ballistic transport in silicon vertical transistors
TRANSCRIPT
Ballistic transport in silicon vertical transistorsK. Nishiguchi and S. Oda Citation: Journal of Applied Physics 92, 1399 (2002); doi: 10.1063/1.1489496 View online: http://dx.doi.org/10.1063/1.1489496 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/92/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigating the impact of source/drain doping dependent effective masses on the transport characteristics ofballistic Si-nanowire field-effect-transistors J. Appl. Phys. 115, 124502 (2014); 10.1063/1.4869495 Tensile strained Ge tunnel field-effect transistors: k·p material modeling and numerical device simulation J. Appl. Phys. 115, 044505 (2014); 10.1063/1.4862806 High-field transport and terahertz generation in GaN J. Appl. Phys. 104, 113709 (2008); 10.1063/1.3032272 Efficient simulation of silicon nanowire field effect transistors and their scaling behavior J. Appl. Phys. 101, 024510 (2007); 10.1063/1.2430786 Si/Si 1−x Ge x heterostructures: Electron transport and field-effect transistor operation using Monte Carlosimulation J. Appl. Phys. 82, 3911 (1997); 10.1063/1.365696
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Ballistic transport in silicon vertical transistorsK. Nishiguchia)
Research Center for Quantum Effect Electronics, Tokyo Institute of Technology, 2-12-1 O-okayama,Meguro-ku, Tokyo 152-8552, Japan
S. Odab)
Research Center for Quantum Effect Electronics, Tokyo Institute of Technology, and CREST, JST2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan
~Received 30 October 2001; accepted for publication 2 May 2002!
Clear evidence for ballistic transport has been observed at 5 K from silicon vertical transistors withwrap around gates. The effect of channel shape was investigated experimentally and accounted fortheoretically by the anisotropy of the Si conduction band. A reduction in conductance and theappearance of multiple steps were observed when a magnetic field was applied perpendicular to thechannel. These results were successfully modeled within the effective mass approximation byincluding the magnetic vector potential and effects due to series resistance and the spin and valleydegeneracy. ©2002 American Institute of Physics.@DOI: 10.1063/1.1489496#
I. INTRODUCTION
A shrinkage in transistor size is the major trend forhigher performance integrated circuits. However, a continuedshrinkage may lead to effects disturbing the operation of cir-cuits, such as electron tunneling, fluctuation in the number ofelectrons, and the short channel effect. Moreover, it is hard toreduce the power supply voltage without sacrificing devicereliability, making power consumption a serious problem.The ballistic transport effect is promising for future electrondevices because of an absence of fluctuation and a highertransconductance compared with conventional devices.1–4
These merits enable operation at a low supply voltage, lead-ing to low power consumption.
Ballistic transport in silicon has thus far been observedonly when the sample was cooled in a dilution refrigerator orby ac measurement using a lock-in technique, because of itsshort electron mean free path.5–7 Previously, we observedclear quantized conductance in silicon at 3 K in direct current~dc! mode using a vertically structured device.8 In the verti-cal device, the channel length was defined by the thicknessof a chemical-vapor deposited~CVD! film and the channelwidth was controlled by electric field confinement, using awrap-around gate. Electric confinement reduced electronscattering that caused smeared ballistic transport. The poly-crystalline Si~poly-Si! channel, which was prepared by solidphase crystallization~SPC! of a CVD amorphous Si~a-Si!film, was of very high quality with a grain size of 200 nmand a mean free path of 45 nm. These characteristics enabledthe observation of a clear ballistic transport effect.8 Verticaldevices with little electron scattering and a strong confine-ment effect are also expected to be useful for the investiga-tions of characteristics under magnetic field, i.e., the spin andvalley splitting effect.
In this article, we present fabrication and properties ofvertical silicon transistors which show the ballistic transport.
In Sec. II, we describe fabrication processes of this device.All fabrication processes involve conventional techniquesonly. In Sec. III, we discuss electrical characteristics withoutmagnetic field. First we theoretically analyze the dependenceof ballistic transport effect on the geometry of a poly-Sichannel. Then, we investigate the structure of the measuredsample. In Sec. IV, we investigate the effect of magnetic fieldin vertical devices made of silicon material, and discuss themeasured characteristics in terms of spin and valley splittingof the Si conduction band. In Sec. V, the effect of the channelgeometry longitudinal to the current path is investigated. InSec. VI, we discuss vertical transistors from the viewpoint ofactual applications. Finally, in Sec. VII, we present a sum-mary and concluding remarks. Methods are proposed to in-crease the conductivity and the operation temperature ofthese devices.
