Possibility of a Metallic Field-Effect Transistor

0.27 MB·English

by  Slava V. Rotkin

Possibility of a Metallic Field–Effect Transistor
Slava V. Rotkin, Karl Hess
Beckman Institute for Advanced Science and Technology,
University of Illinois at Urbana–Champaign,
405 N.Mathews, Urbana, IL 61801, USA;
4 Email: [email protected]
0 Fax: [phone]
0
2 Wedeveloptheoreticalargumentsthatdemonstratethepossibilityofmetallicfield–effecttransis-

n tors (METFET’s) in one–dimensional systems and particularly in armchair carbon nanotubes. A
a very inhomogeneous electric field, such as the field of a tunnelling tip, can penetrate the relatively
J weakly screened nanotubes and open an energy gap. As a consequence, an energy barrier forms

9 that impedes electron flow and thus permits transistor action. This type of metallic field effect is
advantageous because of the high conductance of themetallic tubesin theON–state.

]
i
c
s Field–effect transistors in current use are semiconduc- (cid:22)
(cid:40)(cid:18)(cid:74)
- tor devices. The scaling trend to nanometer dimensions
l (cid:21)
r calls for ever higher doping and channel conductance of
t
m these devices [1]. Ultimately one desires a conductance (cid:20)
. close to that of a metal if one wants to scale devices to (cid:16)(cid:83)(cid:18)(cid:151)(cid:22) (cid:19) (cid:37) (cid:36) (cid:83)(cid:18)(cid:151)(cid:22)
t
a the smallest possible size. However, a metallic conduc- (cid:78)(cid:69)
m tance also prevents the penetration of the electric field (cid:16)(cid:20)
- exceptforextremelyshortdistances;ordinarilytooshort
d
to achieve device function.
n
We propose here an innovative approach to control
o
c electrontransportinmetallic one–dimensional (1D)sys-
[ tems by use of the inhomogeneous electric field of a low– FIG. 1: Band structure of armchair [10,10] M–SWNT. Left
1 dimensional highly localized ”modulating” gate (MG) Inset shows zoom view of the Fermi point with opening of a
v suchasnanometertipsormetalnano-interconnects. Use gap in the M–SWNT. Right Inset shows how the gap grows
2 ofahighlylocalizedgateresultsinastrongenhancement linearly with applied potential, cf. Eq.(1).
6
of the electric field in a narrow region. Thus any deple-
1
tion of charge is easier to achieve since it occurs only in
1
0 extremely small volumes. In addition, the weak screen- virtually ballistic conductance [3, 7, 8].
4 ingofelectricpotentialsin1Dchannelsfurtherenhances The band structure of an armchair SWNT, as shown
0 possible field effects. inFig.1,hastwosubbandsAandB(shownasboldgreen
t/ The purpose of this letter is to show that metallic and blue curves) crossing at the Fermi level. Absence of
a
carbon nanotubes may be suitable for use as metallic thegapisultimatelyrelatedtotheSWNTmirrorsymme-
m
field–effect transistors (METFET’s). Key to this novel try [9], which forbids any mixing of these subbands. We
d- metallic field effect is the opening of a band gap due to propose to break this mirrorsymmetry by applying very
n the breaking of the mirror symmetry by the MG. Note, non–uniform electric fields by use of a MG. Two possi-
o that a region of spatially non-uniform electric field also blemechanismsforgapopeninginarmchairSWNTsare:
c represents a barrier that reflects electrons, while the ho- (1) by a direct mixing of electron and hole states of the
:
v mogeneous field of an extended gate changes predomi- A and B subbands, which happens if the external po-
i nantly the carrier density. These qualitative factors are tential has an atomic scale corrugation, and (2) by the
X
discussedbelowinamorequantitativewayandtheirpo- indirect subband mixing in a higher order of the pertur-
r
a tential for device applications is illustrated. bation theory, which essentially can be realized for any
Metallic field effect in Armchair NT’s. – IBM non–uniform potential. Both mechanisms require (i) a
researchers have enunciated the vision of combining relatively high electric field (order of 107 V/cm) and (ii)
metallic and semiconducting single–wall nanotubes (M– SWNT symmetry breaking.
