Zener Tunneling in Semiconducting Nanotube and Graphene Nanoribbon p-n Junctions
ZZener Tunneling in Semiconducting Nanotube and Graphene Nanoribbon p-n Junctions
Debdeep Jena, Tian Fang, Qin Zhang, & Huili Xing
Department of Electrical EngineeringUniversity of Notre DameIN, 46556 (Dated: November 10, 2018)A theory is developed for interband tunneling in semiconducting carbon nanotube and graphenenanoribbon p-n junction diodes. Characteristic length and energy scales that dictate the tunnelingprobabilities and currents are evaluated. By comparing the Zener tunneling processes in these struc-tures to traditional group IV and III-V semiconductors, it is proved that for identical bandgaps,carbon based 1D structures have higher tunneling probabilities. The high tunneling current magni-tudes for 1D carbon structures suggest the distinct feasibility of high-performance tunneling-basedfield-effect transistors.
PACS numbers: 73.40.-c, 73.40.Kp, 73.40.LqKeywords: interband, Zener, tunneling, graphene, nanoribbon, carbon, nanotube, diode, transistor
Carbon-based 1D materials such as nanotubes (CNTs)and graphene nanoribbons (GNRs) are currently underextensive investigation for the novel fundamental physicsthey exhibit, as well as possible applications they mighthave in the future . A large class of traditional semi-conductor devices rely on the quantum mechanical tun-neling of carries through classically forbidden barriers.Among these, the Esaki diode, resonant tunneling diode,and backward diode are the prime examples . In ad-dition, the high-field electrical breakdown in a numberof semiconductors occurs by interband Zener tunneling.The phenomena of tunneling has been studied extensivelyfor traditional parabolic-bandgap semiconductors in 3Dbulk, as well as quasi-2D and quasi-1D heterostructures.Semiconducting CNTs and GNRs do not have parabolicbandstructures, and the carrier transport in them ap-proaches the ideal 1D case. In that light, it is timely toexamine the phenomena of tunneling in these materials.Tunneling currents in semiconducting CNT p-njunctions have been measured and analyzed recently(see ). For tunneling probabilities, an energy-dependent carrier effective mass has been used earlier , totake advantage of previously existing results of parabolicbandstructure semiconductors. In this work, we evalu-ate the Zener tunneling probabilities of CNT and GNRbased p-n diodes starting from their intrinsic bandstruc-tures, which removes the need to define an effective mass.In addition to interband tunneling probabilities, a num-ber of fundamental associated parameters characterizingthe tunneling process are found.The bandstructure of the nth subband of a semicon-ducting CNT or GNR is given by E = s (cid:126) v F (cid:112) k x + k n , (1)where 2 π (cid:126) is the Planck’s constant, and v F ∼ cm/sis the Fermi velocity characterizing the bandstructure ofgraphene. s = +1 denotes the conduction band, and s = − (cid:126) k x .For GNRs, the transverse momentum is quantized by the ribbon width ; k n = nπ/ W where n = ± , ± , ± , ± , ± , ± ... for a GNR of dimensions( x, y ) = ( L, W ) where
W << L . The correspondingbandgap is E g = 2 (cid:126) v F k = 2 π (cid:126) v F / W ∼ . /W eV(where W is in nm). For comparison, a semiconductingCNT of diameter D has the same bandstructure with k = 2 / D , and a bandgap of E g = 4 (cid:126) v F / D ∼ . /D eV, where D is in nm. If W = πD/
