level resonances18,19,20,21,22,23 in various materials24,25.

Theoretical understanding of these effects are based on two considerations. One relies on the classical charging energy Ec = e2/2 C of multiple electrons in a parallel-plate capacitor that has been unanimously confirmed in constant peak spacings. The other depends on size quantization that appears to be more complicated because of the geometrical anisotropy in a three-dimensional (3D) QB. This property can lead to multiple charging energies En which will modulate peak’s amplitudes26 and spacings. Luckily enough, when the geometrical symmetry is introduced into the 3D QBT, the ordering of En can be brought out and reflects on the sub-quantum number ni where i denotes the three independent variables x, y, z in a Cartesian coordinate; z, r, θ in a cylindrical coordinate and r, θ, ϕ in a spherical coordinate. Based on the circular symmetry in a two-dimensional (2D) electron gases, Tarucha et al. performed a study27,28 on a disk QBT in zero magnetic field and found the one-to-one correspondence between the ionization energy of the Nth electron discharging from the Nth tunnel peak to uncover two magic quantum series. The first is all even numbers of 2, 6, and 12 claimed to be the full filling of electrons in the first, second, and third shells in a two-dimensional (2D) harmonic potential; the second is mixture of even and odd numbers at 4, 9, and 16 being claimed the half filling in the second, third and fourth shells to mimic three-dimensional (3D) spherically symmetric shells of 1s, 2s, 2p, 3s, 3p, etc. in real atoms to demonstrate the existence of artificial atoms.

These outstanding numbers no doubt prove the effectiveness of Pauli exclusion principle for spin-spin alignments between electrons and of Hund’s rule for charge-charge interactions among electrons in different channels. However, the dynamical symmetry of ni electrons are not explored, leading to the paucity of understanding how single electron tunnel is generated in the QBT. In our view, the N electrons will not be self-confined in the 2D x-y plane although they were assumed to be frozen at a very low temperature ∼50 mK26. Some of the electrons ought to be excited and stabilized in the conduction channel and that will make the interactions between the electrons in the x-y channels and those in the z channel critical which are believed to be the key of understanding the dynamical effects of SET in a deeper level; so does to the aforementioned seven effects.

In this paper, we will demonstrate such picture of ni with the focus aiming on how the electrons excited in the narrow conduction z channel can be stabilized firstly, then on how they interact with the nx and ny electrons. It is found that the interactions of electron-electron, spin-spin, electron-phonon and electron-hole (exciton) in the z channel are vital in helping nz becomes the most decisive quantum number. The electrons of nx and ny collaborate with the mechanical vibrations to assist the indirect tunnel of single electron. Each ni is found to obey the fundamental rules of even number in doing electro-quantum mechanical stability and the odd number is in doing instability. As a harmony effect, the first peak generated in the stabilized SET is triggered by the addition of the third electron into the easy x and y channels when two (or four) electrons are pre-excited in the z channel. With all the ni being assigned to all the peaks in the I-V spectrum, a comprehensive model is established to map out a full chart of ni and N that eventually leads to the conclusion that nz is the principle quantum number in characterizing the transport quality of single electron tunnel.

Mirror Symmetry of Moving Electrons in a Box

Once the importance of the dynamical symmetry is introduced29, its influential effects must be established based on the degrees of balance and stability in a multiple electrons system by checking whether the moving electrons obey the symmetry of mirror images. When two electrons with antiparallel spins30 and equally apart from the box’s center move toward (or away from) each other in a one-dimensional well, the symmetry is called conserved; whereas when the condition is of one moving electron only, the symmetry has no chance to be conserved. Extending such an idea to 3D, the total N acting electrons and the sub acting ni electrons in each channel clearly are periodical functions of the number 6 and 2, denoted as Ne and nie. At these numbers, the box is at the peak of stability. While at the middle odd numbers denoted as No = 3 and nio = 1, the box is at the valley of stability. Assuming the N is increased at the steady pace of +1 for each tunnel peak, the Ne and nie, No and ni0 will alternatively play conservation and breaking of the global-3D and local-1D symmetries. In accordance, the Ne comes as the series of 0, 6, 12, 18…; the No as 3, 9, 15, 21…; the nie as 0, 2, 4, 6… and the nio as 1, 3, 5, 7…etc. The combination of Ne and nie is possible and it becomes the series of 2, 8, 14, 20…; the Ne + nio of 1, 7, 13, 19…; the No + nie of 5, 11, 17, 23… and the No + nio of 4, 10, 16, 22…etc. The combination of nie + nio no doubt will become the most flexible and popular states. These values will be confirmed later.