SWNT’sandS–SWNT’s)incircuits[2]. TheM–SWNT’s Group theory provesthat the direct matrix element is
would serve as interconnects and the S–SWNT’s as ac- non–zeroiftheFouriertransformofthepotentialU is
q,δm
tive devices of the extremely small size (typically, the afullscalarwithrespecttotherotationsofthesymmetry
SWNT radius is ∼ 0.7 nm) [3, 4, 5, 6]. Here we discuss groupofthe nanotube[9]. Thisis fulfilledforδm=s·n,
the possibility to also use metallic tubes as transistors where s = 2,4,... is an positive even integer and n is
or switching devices, which is appealing because of their the integer appearing in the notation [n,n] for armchair
2
Id, µA T=4K,W=50nm
0.0175
0.015
0.0125
(cid:19)(cid:17)(cid:20)(cid:21) Eg,eV
0.01 (cid:19)(cid:17)(cid:20) (cid:62)(cid:24)(cid:15)(cid:24)(cid:64)
(cid:19)(cid:17)(cid:19)(cid:27)
0.0075
(cid:19)(cid:17)(cid:19)(cid:25)
0.005 (cid:19)(cid:17)(cid:19)(cid:23)
(cid:62)(cid:20)(cid:19)(cid:15)(cid:20)(cid:19)(cid:64)
(cid:19)(cid:17)(cid:19)(cid:21)
0.0025
(cid:19)(cid:17)(cid:24) (cid:20) (cid:20)(cid:17)(cid:24) (cid:21) (cid:21)(cid:17)(cid:24) (cid:22) (cid:22)(cid:17)(cid:24)
V,Volts
g
(cid:19)(cid:17)(cid:19) (cid:19)(cid:17)(cid:19)(cid:21)(cid:24) (cid:19)(cid:17)(cid:19)(cid:24) (cid:19)(cid:17)(cid:19)(cid:26)(cid:24) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:20)(cid:21)(cid:24) (cid:19)(cid:17)(cid:20)(cid:24)
V ,Volts
d
FIG. 2: Setup of an armchair SWNT METFET using the
STM tip as a modulation gate. FIG. 3: Current of a METFET versus the drain potential.
Thegatewidthis50nm. T =4K.Curvesfromtoptobottom
correspond to Vg from 0 to 3 V (in color: from red to blue).
SWNT’s [10]. Then, the energy dispersion of two new Insetshowscorrespondingbandgapopeningforthesamegate
subbands|A±Biinvicinity ofthe Fermipointreadsas: voltage range for [5,5] and [10,10] SWNTs.
E± = ±qEA(q)2+Uq2,δm,. Here EA(q) = −EB(q) =
EB(−q) is the dispersion in A/B subband at zero gate anexample oftwometaltips atoppositesidewallsofthe
voltage. At the Fermi point EA(0) =0 and, thus, a gap SWNT: one at +2Vg and the other at −Vg bias. The
opens: dependence of the bandgap on V is shown for [5,5] and
g
[10,10] armchair SWNTs in the Inset of Fig.3. The gap
E(sn) =2|U | (1)
g q,sn E first increases with V and decreases beyond πγ/(en)
g g
This is a direct mixing of the subbands A and B. The where γ ≃3 eV is a hopping integral. Perturbation the-
bandgapis linear in the applied gate potential (see Inset ory [10, 11, 13] predicts a maximum gap that depends
inFigure1). Afullcalculationpresentingaselfconsistent onlyonthe size ofthe tube andscalesinauniversalway
for nanotubes of arbitrary chirality:
solution of joint Schroedinger and Poissonequations fol-
lowing the derivation of Refs. [11, 12] will be published
¯hv b γ
elsewhere. Except for a natural condition that the field E(c) ∼ F ∝ , (2)
g 2R2 n2
mustnotcauseanelectricbreakdown,thereexistsnoup-
per limit for the magnitude of the gapthat opens in this ThiswasconfirmedbynumericalcalculationsinRefs.[11,
case and we estimate the possibility of a gap of several 13]. Here v = 3γb/2h¯ is the Fermi velocity for a M–
F
eV. SWNT,andb≃0.14nmisthebondlength. Eventhough
Thus direct mixing of the crossing subbands allows to the maximumgapis smallinthis caseand thus has con-
open a large gap. It does requires, however, a very high sequences for the conductance only atlow temperatures,
multipole moment of the potential in order to break the the gap opening should be much easier to achieve ex-
symmetry (we must at least have δm = 2n) and this perimentally e.g. by use of the inhomogeneous electric
is difficult to achieve because the amplitude of any high field of a tunnelling tip (Fig.2). Also, ultra narrow leads
multipolecomponentwillbetypicallymuchsmallerthan [14], fabricated closely to the nanotube channel, special
the gate voltage, U << V . Nevertheless, it may be (electro–)chemical function groups at the tube sidewalls
q,2n g
possible to create such a potential by applying the gate or inside the tube [15] may be used as a MG. We notice
voltage to a chemically modified surface of the SWNT. that use of a dual gate (both local and backgate) may
This approachis technologically complex. be beneficial for 1D METFET’s because of the uniform
Fortunately,thereexists amorestraightforwardpossi- backgate controls the Fermi level (charge density) while
bilitytoopenabandgapbyvirtuallyanykindofinhomo- the MG controls the conductivity of the channel.
geneouselectricfieldthatbreaksthemirrorsymmetry. A Field effectmodulation of metallic conductance. –Any
bandgap opens also by indirect mixing of the subbands. opening of the gate induced semiconductor gap along
This bandgap opening is smaller because it arises from the armchair metallic tube will create a potential bar-
higher order perturbation terms (similar to a nonlinear rierfortheelectronsandthereforemodulateconduction.
