- Email: [email protected]
Study of electron charging by voltage pulses in nanopillar transistors at high temperature
Chinese Journal of Physics 55 (2017) 1225–1229
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Study of electron charging by voltage pulses in nanopillar transistors at high temperature Te Chien Wang, Yue-Min Wan∗ Department of Electronic Engineering, I-Shou University, Kaohsiung, 84001, Taiwan
a r t i c l e
i n f o
Article history: Received 10 December 2016 Revised 7 May 2017 Accepted 22 May 2017 Available online 29 June 2017 Keywords: Quantum dot Nanopillar transistor Pulsed voltage Mechanical vibration Room temperature
a b s t r a c t A study of electron transport in nanopillar transistors at 300 K shows that elastic vibration is an intrinsic behavior of the device. The frequency observed in the drain-source current is found to agree with that of the charging pulsed voltages. Given a quantum dot of size 10 × 10 × 9 nm3 , the maximum displacement is estimated to be 0.3 nm. Once the displacement diminishes to zero, single-electron tunneling becomes the dominant effect. A forced vibration model is proposed to explain the correlation between the surface charges and the vibrations. When the distribution of charges is uniform on each SiNx atom, the vibration becomes stable and can yield a homogenous damping current. © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Research in electron transport of mechanical quantum dot transistors [1–4] has made tremendous progress in the last two decades. No doubt it is one of the most fascinating topics for physicists, because it has the potential to revolutionize modern technology in semiconductors. It also provides challenges for theoretical understanding in the field. The discovery of single electron tunneling is a hallmark effect, because it has been incorporated into devices, such as the single-electron pump [5], single-electron memory cell [6], and single-electron detector and counter [7]. In addition to these devices, we have also presented the vertical “nanopillar” transistor [8]. In this device, drain current versus drain-source voltage measurements at room temperature show excellent features associated with single-electron peaks. By matching the peak spacing with the parallel plate charging energy Ec = e2 /2C, a single electron tunnel is then confirmed. However, a critical issue remains unanswered in that paper, which is why the onset of peaks is always at a finite voltage. Early I-V measurements at low temperatures [9–11] had found the same kind of effect, and it was attributed to a Coulomb blockade, where all the charges are totally repelled by the Coulomb interaction and no current could flow. But lately, the concept of a quantum shuttle [12] has been proposed to explain how electrons can move from one electrode to another via the repetitive motion of the central dot at finite temperatures. Consequently, current noise [13] will appear. In our view, in this highly uncertain region other mechanisms, such as dissipation [14], elastic deformation [15], and mechanical feedback [16] can also play a vital role. As such, it is definitely worth doing more careful experiments in this subject. As is expected, we find that at very low charging voltages, the Id -Vd curve shows drastically different behavior compared to what has been observed before; giant oscillations appear and dominate the I-V spectrum from the very beginning. After reaching a maximum value the current quickly decreases to zero. Thereafter, periodic peaks show up again. To understand this unprecedented phenomena, we proposed a forced vibration model. It turns out that this model can explain the observed data very well.
∗
Corresponding author. E-mail address: [email protected] (Y.-M. Wan).
http://dx.doi.org/10.1016/j.cjph.2017.05.030 0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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Fig. 1. (a) Schematic diagram of nanopillar transistor structure. A SiNx /Si/SiNx quantum dot 10 × 10 × 9 nm3 in size is in the center. The Al side gate is ∼6 nm away from the dot. (b) TEM picture of the SiNx /Si/SiNx layers after deposition. Bobbles on the sides are Ga+ accumulated through use of a focused ion beam.