II. FABRICATION
Figure 1 shows the device structure and fabrication pro-cesses. Vertical transistors were fabricated on a SiO2~20 nm!/poly-Si~20 nm!/SiO2~20 nm!/~100! Si substrate as shown inFig. 1~a!. The lower SiO2 was formed by thermal oxidation.A poly-Si layer with phosphorous doping of 1018/cm3 servedas a gate electrode. A low density of electron states is desir-able in order to suppress fluctuations of electron transport,i.e., random telegraph signals. Thus, we performed the SPCmethod to obtain a large grain size of poly-Si. First, ana-Sifilm was deposited by a Hg-sensitized photoinduced CVDmethod because a high density of hydrogen-terminated Sibonds could lead to a large grain size. Then, the sample wasannealed at 700 °C for 4 h for SPC,9,10 followed by anneal-ing at 900 °C for complete impurity activation. This methodgrows poly-Si with a grain size of more than 200 nm. Afterthis operation, the upper SiO2 layer was formed by theplasma enhanced CVD using tetraethoxysilane~TEOS! fol-lowed by annealing at 1100 °C for 2 h.
For channel formation, 60 nm360 nm square holes werepatterned by electron-beam~EB! lithography. An upper SiO2layer was etched anisotropically by electron-cyclotron-
a!Electronic mail: [email protected]!Electronic mail: [email protected].
JOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 3 1 AUGUST 2002
13990021-8979/2002/92(3)/1399/7/$19.00 © 2002 American Institute of Physics
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resonance reactive-ion-etching~ECR-RIE! with CH4 gas.Then, for the formation of a wrap-around gate electrode, thepoly-Si layer was etched isotropically by plasma etchingwith a CH41O2~10%! gas mixture. This isotropic etchcaused side etching, thus expanding the hole size beyond thatpatterned by EB lithography. The difference between the ac-tual and the patterned sizes defines roughly the gate oxidethickness formed at a later process. Then, the lower SiO2 wasanisotropically etched by ECR-RIE. Figure 1~b! shows thestructure after these processes.
Next, a 20-nm-thick TEOS SiO2 layer was deposited asthe gate oxide, followed by annealing. The TEOS SiO2 layerwas etched by ECR-RIE as shown in Fig. 1~c!. The thicknessof the gate oxide was defined by the thickness of gate TEOSCVD SiO2, the size of the etched poly-Si gate electrode, andthe etching of the gate TEOS SiO2. Thus, the actual thick-ness of the gate oxide was not 20 nm. Later, we will estimatethe thickness of the gate oxide from observed electrical char-acteristics. Then the poly-Si channel was formed by SPC ofCVD a-Si, at the condition similar to the formation of thegate poly-Si, as shown in Fig. 1~d!. A grain size of largerthan 200 nm, as mentioned before, practically ensured thatthe device channel was free of grain boundaries, whichwould affect the electron mean free path strongly. Althoughgrains generally crystallize froma-Si with random orienta-tions, the SPC technique can enhance the probability for aboundary-free poly-Si channel due to the following reason.Before the deposition ofa-Si, the surface of the substratewas covered with little or no SiO2 . Thus,a-Si was expectedto crystallize epitaxially, through exposed Si~100! at the bot-tom of the holes, leading to an absence of grain boundariesin the channel. In fact, the observed electric characteristicswere consistent with an absence of grain boundaries in thechannel of most samples.
Finally, gold films were deposited on the top poly-Si andthe bottom Si substrate for Ohmic contact after an isolationprocess for each device.