Transport measurements are performed on nanopillar transistors. The QB size in the devices is ∼8.5 × 8.5 × 3 nm3 that makes the lateral to normal length ratio 3 to 1, better than the disk QB of 8 to 1. The method adopted to analyze the Id − Vds characteristics is unprecedented. Both the N and the ni of each peak are determined. With these information, the total electronic numbers determined are up to nz = 4 and N = 28. Given the sizes, Eq. (1) is used to calculate all the possible single-particle states, denoted as [nx, ny, nz], along with the associated energies En within 1 eV31. Table 1 summarizes all of them. Devices A and B were selected for discussions. Their qualities were checked by the as-fabricated I-V plots shown in the inset of Fig. 1. Both curves clearly exhibited symmetric effects of Coulomb blockades with CVds = e at Vds ≤ ±0.13 V of Id = 0 A. The value of C was determined to be ∼1.2 aF which exactly matched the net value of the SiNx-Si-SiNx trinature frequency and-capacitors in demonstrating their high qualities.

$$E({n}_{x},{n}_{y},{n}_{z})=\frac{{\pi }^{2}{\hslash }^{2}}{2m}(\frac{{n}_{x}^{2}}{{L}^{2}}+\frac{{n}_{y}^{2}}{{W}^{2}}+\frac{{n}_{z}^{2}}{{H}^{2}})$$
(1)
Table 1 Three-dimensional quantized states and energies of a 8 × 8 × 3 nm3 box.
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Figure 1
figure 1

Id versus Vgs of device B at 298 K. Giant Coulomb currents are clear near 0.1 V. The leading 3D electronic states of nz = 3 dominate the tunnel peaks at high bias. Inset shows Coulomb blockades of device A and B as fabricated.

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Results and Discussions

Coulomb-blockade oscillations in nz = 0

As shown in Fig. 2(b), the first two electrons of the [1, 1, 0] state generate a moderate electron-phonon interaction by yielding Coulomb vibrations and subsequent currents. The alternating currents at Vds < 30 mV suggest that the nx and ny electrons which are scattering created from the Electric-field in the z-channel. Their circular motions are dissipative and force free that can be described by the classical equation of motion with the induced current32 as

$${\rm{I}}({\rm{t}})=neA\alpha \omega {e}^{-\xi {\omega }_{n}t}\,\cos (\omega t+\varphi )$$
(2)

where n is silicon density, A box area, α maximum displacement, 𝜉 damping ratio c/cc, cc = 2 mωn, ω = ωn(1 − ξ2)1/2, ωn = (k/m)1/2 nature frequency and φ the initial phase. The good numerical fit to data in t ≤ 0.3 s proves that the paired electrons are indeed doing harmonic motions and the phase φ is also determined to be π/2.

Figure 2
figure 2

Device structure and current-voltage characteristics of device A at 298 K. (a) A schematic view of nanopillar transistor. (b) A theoretical fit and data. The blue curve is the damping current calculated from Eq. (2) (see main text). Parameters used in the fit are n = 5 × 1022/cm3, A = 72 nm2, α = 3 Å, ω = 1.16 × 103 Hz, φ = π/2 and ξ = 6 × 10−3. (c) Id–Vds at large bias. The 3D electronic states of each tunnel peak are labeled in brackets, referring to Table 1. Odd peaks are labeled on top and even peaks are below. The first series of single-electron tunnel at nz = 1 is in dark cyan, the second series of nz = 3 is in blue. The states at nz = 2 and 4 of stop zones are in red and orange.

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When the third electron is added to the QB for the state [1, 1, 1]. The dynamics of nz = 1 electron show up. As illustrated in Fig. 3(a,b), the E-field generated free electron can lead to significant deformations and mechanical feedbacks in the QBT33. At the very first moment of a small bias being applied onto the box, as shown in Fig. 3(a), surface charges of positive and negative polarities will be built up on all interfaces34, denoted as Qs and qs, respectively. When the eVds reaches E(nz = 1) ∼42 meV, the electron generates a large Coulomb force to bend the box into ±z directions sequentially as illustrated in Fig. 3(b). Note that the initial repulsive force is on the qs−, and it will then result in a forward current +I(t). The elasticity of the QB will restore it to its initial position, and excite the hole to provide a counterforce on the other end of the QB for a backward current −I(t). As results, zig-zag currents35,36 are created between [1, 1, 0] and [1, 1, 1] states.