The transistor, schematically drawn in Fig. 1(a), was fabricated on p-type (100) silicon wafers. It basically features a central polysilicon layer separated from the top and bottom contacts by two nitride barriers. This dot cavity has a critical length of 3 nm and is closely coupled to a gate electrode at the side. The fabrication of the device proceeds as follows; we first deposited the multilayer structure of SiNx (3-nm)-polysilicon (3-nm)-SiNx (3-nm) sequentially via low-pressure chemical vapor deposition (Fig. 1(b)), and then chemically etched it to create a nominal plateau of 200 × 140 × 210 nm3 . The source electrode is located at the bottom of the 200 nm thick structure, and has a sheet resistance of ∼30 /cm2 achieved via doping with 1x1019 cm−3 P+ in the mixed gas of SiH4 and PH3 . To prevent electrical short circuits, a short oxidation time using rapid thermal annealing for 30 s was then carried out to seal the nanopillars (creating ∼1.5 nm oxide). Next, a contact window (∼20 nm wide) was opened on the top nitride layer to form the drain contact. To do this, a layer of tetraethylorthosilicate (TEOS) ∼200 nm thick was first grown to level the height of the plateau, and then spun coated with a layer of photoresist. After development, the exposed TEOS in the upper-right area was cleaned off. Following another chemical etch to further miniaturize the nanopillars, a short oxidation was applied. The photoresist which remained at this stage was used as a hard mask for wet etching in H2 O and HF (ratio 50 to 1) for about 1 min. This lateral etch creates a cut underneath, and opens an active zone. After the strip of the photoresist, polysilicon was defined at a normal angle with respect to the source. The overlap region of both electrodes therefore defines the nanopillars with an outside dimension of ∼20 × 20 × 9 nm3 . To further squeeze the cavity, the technique of self-aligned oxidation was used to add another ∼6 nm layer of oxide, giving a total thickness of ∼9 nm, and resulting in a quantum dot of 10 × 10 × 9 nm3 in size. The last step of the fabrication process consisted of the sputter deposition and etching of Al (300 nm) to provide a side gate next to the cavity. The device was then loaded into a probe station (PS150, Cascade Microtech, USA) for current-voltage measurements. A HP 4156 C three-terminal meter, with a resolution of 1 mV and 10 fA, was used. Since our model is intimately related to the manner of electrical charging, the pulsed voltages are schematically drawn in Fig. 2(a). Within a time period of 1 mS, 1 mV was applied in the first half cycle. After that, another 1 mV was added (for a total of 2 mV on the device) in the next period. Such a type of charging is commonly used by experimentalists, as it can avoid overheating. However, it is also the cause of mechanical vibration. In Fig. 2(b), when certain amounts of charge Qs are accumulated on the surface of the bottom nitride layer, a force it will be created which can bend the layers. At the other end of the layer, a positive current +I(t) is then produced. The elasticity of all the layers, here marked as the energy ࢞E, will then force a backward movement to create a negative current –I(t). As a consequence, the current starts to alternate. Correspondingly, the displacement x(t) will change between +α and –α , as illustrated in Fig. 2(c). ¨ +cx(t) ˙ +kx(t)=f(t) [17], where The theoretical model best suited to describe such behavior is the dynamical equation mx(t) ¨ ˙ x(t) is the second order time derivative of x(t); x(t) is the first order derivative; x(t) is the displacement; m represents the mass of the system, as shown in Fig. 3, c denotes the damping coefficient, k is the spring constant, and f(t) is the applied force.
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Fig. 2. (a) Pulsed voltages used for electron charging, of value 1 mV and 2 mV respectively, in a 1 mS period. (b) Electro-mechanical model, a positive current +I(t) and a negative current –I(t) are created during a period by the surface charges Qs and the elastic energy E of the transistor. (c) Maximum displacements, of value α corresponding to the current.
Fig. 3. Forced vibration model. f(t) is the applied force, m the system mass, c the damping coefficient, k the spring constant, and x the displacement.