III. RESULTS AND DISCUSSION OF CONDUCTANCECHARACTERISTICS
For the electrical measurement, the Si substrate waselectrically grounded and a source-drain bias voltage wasapplied to the poly-Si layer. We could control the current bydepleting the channel with an applied gate voltage. Figure 2shows the channel conductance which displayed staircaselikecharacteristics with a step height of 4e2/h at a low source-drain bias voltage of 1 mV under various magnetic field con-ditions. These measurements were performed at 5 K, and themagnetic field was applied perpendicular to the channel.These characteristics can be explained by ballistic transport,which requires an electron mean free path longer than thesize of the channel. The channel length here was defined bythe thickness of the CVD film, and the channel width wascontrolled by the gate voltage. So a very small channel waseasily realized. The electron mean free path in a large area~.100 mm! poly-Si film, which was considered to containmany grain boundaries, was measured to be 45 nm by Hallmeasurement at 5 K. In the vertical structure device, theelectron mean free path is expected to be longer than evalu-ated length due to the small device size, compared to a largegrain size of a poly-Si channel. Thus, these investigationsindicated that the electron mean free path was longer than thedevice size. Furthermore, since the depletion layer separatedthe channel away from the gate oxide interface, the scatteringprobability at the oxide interface, caused by surface rough-ness and interface states, was reduced significantly.1,2
The heavy doping of the channel enhances the confine-ment effect in the channel. The depletion layer thickness,which is narrow in the heavily doped poly-Si channel, can becontrolled by the gate voltage. The low electric field in thedepletion layer, which depends on the density of the dopingimpurity in the channel, makes the potential profile in thechannel parabolic. In that case, energy differences betweensubband levels near the depletion layer are reduced, leadingto scattering between subbands. The strong electric field inthe more heavily doped channel makes the potential profilesteep in the channel with the same depletion length and pre-vents subbands near the depletion layer from being smeared.Thus, the scattering between energy subbands can be re-
FIG. 1. Fabrication processes.~a! The initial structure before lithographyprocess.~b! The structure after the formation of the channel hole by EBlithography and etching processes.~c! The formation of the gate oxide.~d!The final structure of the vertical structure transistor.
FIG. 2. The conductance characteristics as a function of the gate voltage atvarious magnetic fields. The measurement is performed at 5 K with asource-drain bias voltage of21 and2100 mV.
1400 J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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duced. These features of the vertical structure device canfacilitate the observation of the ballistic transport effect.
All curves in Fig. 2 saturated over the gate voltage of220 mV. We tentatively ascribe this saturation effect not toballistic transport but to the shielding effect caused by ion-ized impurities in the gate oxide, which may be includedduring the etching process of the gate oxide, because thesaturation was observed even at a source-drain bias voltageof 100 mV, where no ballistic transport was observed. Incomparison, a higher current drivability of ballistic transportwas evident. With a source-drain bias of 100 mV, the deviceworked like a normally on field effect transistor~FET!, thusthe device dimensions could be evaluated. From the esti-mated dimensions of the device structure, we can calculatequantized subband energy based on effective mass approxi-mation, and compare with the characteristic in ballistic trans-port regime. First, let us discuss the effect of the channelshape. The channel shape patterned by EB lithography wassquare. However, it was difficult to faithfully reproduce inactual etching the patterned shape due to the very small size.If the channel shape were symmetric, subbands in the chan-nel would be degenerate with energies given by the one-dimensional confinement model. In that case, the second stepheight of the observed conductance in the ballistic transportregime would become twice the unit value; 4e2/h. However,all observed step heights were the same. This experimentalresult indicated that the channel shape was asymmetric.
Another deviation between the patterned and the actualstructure which is of great consequence to ballistic transport,is an angular misorientation of the channel against the Sisubstrate.11 The direction of the current path is expected tobe the@100# axis of Si because the poly-Si film is depositedon the~100! Si substrate. However, the planar view orienta-tion of the channel depends not on the Si substrate but on themechanical setting on the EB instrument. In the conductionband of silicon material, there are six ellipsoidal valleys inkspace. We define axes inxyz space as follows:x axis is@100#, y axis @010#, andz axis @001#. In Si, the valley ink
space is not atG but nearX. Thus, effective mass equationsin each axis become
F 1
2mlS 2 i\
]
]x6\kSiD 2
2\2
2mt
]2
]y22
\2
2mt
]2
]z2
1V~y,z!Gf5Ef for the x axis,
F2\2
2mt
]2
]x21
1
2mlS 2 i\
]
]y6\kSiD 2
2\2
2mt
]2
]z2
1V~y,z!Gf5Ef for the y axis,
F2\2
2mt
]2
]x22
\2
2mt
]2
]y21
1
2mlS 2 i\
]
]z6\kSiD 2
1V~y,z!Gf5Ef for the z axis,
wherekSi is the distance of the center of the ellipsoids fromthe G point in k space,m50.19m0 the transverse mass,ml
50.98m0 the longitudinal mass,\ the Planck’s constant,Ethe energy eigenvalue, andf the wave function.V(y,z) in-dicates the cross sectional potential in the channel. The chan-nel is surrounded by the depletion layer with high electricfield and SiO2 . Thus, we assume that the channel is a one-dimensional confinement system surrounded by infinite po-tential barrier. By solving these equations, we can estimatesubbands in the narrow channel at various shapes of thechannel. In this case, the most critical dimension is that ofthe narrowest region in the channel. Figure 3~a! indicatessubbands at various ratios ofLy to Lz at the constant crosssectional square (Ly3Lz510310 nm2) whereLy andLz arethe width of the channel at they and z axis, respectively.