Figure 3
figure 3

Dielectric response under the action of a drain-source voltage. (a) An electric field E is initiated inside of the box with accumulated surface charges Qs−, qs+, qs− and Qs+. (b) Single electron excited to state nz = 1 and box deformations. A hole is also created. Coulomb forces between electron and qs− and qs+ generate forward +I(t) and backward −I(t) currents.

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Unstable single electron tunnel in nz = 1, 3

At the state [3, 2, 1] of six electrons, the first SET shows up in Fig. 2(c) and it is triggered by the addition of the nx = ny = 3 electron into the [2, 2, 1] state (see Table 1). After that, the first series (#1) of SET develops as the N is increased from 6 to 9, [6, 2, 1], for a total of seven sharp peaks. Each peak is clean in singlet state, suggesting a unique vibration mode is involved, and they are separated by a seemingly constant distance. For the QB of A ≈8.5 × 8.5 nm2 and H = 3 nm, the Ec = e2/2 C, C = εrεoA/D ∼3.5 aF, εr = 11.7, εo = 8.85 × 10−12 C2/Nm2 is estimated to be ∼30 meV. This value is expected to match the peak spacing ΔEn well and as matter of fact, in Fig. 4(a), the ΔEn varies strongly from peak to peak with the 1st ∼30 meV and the 6th ∼18 meV that yields a statistical value ∼24 ± 6 meV and a mean deviation over 20%. The large deviation indicates that the nz = 1 SET of is not stable, with spin 1/2. As a consequence, the SET stops at the states [4, 2, 2]/[2, 4, 2] and [4, 4, 2] of nz = 2 which have s = 0.

Figure 4
figure 4

Single electron tunnel peak dependence of charging energy and total electron number N. (a) Peak number versus charging energy. The first point represents the energy gap between [2, 2, 1] and [3, 2, 1] states. The remaining points represent the energy spacing between two neighboring peaks in Fig. 2(c). (b) Peak number versus N. The instability of N is clear as it comes across the number of 9.

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Notably, at a higher eVds ∼0.4 eV, the stopped SET revives itself by changing from the singlet to doublet state (meaning L ≠ W), and also by regrouping the ni to increase the nz from 1 to 3 while maintaining the total N fixed at 6. In the following 10 split peaks, the N quickly increases from 12 to 24. Thanks to the extra stability introduced by the doublet states17, the #2 SET runs a little bit longer than the #1. Accordingly, the distributions of ΔEn becomes a uniform value of ∼25 ± 3 meV in Fig. 4(a). Nevertheless, these peaks still show progressively decays in amplitudes for another end near 0.7 V to prove that the odd nz = 3 is still unstable (s = 1/2).

Stable single electron tunnel in nz = 2, 4

To make the nz = 2 SET happen, device B is used. In Fig. 5(a), it is again that giant Coulomb currents dominate the Id–Vds from 0 to −0.23 V. The first peak emerges right at the doublet [3, 2, 2]/[2, 3, 2] states to confirm that the starting nz is indeed 214. After that the SET peaks are well spaced by a ΔEn which is estimated to be ∼33 ± 3 meV. This value matches the Ec ≈34 meV very well. By checking with Table 1, the ground state of all the two-level resonances is at the most balanced Ne = 6, [2, 2, 2] state. The gap ΔEn ∼29.4 meV between [2, 2, 2] and [3, 2, 2]/[2, 3, 3] is an excellent value near Ec. No wonder that the stable SET runs very long for a total of 21 doublet peaks24. In fact, they are triggered by the N = Ne + nio = 6 + 1 = 7 electron in each mode with 2 N = 14 in total. In another words, it is the third electron added to the stable nx = ny = 2 states creating the weak dynamical break to assist one electron tunnel in the z-channel. Inset of Fig. 5(b) illustrates how this nx/ny = 3 electron can take advantage of the two-level vibrations18,19,20,21,22,23; it is firstly charged into the nx/ny channels, then propelled out of them into the main channel for a current peak.

Figure 5
figure 5

Id versus Vds of device B at 298 K. Note that the polarity of current is changed. (a) Electron charging from 0 to 1 V. The first peak is at [3, 2, 2]/[2, 3, 2] doublet. Double electron tunnel begins at [6, 2, 4]/[2, 6, 4] states. Inset (left) shows two-level resonance of one electron tunnel; (right) two-channel, two-level resonance tunnels of double electrons. (b) Reversed charging from −1 to 0 V. Transition from 2e to 1e tunnel is at [4, 4, 4] state. Inset shows the vibration assisted electron tunnel; from left to right, prior to the entrance of 1e, resonances with 2e in the ny/nx channels, and repel of the charged electron.