To solve this equation, two situations must be considered. One is the zero force condition with f(t) = 0, and the other is a finite force with f(t)=0. The solution xg (t) (general) of the force-free condition is a textbook example [13], and it is
α e−ξ ωn t sin(ωn 1 − ξ 2 )t + ϕ ). Here α is the maximum displacement, ξ is the ratio ccc , cc is 2mωn and ωn =( mk )1/2 is the natural frequency. The term inside the bracket of the sinusoidal function ω=ωn 1 − ξ 2 is the response frequency, which is also the frequency measured in an experiment. ϕ is the initial phase, which can be set to zero. Overall, xg (t) is simplified as α e−ξ ωn t sin(ωt ). With this quantity, we can calculate the induced current. Here we stress that although the device is very small, the transportation of electrons is expected to obey the fundamental law of conduction, namely, Ig (t)= neAυ d , where n is the ˙ volume density of electrons, A is the layer area, and υ d is the drift velocity, as defined by x(t). Given all these parameters, the current created by the damping vibration is found to be Ig (t)= neA{αωe−ξ ωn t cos(ωt )-αξ ωne−ξ ωn t sin(ωt )}. Notice that there are two components; an even cosine term and an odd sine term. As will be demonstrated later, the second component is negligible due to the small value of ξ = 0.006. As a consequence, one has
Ig (t ) = neAαωe−ξ ωn t cos(ω t )
(1)
Apart from Ig (t), there is another part of the current known as the particular solution Ip (t). Derivation of it is different from Ig (t), because one has to go back to the original equation mx¨ +cx˙ +kx=f(t). This equation is complicated in mathematics due to the presence of the time-dependent term f(t). As a result, an analytical solution is almost impossible (if not reachable by numerical calculations). So, one can only expect an approximation solution. Here, we use such an approach. Clearly, the resistive term has to be responsible for counteracting the applied force, similar to the terminal velocity of an object free falling in a gravitational field. Accordingly, the identity becomes easy, i.e, cx˙ p =f(t), and Ip (t) can be derived. By taking the parameters of x˙ p (t)= f (ct ) , f(t) =eE, E= Vd , polysilicon thickness (d), and applied voltage (V) into neAx˙ p (t), one obtains
Ip ( t ) =
nAe2V . cd
(2)
This formula shows that Ip (t) is predicted to be proportional to the applied voltage V. With the theoretical discussions finished, we now analyze the data. In Fig. 4(a), a typical Ig (t) curve is plotted for comparison to the data. This result is obtained after several estimates for ωn and ξ , which appear to be the most critical parameters in the model. Notice that this alternating and slightly modulated damping curve mimics the main features of all the data very well. In Fig. 4(b), the best fit happens in the range between 0 V and 0.12 V, where a smaller damping module appears and is interrupted by a giant peak in the middle. A similar situation is shown in Fig. 4(c), except that the interruption is wider and has smaller peaks. As shown in Fig. 4(c), the interruption disappears and the good fit goes all the way from 0 V to 0.17 V. This result is outstanding as it not only answers the key query of this paper, but also proves that mechanical stability is an essential part for the upcoming quantum tunneling of single electron.
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Fig. 4. (a) Model predictions of the general current Ig (t). This data is obtained by using the fitting parameters n = 5×1022 /cm3 , e = 1.6×10−19 C, A = 100 nm2 , α = 0.3 nm, ω =103 Hz, and ξ = 0.006. (b) – (d) Drain current versus drain-source voltage at 300 K.
Other than these major effects, some minor features are also observed in the data. In Fig. 4(b), the appearance of a giant wave packet near 0,04 V is clearly due to coupling to a stronger vibrational mode. Similar behavior is seen in Fig. 4(c). However, in Fig. 4(d) when the gate bias becomes close to 100 mV, this odd mode is eventually eliminated as the multilayers become much stiffer. As to the influence of Ip (t), the impact is quite interesting. As discussed earlier, the high voltage squares should appear and dominate the Id -Vd spectrum. Indeed, those effects are seen in the data. In Fig. 4(b), pulses are clearly seen between 0.2 V and 0.3 V and in Fig. 4(c), they do emerge between 0.22 V and 0.26 V. In Fig. 4(d), they show up between 0.2 V and 0.35 V. Their appearances at high Vd indeed agree with the model prediction, however, their peak heights seem to be overwhelmed by the normal damping. One thing worth pointing out is that the giant e2 /2C peaks prove that they are due to the collective effects of electrons. Finally, we estimate the value of α . This can be done at t = 0 s (Vd = 0 V). At this moment Id is roughly about 0.2 pA (see Fig. 4(b)) and it also equals neAα . By taking a doping concentration of n = 1019 /cm3 , e = 1.6×10−19 C, A = 100 nm2 , and ω = 103 Hz, we find that α is ∼1 μm. This is obviously too large to be correct. A proper α should fall in the nanometer range. So we replace n with a value of 5 × 1022 /cm3 [18], which is the atomic volume density of Si (SiNx as well), and find that α becomes ∼0.3 nm. This more realistic value appears to be much better at suggesting that a uniform distribution of electrons on each Si atom (or Qs ) is the underlying mechanism for the stable vibration. In conclusion, we have studied electron transport in a nanopillar transistor at 300 K. Current-voltage measurements at various gate bias voltages show that damping currents are an intrinsic behavior at zero and low voltages. An electromechanical model is proposed to explain the experimental data, and good agreement is found between it and the data. We also find that mechanical vibration is an essential part of the transport, and once it has ceased single-electron tunneling takes over. A uniform distribution of electrons on the device is critical, as it can provide a balanced force to bend the layers and to eliminate the intrusion of odd vibrational modes.
