Symmetry selected quantum dynamics of few electrons in nanopillar transistors

Study on single electron tunnel using current-voltage characteristics in nanopillar transistors at 298 K show that the mapping between the Nth electron excited in the central box ∼8.5 × 8.5 × 3 nm3 and the Nth tunnel peak is not in the one-to-one correspondence to suggest that the total number N of electrons is not the best quantum number for characterizing the quality of single electron tunnel in a three-dimensional quantum box transistor. Instead, we find that the best number is the sub-quantum number nz of the conduction z channel. When the number of electrons in nz is charged to be even and the number of electrons excited in the nx and ny are also even at two, the adding of the third electron into the easy nx/ny channels creates a weak symmetry breaking in the parity conserved x-y plane to assist the indirect tunnel of electrons. A comprehensive model that incorporates the interactions of electron-electron, spin-spin, electron-phonon, and electron-hole is proposed to explain how the excited even electrons can be stabilized in the electric-field driving channel. Quantum selection rules with hierarchy for the ni (i = x, y, z) and N = Σni are tabulated to prove the superiority of nz over N.

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 n i 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 n x and n y 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 n z becomes the most decisive quantum number. The electrons of n x and n y collaborate with the mechanical vibrations to assist the indirect tunnel of single electron. Each n i 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 n i being assigned to all the peaks in the I-V spectrum, a comprehensive model is established to map out a full chart of n i and N that eventually leads to the conclusion that n z 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 introduced 29 , 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 spins 30 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 n i electrons in each channel clearly are periodical functions of the number 6 and 2, denoted as N e and n ie . At these numbers, the box is at the peak of stability. While at the middle odd numbers denoted as N o = 3 and n io = 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 N e and n ie , N o and n i0 will alternatively play conservation and breaking of the global-3D and local-1D symmetries. In accordance, the N e comes as the series of 0, 6, 12, 18…; the N o as 3, 9, 15, 21…; the n ie as 0, 2, 4, 6… and the n io as 1, 3, 5, 7…etc. The combination of N e and n ie is possible and it becomes the series of 2, 8, 14, 20…; the N e + n io of 1, 7, 13, 19…; the N o + n ie of 5, 11, 17, 23… and the N o + n io of 4, 10, 16, 22…etc. The combination of n ie + n io 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 nm 3 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 I d − V ds characteristics is unprecedented. Both the N and the n i of each peak are determined. With these information, the total electronic numbers determined are up to n z = 4 and N = 28. Given the sizes, Eq. (1) is used to calculate all the possible single-particle states, denoted as [n x , n y , n z ], along with the associated energies E n within 1 eV 31 . 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 CV ds = e at V ds ≤ ±0.13 V of I d = 0 A. The value of C was determined to be ∼1.2 aF which exactly matched the net value of the SiN x -Si-SiN x trinature frequency and-capacitors in demonstrating their high qualities. state generate a moderate electron-phonon interaction by yielding Coulomb vibrations and subsequent currents.
The alternating currents at V ds < 30 mV suggest that the n x and n y 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 current 32 as where n is silicon density, A box area, α maximum displacement, damping ratio c/c c , c c = 2 mω n , ω = ω n (1 − ξ 2 ) 1/2 , ω n = (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. When the third electron is added to the QB for the state [1, 1, 1]. The dynamics of n z = 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 QBT 33 . 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 interfaces 34 , denoted as Q s and q s, respectively. When the eV ds reaches E(n z = 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 q s − , 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 currents 35 Fig. 2(c) and it is triggered by the addition of the n x = n y = 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 nm 2 and H = 3 nm, the E c = e 2 /2 C, C = ε r ε o A/D ∼3.5 aF, ε r = 11.7, ε o = 8.85 × 10 −12 C 2 /Nm 2 is estimated to be ∼30 meV. This value is expected to match the peak spacing ΔE n well and as matter of fact, in Fig. 4(a), the ΔE n varies strongly from peak to peak with the 1 st ∼30 meV and the 6 th ∼18 meV that yields a statistical value ∼24 ± 6 meV and a mean deviation over 20%. The large deviation indicates that the n z = 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 n z = 2 which have s = 0.
Notably, at a higher eV ds ∼0.4 eV, the stopped SET revives itself by changing from the singlet to doublet state (meaning L ≠ W), and also by regrouping the n i to increase the n z 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 states 17 , the #2 SET runs a little bit longer than the #1. Accordingly, the distributions of ΔE n 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 n z = 3 is still unstable (s = 1/2).
Stable single electron tunnel in nz = 2, 4. To make the n z = 2 SET happen, device B is used. In Fig. 5(a), it is again that giant Coulomb currents dominate the I d -V ds 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 n z is indeed 2 14 . After that the SET peaks are well spaced by a ΔE n which is estimated to be ∼33 ± 3 meV. This value matches the E c ≈34 meV very well. By checking with Table 1, the ground state of all the two-level resonances is at the most balanced N e = 6, [2, 2, 2] state. The gap Table 1. Three-dimensional quantized states and energies of a 8 × 8 × 3 nm 3 box. The data are calculated from Eq. (1) (see main text). The states in magenta, dark cyan, red, blue and orange colors correspond to those of n z = 0, 1, 2, 3 and 4. It is evident that in the highly symmetric device B, the number of states selected is very few ∼13, while in the less symmetric device A, the states selected are vast, more than 40.  www.nature.com/scientificreports www.nature.com/scientificreports/ At eV ds ∼ −0.92 eV, the SET makes a transition to double electron tunnel (DET) by presenting a sharp increase in the magnitude of I d that resembles the Coulomb staircases which have been spotted elsewhere 9,10 . The cause of this transition is created by the same ΔE n ∼29.4 meV which is amazing. Notice that although the DET occurs at the next even n z = 4, the same n x /n y = 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 4N e + 4n o = 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 V ds = 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 E c is an excellent indication of stable electron charging energy and it is only when the E n 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 holes 37 . As results of the holes excitations illustrated in Fig. 3(b), negative mechanical  www.nature.com/scientificreports www.nature.com/scientificreports/ feedbacks will constantly act on the drain electrode for negative currents to lower the level of I d . 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 n z = 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 +∆eV ds , 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 n ie + n io . Here the n ie represents the sum of n xye and n xyo , and the n io is the n zo . 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 acting 38 , 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 V gs ∼ 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 N e , the starting N o is at 3, then is the isotropic state of N o = 9, [4,2,3], the N then fluctuates down and up in the states of n z = 3. At the [6,3,3], an isotropic state of N = 12, a sharp peak appears in reminding of the end in SET #2. Table 2, the hierarchy of symmetry in terms of the N e , N o , n ie , and n io becomes very clear and the result is N e > n ie > n io > N o . N o 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 N e provides the highest stability in [2,2,2] and [4,4,4] states for an activated QBT. The primary combination of N e + n xyo then creates the most needed tunneling of single electron. The secondary combination of n ie + n xyo generates the most urgently needed two-channel, two-level resonance tunneling of double electrons. In between, the combination of n xye + n xyo + n zo creates most of the unstable SET and they will be temporarily stopped at the states of n xye + n xyo + n oe . 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 n xye + n z with the latter being the incremental number of 0, 1, and 2. After subtracting the n z , that leaves the n xye = 4, 8, and 14 which fit perfectly into the claimed all even numbers of 2, 6, and 12.  [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 n y /n x channels, and repel of the charged electron.  Fig. 2(a). Its ν z is fitted to be ∼1.85 × 10 2 Hz and the maximum displacement of α z is fitted to be ∼3 Å. At N = 7, the values of ν y and ν x 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 × 10 13 Ω that leads the RC ∼4.4 × 10 −5 s and the ν x /v y ∼2.2 × 10 4 Hz. By taking this value into I d = eν y , a peak current of 0.15 pA is obtained, which agrees with the data very well. With the vibrating frequency being increased at the n z = 3 states, the α y or α x will become smaller. For an isotropic electron mobility in the QB, the I d equals neAυ d with υ d = αω y /2π. Given the n = 10 19 /cm 3 and A = 64 nm 2 , the α y (or α x ) is calculated to be ~1 Å.