FIG. 3. ~a! Calculated subbands as a function ofLz /Ly at Ly3Lz5100 nm2. The solid, dashed, and dotted curves indicate the subbands resulted from thex,y, andz axis. ~b! Calculated subbands as a function ofu at Ly512 andLz58 nm.u is an angle betweeny axis andLy as shown in the inset.
1401J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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Figure 3~b! indicates subbands as a function of the angleushown in the inset. At an angle of 45°,y and z valleys aredegenerate because of symmetry in theyz plane. This simu-lation shows that the channel shape and angle are very im-portant parameters. Thus, a high reproducibility in the devicecharacteristics requires an accurate fabrication technology.
Next, we estimate the device geometrical parametersfrom characteristics in the nonballistic transport regime,where the device works like a normally on FET, and in bal-listic transport without magnetic field, where the step ofquantized conductance is given by 4e2/h. When the channelprofile is assumed to be as depicted in Fig. 4~a!, the calcu-lated conductance characteristics qualitatively reproducedexperimentally observed characteristics as shown in Figs.4~b! and 4~d!. As shown in Fig. 4~a!, the cross sectionalshape of the channel is modeled to be asymmetric, and theplanar view orientation is best-fitted to be tilted to 12°against the Si substrate orientation flat. We prepared samplesafter cutting Si wafer into pieces and set them to an EBlithography system. Thus, a deviation of the shape and theangle from the designed pattern was plausible. Figure 4~b!shows also the dependence of the channel size on the gatevoltage. The estimated channel size renders subbands in thechannel, from the effective mass approximation in the fash-ion, as shown in Fig. 4~c!. The Fermi energy shown in Fig.4~c! is estimated from the step position of the observed char-acteristic atB50 T. The doping level and the resistivity,which were used as fitting parameters, are 431017/cm3 and0.0045 V cm, respectively. As we mentioned before, theshielding effect in SiO2 was expected to shift conductancecharacteristics by a gate voltage of220 mV. Thus, we in-
clude this gate voltage shift in the simulation. In Fig. 4~b!,the calculated curve agrees quite well with experimental ob-servation. However, the calculated curve in the ballistictransport regime agrees only for the height of the conduc-tance plateau and the approximate position of the voltagestep as shown in Fig. 4~d!. The difference is speculated to becaused by the thermal fluctuation of electrons and the source-drain bias voltage.12 At finite temperatures, electrons enterdiscrete subband levels of the point contact with a range ofelectron energy gained from finite temperature and thesource-drain bias voltage. The electron transport is thusbroadened by the energy spread of the incoming electrons,leading smeared experimental curves. In another sample, weobserved characteristics smeared dramatically with an in-crease of measurement temperature as shown in Fig. 5~b!.Taking into account the experimental condition where thethermal energy is 4kBT of 1.7 meV and the voltage-gainedenergy is 1 meV, it is plausible to conclude that the differ-ence between the experimental and the simulated curves isdue to the thermal fluctuation of electrons.
IV. CONDUCTANCE CHARACTERISTICS UNDERMAGNETIC FIELD
With increasing magnetic field, the conductance de-creased in such a way that values of the quantized conduc-tance became half or quarter as shown in Fig. 2. These char-acteristics likely resulted from spin and valley splitting in astrong magnetic field.13–15Spin splitting follows the Zeeman
FIG. 4. ~a! Plan view structure of thevertical channel.~b! The experimentaland calculated conductance character-istics as a function of the gate voltageat the source-drain bias voltage of 100mV. The channel size is also shown atthe source-drain bias voltage of 1 mV.~c! The calculated subbands in thepoint contact as a function of the gatevoltage. Based on the consideration ofthe shielding effect from ionized im-purities in the gate oxide, the curvesare shifted by220 mV in thex axis.~d! The experimental and calculatedconductance characteristics as a func-tion of the gate voltage at the source-drain bias voltage of21 mV in ballis-tic transport regime. The bold and thedashed line indicate the calculated andexperimental characteristic, respec-tively.