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Acknowledgments We thank the National Nano Device Laboratory for much assistance with device fabrication. The work was supported by the National Science Council of the Republic of China under Contracts NSC97-2112-M-214-003 and NDL97-C05SG-140. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
H. Park, J. Park, A.K.L. Lim, E. Anderson, A.P. Alivisatos, P.L. McEuen, Nature 407 (20 0 0) 57 (London). M.D. LaHaye, O. Buu, B. Camarota, K.C. Schwab, Science 304 (2004) 74. E.M. Weig, R.H. Blick, T. Brandes, J. Kirschbaum, W. Wegscheider, M. Bichler, J.P. Jotthaus, Phys. Rev. Lett. 92 (2004) 046804. D.V. Scheible, R.H. Blick, Appl. Phys. Lett. 84 (2004) 4632. Y. Ono, Y. Takahasi, Appl. Phys. Lett. 82 (2003) 1221. N.J. Stone, H. Ahmed, Appl. Phys. Lett. 84 (2004) 4632. W. Wu, J. Gu, H. Ge, C. Keimel, S.Y. Chou, Appl. Phys. Lett. 83 (2003) 2268. Y.M. Wan, H.T. Lin, C.L. Sung, S.F. Hu, Appl. Phys. Lett. 87 (2005) 123506. T.A. Fulton, G.J. Dolan, Phys. Rev. Lett. 59 (1987) 109. J. Weis, R.J. Haug, K. v. Klitzing, K. Ploog, Phys. Rev. Lett. 71 (1993) 4019. D.V. Averin, K.K. Likhavev, Mesoscopic Phenomena in Solids, in: B.L. Altshuler, P.A. Lee, R.A. Webb (Eds.), Elseiver, Amsterdam, 1991 edited by Single Charge Tunneling, edited by H. Grabert and M. H. Devoret, Plenum Press, New York (1992). D. Fedorets, L.Y. Gorelik, R.I. Shekhter, M. Jonson, Phys. Rev. Lett. 92 (2004) 166801. T. Novotny, A. Donarini, C. Flindt, A.P. Jauho, Phys. Rev. Lett. 92 (2004) 248302. T. Nord, L.Y. Gorelik, R.I. Shekhter, M. Jonson, Phys. Rev. B 65 (2002) 165312. N. Nishiguchi, Phys. Rev. B 68 (2003) 121305. Ya. M. Blanter, O. Usmani, Yu.V. Nazarov, Phys. Rev. Lett. 93 (2004) 136802. A.A. Shabana, Theory of Vibration, Springer-Verlag, New York, 1996 2nd edition. Semiconductor Physics and Devices, 3rd edition, written by A. Neamen Donald, McGraw-Hill, New York, Inc, 2003, p. 11.