Stark effect). In particular, any potential that contains (NotethatthismodulationmaybeenhancedbyCoulomb
both a uniform field and a higher order (quadrupole) blockade [16].) Inclusion of tunnelling is important for
componentwill leadto a gapwhichscales with the third the calculation of the current because the effective mass
power of the gate voltage. In what follows we consider oftheelectronsinSWNT’sisverysmall(∼0.06m ,simi-
o
3
logI /I (cid:19)(cid:17)(cid:19) (cid:19)(cid:17)(cid:24) (cid:20)(cid:17)(cid:19) (cid:20)(cid:17)(cid:24) (cid:21)(cid:17)(cid:19) (cid:21)(cid:17)(cid:24) (cid:22)(cid:17)(cid:19) Id(cid:15)µA curve). IntheRightInsetofFig.4,theMETFETcurrent
ON OFF
100 Vg,Volts 0.0175 is plotted vs. the Vg for the dipole–quadrupole potential
(twotipgeometry)asdescribedabove. Atlowdrainbias
0.015
(bluecurve)theMETFETisswitchedOFFatV ∼3V.
10-1 G/G 0.0125 g
0 The total transmission coefficient of the METFET,
1.0 0.01
T(E) = G(E)/G , assuming transparent contacts, is
10-2 0.8 0.0075 plotted in the Lefot Inset of Fig.4. Several families of
0.6 0.005 curvesaregivenfor different temperatures. Inthe upper
10-3 0.4 T=295K 0.0025 family of curves (room temperature) the suppression of
0.2 T=77K transmission through the gated region is clearly seen as
10-4 0.0 T=4K a function of an increased gap: from 0 to 0.12 eV (from
0.0 0.02 0.04 0.06 V,Volts 0.10 green to red, top to bottom). At lower temperatures
d
0.0 0.1 0.2 0.3 0.4 0.5 T =77K and 4K this effect is naturally sharper.
E ,eV Thecurrentisanexponentialfunctionofthegap(and,
g
hence, of the gate voltage). Results are shown in Figure
4: by opening a gap of about 0.5 eV (which would cer-
FIG.4: TheMETFET currentvstheopenedbandgap. W =
50 nm. The red (upper) curve is at 295K, the blue (lowest) tainlyrequiresophisticatedtechnology)onecandecrease
curve is at 4K. The drain voltage dependence of the barrier the current by five orders of magnitude.
transmittance is shown in the Left Inset. The families of In conclusion, we have proposed a novel type of
curves correspond different temperatures: T =4K, 77K and electronic switching device based on carbon nanotubes:
295K from bottom to top. Within each family, the gap is
a one–dimensional metallic field–effect transistor (1D
increasedfrom0to0.12eVfromtoptobottom(incolor: from
METFET). We have described armchair SWNT’s in de-
blue–green topink–red). Right Inset: METFET currentasa
tailbecauseoftheirspecialsymmetrythatalsogivesrise
function of Vg (at the bias Vd from 0 to 0.15 V from bottom
to top, in color: from blueto red). to virtually ballistic transport. Our calculations demon-
strate, at least principle, the possibility to open a band
gap by application of inhomogeneous electric fields that
lartothein–planeeffectivemassofgraphite[18]). Atnon are created by special gates (modulation gates) that in
zerotemperaturethermionicemissionmustbetakeninto the simplest case resemble a tunnelling tip to break the
accountaswell. Althoughthe rateoftunnelling through SWNTsymmetry. Assumingballistictransport,wehave
the classically forbidden (gated) region is high, a signif- calculatedIVcurvesfortheMETFET’swithgatewidths
icant metallic field effect can be achieved by increasing of the order of 15–50 nm and found significant modu-
the width ofthe gatedregion,orby operatingthe device lation as well as reasonably large ON/OFF current ra-
at low source–drain voltage, V , and low temperature, tios. We finally note that one can expect excellent scal-
d
T. (The ON/OFF current ratio is controlled by the pa- ing properties of the SWNT METFET because of the
rameter E /V .) Because the transport in the armchair metallic conductivity enhanced by ballistic transport.
g d
SWNT METFET is ballistic, its conductance in the ON Authors are indebted to Mr. B. Grosser for help with
stateislimitedonlybyreflectionsatthecontact. Wewill image preparation and Ms. Y. Li for help in numerical
use in what follows (mainly for normalization purposes) computations. This work was supported by NSF Grant
a maximum conductance of 4G =2e2/h [19]. No. 9809520, by the Office of Naval Research grant
o
For the OFF state, we estimate a semiclassical WKB NO0014-98-1-0604 and the Army Research Office grant
transmission coefficient, T(E), by assuming a uniform DAAG55-09-1-0306. SVR acknowledges partial support
gap of width W in the gated region. In order to extend ofDoEGrantNo. DE-FG02-01ER45932,andNSFgrant
the validity of our results to non–zero temperatures we No. ECS–0210495.
calculate the total transmission of the channel by inte-
grating the partial current, ∼ T(E), multiplied by the
difference of the Fermi distribution functions of the left
and right electrodes.