1402 J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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effect;mBg* B, wheremB is the Bohr magnetron, andg* theeffective Lande´ g factor. The thermal energy at 5 K~0.43meV! is smaller than the difference of spin splitting of 0.7meV at a magnetic field of 6 T. Thus, it is possible indeed toobserve spin splitting. On the other hand, the condition forobservation of the Landau level (Bm.1; m is electron mo-bility ! is satisfied. In Si, there are three pairs of valleys. Sothere are two kinds of subband shift on each pair of valleys,positive and negative. There are many models for valley
splitting in strong inversion layers in metal–oxide–semiconductor structures, i.e., electron tunneling effect andsurface scattering.15 For a simple description of valley split-ting, we add the vector potentialA to the effective massequations. Since the magnetic field was applied perpendicu-lar to the channel, we DefineA5(zBy2yBz,0,0), whereBy
and Bz are the magnetic fields alongy and z axes, respec-tively, and they satisfy the conductionB25By
21Bz2. In this
case, effective mass equations are
H 1
2mlF2 i\
]
]x6\kSi1e~zBy2yBz!G2
2\2
2mt
]2
]y22
1
2mt\2
]2
]z21V~y,z!J f5Ef for the x axis,
F21
2mtH 2 i\
]
]x1e~zBy2yBz!J 2
11
2mlS 2 i\
]
]y6\kSiD 2
2\2
2mt
]2
]z21V~y,z!Gf5Ef for the y axis,
F21
2mtH 2 i\
]
]x1e~zBy2yBz!J 2
2\2
2mt
]2
]y21
1
2mlS 2 i\
]
]z6\kSiD 2
1V~y,z!Gf5Ef for the z axis.
Figures 6~a! and 6~b! show subband levels as a function ofthe gate electrode atBz56 T and 9 T, respectively. TheFermi energy, derived from the conductance characteristicswithout magnetic field in Sec. II, is also shown. These curvesindicate that there are more than eight subbands at the gatevoltage of220 mV, and, therefore, the conductance shouldbe more than 83e2/h. However, the observed curves indi-cated that conductance at 6 and 9 T was less than 83e2/hand 43e2/h, respectively. These deviations are quite largeand worth further discussion. A plausible explanation is theincrease of the series resistance connecting to the smallpoint, which was contact caused by magnetic resistance orelectron scattering resulting from cyclotron movement, asshown in Fig. 7~a!. Usually, cyclotron movement can reducethe electron scattering from the impurities. However, whenthe radius of the cyclotron movement is comparable to thesize of leads connecting to the point contact, electron scat-tering at channel edges and reflection at the point contactincrease.16 In that case, the measured conductanceGm isrelated to the conductance of the point contactGp and theseries resistanceGs
21 by
Gm51
Gp211Gs
21.
From this equation, we can obtainGp . Since experimentalvalues ofGm are not available, we treated them as a fittingparameter. Figure 7~b! shows the corrected conductances ofthe point contact under various magnetic fields, withG0
54e2/h. Calculated curves are roughly the same as the ob-served curves. The deviation results from the smearing byfinite temperature. Thus, it suggests that the observed char-acteristics were influenced by both the ballistic transport ef-fect from the point contact and the series resistance from the
Si film. Further discussion, however, is needed for the as-sumed values ofGs because the series resistance may bechanged not only by magnetic field but also the channel sizecontrolled by the gate electrode. The model of the calculationalso should be discussed because there is, as yet, no com-pletely satisfactory model of valley splitting even thoughmany models are proposed, e.g., the one-dimensional zeroeffective mass theory can account for the coupling of twovalleys.14
V. CONDUCTANCE CHARACTERISTICS FROM ATAPERED CHANNEL
Next, we investigate the effect of the cross sectionalchannel shape perpendicular to the substrate. The channel isformed after the etching process. In some conditions, theetched shape becomes tapered like Fig. 5~a!. The samplewith tapered channel shows quantized conductance charac-teristics resulting from ballistic transport effect as shown inFig. 5~b!. However, the step height of quantized conductancebecame one-half of the height for a point contact. The reasonfor this phenomenon is not clear. One speculation is the for-mation of double point contact, leading to half-conductance.Figure 8 shows the change of depletion layer by the gateelectrode. In proper device structures, two point contacts areformed near the top and the bottom of poly-Si gate electrode.When the distance between two point contacts is comparableto the electron mean free path, the conductance becomeshalf.17 However, the probability of double point contact for-mation is expected to be small. Here we speculate on anotherpossible reason for the observed step height. Figure 9 depictsthe energy profile in the device.En andEn11 indicate sub-band energy with indexn andn11, respectively, in the pointcontact.EFL and EFR indicate the Fermi energy of the left
1403J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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and right leads connecting to the point contact, respectively.At the small source-drain bias voltageVb as in Fig. 9~a!, theconductance isnG0 , whereG0 is the unit conductance ofballistic transport. WhenVb increases and one subbandEn11
is sandwiched between the Fermi energies of two leads con-necting to the point contact as shown in Fig. 9~b!, the quan-tized conductance has a fractional form (n11/2)G0 . Furtherincrease ofVb returnsG to nG0 . This phenomenon stronglydepends on the relation between subband levels and twoFermi energy as shown in Fig. 9. Subband levels in the chan-nel depend on the channel geometry as we mentioned in Sec.