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Study of electron charging by voltage pulses in nanopillar transistors at high temperature Te Chien Wang, Yue-Min Wan∗ Department of Electronic Engineering, I-Shou University, Kaohsiung, 84001, Taiwan
a r t i c l e
i n f o
Article history: Received 10 December 2016 Revised 7 May 2017 Accepted 22 May 2017 Available online 29 June 2017 Keywords: Quantum dot Nanopillar transistor Pulsed voltage Mechanical vibration Room temperature
a b s t r a c t A study of electron transport in nanopillar transistors at 300 K shows that elastic vibration is an intrinsic behavior of the device. The frequency observed in the drain-source current is found to agree with that of the charging pulsed voltages. Given a quantum dot of size 10 × 10 × 9 nm3 , the maximum displacement is estimated to be 0.3 nm. Once the displacement diminishes to zero, single-electron tunneling becomes the dominant effect. A forced vibration model is proposed to explain the correlation between the surface charges and the vibrations. When the distribution of charges is uniform on each SiNx atom, the vibration becomes stable and can yield a homogenous damping current. © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Research in electron transport of mechanical quantum dot transistors [1–4] has made tremendous progress in the last two decades. No doubt it is one of the most fascinating topics for physicists, because it has the potential to revolutionize modern technology in semiconductors. It also provides challenges for theoretical understanding in the field. The discovery of single electron tunneling is a hallmark effect, because it has been incorporated into devices, such as the single-electron pump [5], single-electron memory cell [6], and single-electron detector and counter [7]. In addition to these devices, we have also presented the vertical “nanopillar” transistor [8]. In this device, drain current versus drain-source voltage measurements at room temperature show excellent features associated with single-electron peaks. By matching the peak spacing with the parallel plate charging energy Ec = e2 /2C, a single electron tunnel is then confirmed. However, a critical issue remains unanswered in that paper, which is why the onset of peaks is always at a finite voltage. Early I-V measurements at low temperatures [9–11] had found the same kind of effect, and it was attributed to a Coulomb blockade, where all the charges are totally repelled by the Coulomb interaction and no current could flow. But lately, the concept of a quantum shuttle [12] has been proposed to explain how electrons can move from one electrode to another via the repetitive motion of the central dot at finite temperatures. Consequently, current noise [13] will appear. In our view, in this highly uncertain region other mechanisms, such as dissipation [14], elastic deformation [15], and mechanical feedback [16] can also play a vital role. As such, it is definitely worth doing more careful experiments in this subject. As is expected, we find that at very low charging voltages, the Id -Vd curve shows drastically different behavior compared to what has been observed before; giant oscillations appear and dominate the I-V spectrum from the very beginning. After reaching a maximum value the current quickly decreases to zero. Thereafter, periodic peaks show up again. To understand this unprecedented phenomena, we proposed a forced vibration model. It turns out that this model can explain the observed data very well.
∗
Corresponding author. E-mail address: [email protected] (Y.-M. Wan).
http://dx.doi.org/10.1016/j.cjph.2017.05.030 0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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Fig. 1. (a) Schematic diagram of nanopillar transistor structure. A SiNx /Si/SiNx quantum dot 10 × 10 × 9 nm3 in size is in the center. The Al side gate is ∼6 nm away from the dot. (b) TEM picture of the SiNx /Si/SiNx layers after deposition. Bobbles on the sides are Ga+ accumulated through use of a focused ion beam.
The transistor, schematically drawn in Fig. 1(a), was fabricated on p-type (100) silicon wafers. It basically features a central polysilicon layer separated from the top and bottom contacts by two nitride barriers. This dot cavity has a critical length of 3 nm and is closely coupled to a gate electrode at the side. The fabrication of the device proceeds as follows; we first deposited the multilayer structure of SiNx (3-nm)-polysilicon (3-nm)-SiNx (3-nm) sequentially via low-pressure chemical vapor deposition (Fig. 1(b)), and then chemically etched it to create a nominal plateau of 200 × 140 × 210 nm3 . The source electrode is located at the bottom of the 200 nm thick structure, and has a sheet resistance of ∼30 /cm2 achieved via doping with 1x1019 cm−3 P+ in the mixed gas of SiH4 and PH3 . To prevent electrical short circuits, a short oxidation time using rapid thermal annealing for 