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At eVds ∼ −0.92 eV, the SET makes a transition to double electron tunnel (DET) by presenting a sharp increase in the magnitude of Id that resembles the Coulomb staircases which have been spotted elsewhere9,10. The cause of this transition is created by the same ΔEn ∼29.4 meV which is amazing. Notice that although the DET occurs at the next even nz = 4, the same nx/ny = 2 ↔ 3 resonance remains active. Ideally, the resonance should happen at the global ground state of [4, 4, 4] (just like [2, 2, 2]) and then jump to the higher states of [5, 4, 4]/[4, 5, 4], but that would require an much higher energy ∼53 meV which is almost impossible. As an alternation, the QBT skips three energy levels until the meet of the doublet ground states [6, 2, 4]/[2, 6, 4]; therein the QBT makes the transition to the doublet [6, 3, 4]/[3, 6, 4] states. As a reward, one more conduction channel is created, as shown in the inset of Fig. 5(a), making electrical conduction more efficient. In the meantime, the total N reaches the highest value of 4Ne + 4no = 28.

The robust SET and DET are reproducible from a high bias. In Fig. 5(b), when the bias begins with the −1 V, the DET shows no sign of fading at all in the beginning. The EQM coherence remains firmly locked with the N = 24 ↔ 28 transitions, with the only difference being the φ shifted from the π/2 (t = 0 s) to π. At the all even [4, 4, 4] state, the DET makes a decay to the SET. After that, the N = 12 ↔ 14 resonances take over and run the SET all of the way to Vds = 0 V to convince the initial Coulomb-blockade oscillations observed in Fig. 2(b). These data also explain the long-time search of why the classical Ec is an excellent indication of stable electron charging energy and it is only when the En matches its value, the coherent SET will show up. Other than the outstanding finding, the breakdown of the one-to-one correspondence between the N and the tunnel peaks is also remarkable as it indicates that there will have multiple peaks in corresponding to the same electron numbers and their effects will be discussed in the following paragraph.

Excitation of holes to neutralize electrons

The transition from DET to SET drastically changes the value of N. Here, the ΔN is as large as 16, from 28 to 12 at the state of [4, 4, 4] that makes 16 excited electrons left over to be annihilated by holes37. As results of the holes excitations illustrated in Fig. 3(b), negative mechanical feedbacks will constantly act on the drain electrode for negative currents to lower the level of Id. As a consequence, after the largest drop at [4, 4, 4], the next drop develops at the doublet [4, 2, 4]/[2, 4, 4], then is the hidden nz = 3 [6, 3, 3]/[3, 6, 3] doublet states, following by the all odd singlet [3, 3, 3] right in the middle of the SET series. Notably, in Fig. 5(a) of electron charging in device A, the [3, 3, 3] state is also becoming visible in a dip to signal that the weakly bonded pairs of electrons and holes (excitons) are important dynamical entities. The same effect is also detected in Fig. 2(c), when the ΔN is not increased in pace with the +∆eVds, three drops appear at the [4, 2, 2], [4, 3, 2]/[3, 4, 2], [4, 4, 2] and [5, 2, 2]/[2, 5, 2] states.

Interferences in two conduction channels

Electron charging from the side gate confirms the high flexibility of nie + nio. Here the nie represents the sum of nxye and nxyo, and the nio is the nzo. The reason is simple because the driving E-field will be separated into two components; one is along the side channel and the other is along the main channel. As results of their interference acting38, the states excited are most likely to be the highest probability but with the lowest symmetry. As shown in Fig. 1, The first giant currents of [1, 1, 1] again appear at Vgs ∼ 0.1 V. When compared to the noise at 50 mV in Fig. 2(b), the gate-dot coupling strength is determined to be 0.5. Based on this value, the successive leading peaks for SET are identified to be the following states; [4, 2, 3], [5, 1, 3], [5, 2, 3], [5, 3, 3], [6, 1, 3], [6, 2, 3] and [6, 3, 3]. Notice that there are no all even states like Ne, the starting No is at 3, then is the isotropic state of No = 9, [4, 2, 3], the N then fluctuates down and up in the states of nz = 3. At the [6, 3, 3], an isotropic state of N = 12, a sharp peak appears in reminding of the end in SET #2.