Figure3showstypicalIVcurves(IVC)foraMETFET
using a [5,5] armchair M–SWNT of the diameter 0.7 nm
at T =4K, W = 50 nm. The upper (red) IVC corre-
sponds to a zero gate voltage (no gap). The channel is
fullyopenandtheONcurrentisdeterminedbyinjection
from a contact and thus by the quantum conductance
4G [19]. With increasing gate voltage one observes a
o
substantialdecrease of the currentdue to the opening of
the gap and a depletion of electrons in this region (blue
4
March 7-11, 2004 (in press).
[14] J.-F. Lin, J.P. Bird, L. Rotkina, P.A. Bennett, Appl.
Phys. Lett. 82 (5), 802, 2003.
[1] Advanced Theory Of Semiconductor Devices// Karl [15] S.V. Rotkin, I.Zharov, Int. Journal of Nanoscience 1
Hess. New York: IEEE Press, 2000. (3/4), 347, 2002.
[2] J.Appenzeller, DeviceResearch Conference, Utah,June [16] At the points of metal–semiconductor junction (MSJ)
26, 2003 a charge accumulation may happen. This MSJ bound-
[3] J. Appenzeller, J. Knoch, R. Martel, V. Derycke, S.J. ary becomes a side of a1D capacitor with a capacitance
Wind,Ph.Avouris, IEEE Transactions on Nanotechnol- ∼W/2ln(W/R).Neglectingthelogarithmicterm,aclas-
ogy, 1(4), 184, 2002. sicalchargingenergyofthiscapacitorise2/2W,whichis
[4] J.Appenzeller,R.Martel,V.Derycke,M.Radosavljevic, largeforanarrowgate.However,inthisworkwedidnot
S.Wind,D.Neumayer,andPh.Avouris,Microelectronic consider a Coulomb blockade because an effective gap is
Engineering, 64, 391, 2002. exponentially renormalized at ahigh conductanceof the
[5] Ph.G. Collins, Ph. Avouris, Sci.Am., 12, 62, 2002. Ph. tunneljunction[17].Largecouplingbetweensidesofthe
Avouris,Chemical Physics, 281, 429, 2002. MSJ and quantum fluctuations of the charge wash out
[6] S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appen- the correlation and destroy theCoulomb blockade.
zeller, Ph.Avouris, Physical Review Letters 89, 106801, [17] K.A. Matveev, Phys.Rev. B 51, 1743, 1995.
2002. [18] The effective mass near the band edge m∗ =
[7] P.L. McEuen, M. Bockrath, D.H. Cobden, Y-G. Yoon, ¯h2(∂2E/∂k2)−1 = 2¯h2/(9bRγ) is about 0.06 of the free
and S.G. Louie, Phys.Rev.Lett. 83, 5098, 1999. electron mass for theSWNTof theradius R≃0.7 nm.
[8] A. Javey, H. Kim, M. Brink, Q. Wang, A. Ural, J. Guo, [19] For a circuit with macroscopic leads to the M–SWNT
P.McIntyre,P.McEuen,M. Lundstrom,H.Dai, Nature channel the total conductance will be about 4Go =
Materials 1, 241, 2002. 2e2/h, 4 times of the conductance quantum (for 2 spin
[9] T. Vukovic, I. Milosevic, M. Damnjanovic, Physical Re- and 2 space channels). This gives a minimum resistance
view B 65(04), 5418, 2002. of the SWNT device ∼ 6.5kΩ. The lower resistance can
[10] Y.Li, S.V.Rotkin,U.Ravaioli, and K.Hess, unpublished. be expected in the case of entirely nanotube circuit [2].
[11] Y.Li, S.V.Rotkin and U.Ravaioli, Nano Letters, 3(2), The quantum contact resistance will not limit anymore
183, 2003. theONcurrentinthiscase.Thatdevicecanfullyexploit
[12] K.A. Bulashevich, S.V. Rotkin, JETP Letters 75, 205 , all advantages of theMETFET.
2002.
[13] S.V. Rotkin, K. Hess, Proc. of Nanotech 2004, Boston,