30 s was then carried out to seal the nanopillars (creating ∼1.5 nm oxide). Next, a contact window (∼20 nm wide) was opened on the top nitride layer to form the drain contact. To do this, a layer of tetraethylorthosilicate (TEOS) ∼200 nm thick was first grown to level the height of the plateau, and then spun coated with a layer of photoresist. After development, the exposed TEOS in the upper-right area was cleaned off. Following another chemical etch to further miniaturize the nanopillars, a short oxidation was applied. The photoresist which remained at this stage was used as a hard mask for wet etching in H2 O and HF (ratio 50 to 1) for about 1 min. This lateral etch creates a cut underneath, and opens an active zone. After the strip of the photoresist, polysilicon was defined at a normal angle with respect to the source. The overlap region of both electrodes therefore defines the nanopillars with an outside dimension of ∼20 × 20 × 9 nm3 . To further squeeze the cavity, the technique of self-aligned oxidation was used to add another ∼6 nm layer of oxide, giving a total thickness of ∼9 nm, and resulting in a quantum dot of 10 × 10 × 9 nm3 in size. The last step of the fabrication process consisted of the sputter deposition and etching of Al (300 nm) to provide a side gate next to the cavity. The device was then loaded into a probe station (PS150, Cascade Microtech, USA) for current-voltage measurements. A HP 4156 C three-terminal meter, with a resolution of 1 mV and 10 fA, was used. Since our model is intimately related to the manner of electrical charging, the pulsed voltages are schematically drawn in Fig. 2(a). Within a time period of 1 mS, 1 mV was applied in the first half cycle. After that, another 1 mV was added (for a total of 2 mV on the device) in the next period. Such a type of charging is commonly used by experimentalists, as it can avoid overheating. However, it is also the cause of mechanical vibration. In Fig. 2(b), when certain amounts of charge Qs are accumulated on the surface of the bottom nitride layer, a force it will be created which can bend the layers. At the other end of the layer, a positive current +I(t) is then produced. The elasticity of all the layers, here marked as the energy ࢞E, will then force a backward movement to create a negative current –I(t). As a consequence, the current starts to alternate. Correspondingly, the displacement x(t) will change between +α and –α , as illustrated in Fig. 2(c). ¨ +cx(t) ˙ +kx(t)=f(t) [17], where The theoretical model best suited to describe such behavior is the dynamical equation mx(t) ¨ ˙ x(t) is the second order time derivative of x(t); x(t) is the first order derivative; x(t) is the displacement; m represents the mass of the system, as shown in Fig. 3, c denotes the damping coefficient, k is the spring constant, and f(t) is the applied force.
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Fig. 2. (a) Pulsed voltages used for electron charging, of value 1 mV and 2 mV respectively, in a 1 mS period. (b) Electro-mechanical model, a positive current +I(t) and a negative current –I(t) are created during a period by the surface charges Qs and the elastic energy E of the transistor. (c) Maximum displacements, of value α corresponding to the current.
Fig. 3. Forced vibration model. f(t) is the applied force, m the system mass, c the damping coefficient, k the spring constant, and x the displacement.
To solve this equation, two situations must be considered. One is the zero force condition with f(t) = 0, and the other is a finite force with f(t)=0. The solution xg (t) (general) of the force-free condition is a textbook example [13], and it is
α e−ξ ωn t sin(ωn 1 − ξ 2 )t + ϕ ). Here α is the maximum displacement, ξ is the ratio ccc , cc is 2mωn and ωn =( mk )1/2 is the natural frequency. The term inside the bracket of the sinusoidal function ω=ωn 1 − ξ 2 is the response frequency, which is also the frequency measured in an experiment. ϕ is the initial phase, which can be set to zero. Overall, xg (t) is simplified as α e−ξ ωn t sin(ωt ). With this quantity, we can calculate the induced current. Here we stress that although the device is very small, the transportation of electrons is expected to obey the fundamental law of conduction, namely, Ig (t)= neAυ d , where n is the ˙ volume density of electrons, A is the layer area, and υ d is the drift velocity, as defined by x(t). Given all these parameters, the current created by the damping vibration is found to be Ig (t)= neA{αωe−ξ ωn t cos(ωt )-αξ ωne−ξ ωn t sin(ωt )}. Notice that there are two components; an even cosine term and an odd sine term. As will be demonstrated later, the second component is negligible due to the small value of ξ = 0.006. As a consequence, one has
Ig (t ) = neAαωe−ξ ωn t cos(ω t )
(1)