Quantum selection rules in 3D box transistors

As listed in Table 2, the hierarchy of symmetry in terms of the Ne, No, nie, and nio becomes very clear and the result is Ne > nie > nio > No. No of the lowest stability manifests itself as the giant Coulomb currents in [1, 1, 1] and [3, 3, 3]. The [1, 1, 1] state also serves to activate the QBT for later on tunneling of electrons. In sharp contrast, the Ne provides the highest stability in [2, 2, 2] and [4, 4, 4] states for an activated QBT. The primary combination of Ne + nxyo then creates the most needed tunneling of single electron. The secondary combination of nie + nxyo generates the most urgently needed two-channel, two-level resonance tunneling of double electrons. In between, the combination of nxye + nxyo + nzo creates most of the unstable SET and they will be temporarily stopped at the states of nxye + nxyo + noe. According to these rules, the previously claimed not-all odd magic number of 4, 9, and 16 in disk QBT could be either the all odds number of 3, 9, 15, or the composition of nxye + nz with the latter being the incremental number of 0, 1, and 2. After subtracting the nz, that leaves the nxye = 4, 8, and 14 which fit perfectly into the claimed all even numbers of 2, 6, and 12.

Table 2 Three-dimensional electronic states identified in device A and B.
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Vibrational frequencies and maximum displacements

Vibrational frequencies induced by the electron-phonon interactions in the QBTs are estimable. The most outstanding one clearly is the [1, 1, 1] as plotted in Fig. 2(a). Its νz is fitted to be ∼1.85 × 102 Hz and the maximum displacement of αz is fitted to be ∼3 Å. At N = 7, the values of νy and νx are expected to be higher, ∼100 times larger than the initial frequency. Since the C of the two devices is approximate ∼3.5 aF and the R is ∼2 × 1013 Ω that leads the RC ∼4.4 × 10−5 s and the νx/vy ∼2.2 × 104 Hz. By taking this value into Id = eνy, a peak current of 0.15 pA is obtained, which agrees with the data very well. With the vibrating frequency being increased at the nz = 3 states, the αy or αx will become smaller. For an isotropic electron mobility in the QB, the Id equals neAυd with υd = αωy/2π. Given the n = 1019/cm3 and A = 64 nm2, the αy (or αx) is calculated to be ~1 Å.

In conclusion, we study the dynamics of electrons in three-dimensional quantum box, nanopillar transistors. Single electron tunnel Nth peak is found not in one-to-one correspondence with the excited Nth electron. The best quantum number to describe SET is the sub-quantum number nz in the conduction z channel, not the conventionally used total electron number N. Stable SET peaks occur only when the number of electrons excited in nz is even and the number of electrons excited in the normal nx/ny channels are also even at two, robust peaks are then generated one-after-one by the charging/discharging of the third electron in/out of the normal x-y channels. A comprehensive model that incorporates the dynamical interactions of electron-electron, spin-spin, electron-phonon, and electron-hole explains the mechanical stability of even electrons in the electric-field driving z channel. Quantum selection rules with hierarchy of ni and N are tabulated to prove that nz is a better number than the N for characterizing the quality of electron tunnel.

Methods

Device fabrication and current-voltage measurements

The nanopillar transistor as shown in Fig. 2(a) was fabricated on a p-type (100) silicon wafer which featured a central polysilicon layer separated from the top and bottom electrodes by a nitride layer39. This center box had a critical thickness of 3 nm and was coupled to a side gate. Deposition of SiNx (3-nm)-polysilicon (3-nm)-SiNx (3-nm) layers were processed using a low-pressure chemical vapor technique, and then chemically etched it to create a nominal plateau ∼200 × 140 × 210 nm3. Phosphorous of a concentration 1 × 1019 cm−3 was doped in Si layer. The source was located at the bottom with a sheet resistance of ~30 Ω/cm2. To prevent electrical shortage, a short oxidation was carried out to seal the nanopillar (creating another ~1.5 nm oxide). To further squeeze the cavity, the technique of self-aligned oxidation was utilized to add another ~6 nm layer of gate oxide, yielding a total of ~9 nm with the inner box reduced to the size ~9 × 9 × 3 nm3. Finally, an Al (300 nm) side gate was attached. Devices were loaded into a probe station for I-V measurements at 298 K and a three-terminal HP meter with resolutions of 1 mV and 10 fA was used.

Zero magnetic field

To elucidate the authentic electron charging effects, external perturbations were kept to a minimum. No extra magnetic field was applied, except the negligible Earth B-field. Electrical bias was only applied either from the drain-source electrodes or the gate-source electrodes to avoid mutual interference in most cases.

Reduced sizes in parabolic potential

To count for the mutual interactions between multiple electrons in a parabolic potential, the values of L and W are reduced ∼5% smaller than the real values as the effectively electron-electron repulsions will make the QB smaller for a higher charging energy to the entrance of next electron. With these fine tunings, excellent agreements are reached between analysis and data.