III. Thus, only a sample with a tapered channel displayed aballistic transport effect with a step size of one-half quan-tized conductance.
VI. DISCUSSION
For an actual application, an operation at higher tem-peratures and with a higher source-drain voltage, which leadsto the higher device drivability, is required. A small devicesize and a long mean free path are critical. The device size iseasily shrunk by decreasing the thickness of gate electrodefilms. For a long mean free path, a lightly doped channel isdesirable. However, this leads to high parasitic resistance.So, only the channel, which is surrounded by the gate oxide,should be lightly doped. In the present study, the poly-Sichannel is heavily doped of;1018/cm3, which serves as the
FIG. 5. ~a! The cross sectional structure of a vertical transistor with a ta-pered channel and its scanning electron microscopy image.~b! The conduc-tance characteristics from the device with a tapered channel at various tem-peratures. The source-drain bias voltage is 1 mV. For clarity, all curves areshifted by 0.534e2/h.
FIG. 6. The calculated subbands as afunction of the gate voltage at a mag-netic field of 6 and 9 T. The calcula-tion is based on the device structureshown in Fig. 4~a!. The Fermi energyis located at 15 meV.
FIG. 7. ~a! The parasitic resistance elements caused under magnetic field.~b! The experimental and calculated conductance characteristics at a mag-netic field of 6 and 9 T. For clarity, curves at 9 T are shifted by 234e2/h. The bold and the dotted lines indicate the calculations from effec-tive mass equations and the observed characteristic, respectively. Thedashed lines indicated the curves subtracted the series resistance, which isincreased by applying a magnetic field.G0 indicates 4e2/h.
1404 J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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scattering source resulting in a smeared ballistic transporteffect. Thus, the clear observation of ballistic transport mayindicate that there is no doping impurity in the point contactwith a few tens of nanometer size due to a fluctuation inimpurity distribution. This investigation suggests that the finecontrol of the doping profile in the channel is quite impor-tant.
VII. CONCLUSION
We observed clear quantized conductance in silicon ver-tical transistors at 5 K. The vertical structure with the wrap-around gate was amenable to the formation of very smallpoint contacts and reduced scattering. Specifically, the use ofthe SPC method, which provided a large grain size of poly-Sichannel, allowed a long electron mean free path. The ballis-tic transport in Si was shown to depend on the six ellipsoidalvalleys near theX point in k space. This effect has not beenobserved in ballistic transport devices made from compoundsemiconductors. Under magnetic field, electric transportshowed effects due to spin/valley splitting and remarkableseries resistance. Vertical structures with the ballistic trans-port effect have potential application in circuits with higherdevice density, because they require only conventional fab-rication processes and they can be fabricated on Si.
ACKNOWLEDGMENTS
This work was supported in part by the Grant-in-Aid forScientific Research~A! and~B! from MEXT and by CREST,JST.
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FIG. 8. The change of the depletion layer edge for two structures with taperangle of the channel of 70° and 80° when the gate voltage is changed from21.45 to 0 V. The dashed lines indicate the center of the channel. Thearrows indicate a point contact formed by the depletion layer.
FIG. 9. The energy profile in the device.En and En11 indicate subbandenergy with indexn and n11, respectively, in the point contact.EFL andEFR indicate the Fermi energy of left and right leads connecting to the pointcontact, respectively. When the source-drain bias voltageVb is small asshown in~a!, the conductanceG is n3G0 , whereG0 is unit conductance ofballistic transport 4e2/h. The increase ofVb makesG5(n11)3G0 like~b!, and more increase makesG5n3G0 like ~c!.
1405J. Appl. Phys., Vol. 92, No. 3, 1 August 2002 K. Nishiguchi and S. Oda
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