Apart from Ig (t), there is another part of the current known as the particular solution Ip (t). Derivation of it is different from Ig (t), because one has to go back to the original equation mx¨ +cx˙ +kx=f(t). This equation is complicated in mathematics due to the presence of the time-dependent term f(t). As a result, an analytical solution is almost impossible (if not reachable by numerical calculations). So, one can only expect an approximation solution. Here, we use such an approach. Clearly, the resistive term has to be responsible for counteracting the applied force, similar to the terminal velocity of an object free falling in a gravitational field. Accordingly, the identity becomes easy, i.e, cx˙ p =f(t), and Ip (t) can be derived. By taking the parameters of x˙ p (t)= f (ct ) , f(t) =eE, E= Vd , polysilicon thickness (d), and applied voltage (V) into neAx˙ p (t), one obtains
Ip ( t ) =
nAe2V . cd
(2)
This formula shows that Ip (t) is predicted to be proportional to the applied voltage V. With the theoretical discussions finished, we now analyze the data. In Fig. 4(a), a typical Ig (t) curve is plotted for comparison to the data. This result is obtained after several estimates for ωn and ξ , which appear to be the most critical parameters in the model. Notice that this alternating and slightly modulated damping curve mimics the main features of all the data very well. In Fig. 4(b), the best fit happens in the range between 0 V and 0.12 V, where a smaller damping module appears and is interrupted by a giant peak in the middle. A similar situation is shown in Fig. 4(c), except that the interruption is wider and has smaller peaks. As shown in Fig. 4(c), the interruption disappears and the good fit goes all the way from 0 V to 0.17 V. This result is outstanding as it not only answers the key query of this paper, but also proves that mechanical stability is an essential part for the upcoming quantum tunneling of single electron.
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Fig. 4. (a) Model predictions of the general current Ig (t). This data is obtained by using the fitting parameters n = 5×1022 /cm3 , e = 1.6×10−19 C, A = 100 nm2 , α = 0.3 nm, ω =103 Hz, and ξ = 0.006. (b) – (d) Drain current versus drain-source voltage at 300 K.
Other than these major effects, some minor features are also observed in the data. In Fig. 4(b), the appearance of a giant wave packet near 0,04 V is clearly due to coupling to a stronger vibrational mode. Similar behavior is seen in Fig. 4(c). However, in Fig. 4(d) when the gate bias becomes close to 100 mV, this odd mode is eventually eliminated as the multilayers become much stiffer. As to the influence of Ip (t), the impact is quite interesting. As discussed earlier, the high voltage squares should appear and dominate the Id -Vd spectrum. Indeed, those effects are seen in the data. In Fig. 4(b), pulses are clearly seen between 0.2 V and 0.3 V and in Fig. 4(c), they do emerge between 0.22 V and 0.26 V. In Fig. 4(d), they show up between 0.2 V and 0.35 V. Their appearances at high Vd indeed agree with the model prediction, however, their peak heights seem to be overwhelmed by the normal damping. One thing worth pointing out is that the giant e2 /2C peaks prove that they are due to the collective effects of electrons. Finally, we estimate the value of α . This can be done at t = 0 s (Vd = 0 V). At this moment Id is roughly about 0.2 pA (see Fig. 4(b)) and it also equals neAα . By taking a doping concentration of n = 1019 /cm3 , e = 1.6×10−19 C, A = 100 nm2 , and ω = 103 Hz, we find that α is ∼1 μm. This is obviously too large to be correct. A proper α should fall in the nanometer range. So we replace n with a value of 5 × 1022 /cm3 [18], which is the atomic volume density of Si (SiNx as well), and find that α becomes ∼0.3 nm. This more realistic value appears to be much better at suggesting that a uniform distribution of electrons on each Si atom (or Qs ) is the underlying mechanism for the stable vibration. In conclusion, we have studied electron transport in a nanopillar transistor at 300 K. Current-voltage measurements at various gate bias voltages show that damping currents are an intrinsic behavior at zero and low voltages. An electromechanical model is proposed to explain the experimental data, and good agreement is found between it and the data. We also find that mechanical vibration is an essential part of the transport, and once it has ceased single-electron tunneling takes over. A uniform distribution of electrons on the device is critical, as it can provide a balanced force to bend the layers and to eliminate the intrusion of odd vibrational modes.
T.C. Wang, Y.-M. Wan / Chinese Journal of Physics 55 (2017) 1225–1229
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Acknowledgments We thank the National Nano Device Laboratory for much assistance with device fabrication. The work was supported by the National Science Council of the Republic of China under Contracts NSC97-2112-M-214-003 and NDL97-C05SG-140. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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