Quantum selection rules in 3D box transistors. As listed in
In conclusion, we study the dynamics of electrons in three-dimensional quantum box, nanopillar transistors. Single electron tunnel N th peak is found not in one-to-one correspondence with the excited N th electron. The best quantum number to describe SET is the sub-quantum number n z 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 n z is even and the number of electrons excited in the normal n x /n y 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 n i and N are tabulated to prove that n z 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 layer 39 . This center box had a critical thickness of 3 nm and was coupled to a side gate. Deposition of SiN x (3-nm)-polysilicon (3-nm)-SiN x (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 nm 3 . Phosphorous of a concentration 1 × 10 19 cm −3 was doped in Si layer. The source was located at the bottom with a sheet resistance of ~30 Ω/cm 2 . 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 nm 3 . 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.

Device A Device A
[n x , n y , n z ] n z N N e n ie n io N o [n x , n y , n z ] n z N N e n ie n io N o Charging n z = 0, 1  Table 2. Three-dimensional electronic states identified in device A and B. Hierarchy of quantum numbers denoted and classified by N e , n ie , and n io and N o symbols. N e has the highest symmetry in a state with all even n i and n x = n y = n z . n ie comes as the next with at least one even n i in a state, n io is ranked as the third with at least one odd n i in a state. N o is the lowest in ranking with all odd n i and n x = n y = n z in a state. Bold state means doublet states. It is clear that the increment of n z in charging is much steadier than N.