2, the properties(bandgap, bandstructure) of semiconducting CNTs andGNRs are similar. The results derived below are applica-ble to GNRs and CNTs on equal footing. It is assumedthat the length of the GNR (CNT) is much larger thanthe width (diameter) such that the longitudinal momen-tum of carriers in the ribbon are quasi-continuous.We now evaluate the interband tunneling probabilityin a n + − p + GNR or CNT diode of bandgap E g . Weconsider that the doping of the n and p-sides are suchthat the equilibrium Fermi level is at the conduction bandedge ( E c ) in the n-side, and at the valence band edge ( E v )on the p-side. Such doping could be either chemical, orelectrostatic . Under this situation, a forward bias wouldnot lead to current flow (this is similar to the ‘backwarddiode’ ). When a reverse bias eV is applied, the banddiagram looks as shown in Figure 1. Let the electricfield in the junction region be F . We assume that thedepletion region thickness, and the net electric field doesnot change appreciably from the equilibrium values (trueunder small bias voltages). Then the potential energybarrier seen by electrons in the valence band of the p + side is V ( x ) = e F x, (2)in the range 0 < x < d , such that d = E g /e F . Thisis indicated in Figure 1. E is the Dirac point of theunderlying graphene bandstructure, for both GNRs andCNTs, and serves as a convenient reference of energy inthe problem.Since the energy of carriers near the band edge ( k x ≈ a r X i v : . [ c ond - m a t . m t r l - s c i ] J un FIG. 1: Interband tunneling in a reverse-biased GNR/CNTp-n junction, and the potential barrier seen by tunneling elec-trons. valence to the conduction band, the condition − (cid:114) ( (cid:126) v F k x ) + ( E g + E g − e F x = + (cid:114) ( (cid:126) v F k x ) + ( E g (3)holds for the wavevector k x at all x . Within the tunnelingbarrier, the wavevector is imaginary. Denoting this by k x = iκ x where κ x is real, we obtain κ x ( x ) = k (cid:114) − (1 − xd ) , (4)where k = E g / (cid:126) v F is a characteristic wavevector fortunneling. Since (cid:82) d (cid:112) − (1 − xd ) = πd/
4, the WKBband-to-band tunneling (BTBT) probability for the p + − n + junction given by T ∼ exp [ − | (cid:82) + d κ x ( x ) dx | ] leads to T W KB ( k x ≈ ∼ exp (cid:2) − π · E g (cid:126) v F e F (cid:3) , (5)which can be expressed as T W KB ∼ exp( −F / F ),where F = π E g / e (cid:126) v F denotes the characteristic elec-tric field at the junction for the onset of strong tunneling.The corresponding characteristic barrier thickness for theonset of strong tunneling is d ∼ E g /e F = 4 (cid:126) v F /π E g .Using the value of the Fermi velocity, the characteristicfield evaluates to F ∼ . × ( E g ) MV/cm, and the char-acteristic tunneling distance evaluates to d ∼ . / E g nm,where E g is the bandgap of the GNR or CNT expressedin eV in both these expressions. As an example, for aGNR with W = 5 nm, E g ∼ .
275 eV, the characteristictunneling field is ∼ ∼
11 nm.The tunneling probabilities evaluated above are for thefirst subband ( n = 1). For the nth conduction and va-lence subbands of CNTs and GNRs, the effective subband gap scales as ∼ n E g . The tunneling probabilities of car-riers near the respective subband-edges is then given by T W KB,n ∼ exp( − π (cid:126) v F k n / F ), which decay as exp( − n ),indicating a rather strong damping the tunneling proba-bilities of higher subbands. This result turns out to beidentical to one for the nth transverse mode of a zero-gap2D graphene p-n junction, as first evaluated by Cheianovand Fal’ko (see ). Except for the narrowest bandgapCNTs and GNRs, the tunneling is primarily from the1st valence subband in the p-side to the 1st conductionsubband on the n-side.Note that the above derivation uses a triangular bar-rier approximation. It has been shown by Kane that aparabolic barrier more accurately represents the physicsof the tunneling process, and leads to an exponential fac-tor with a different coefficient than from the triangularbarrier approximation. The difference is small, for therest of this work, we will use the result derived above.The tunneling current may now be written in the form I T = e ( g s g v /L ) (cid:80) k x [ f v − f c ] T W KB × v g ( k x ), where g s = 2is the spin degeneracy, g v is the valley degeneracy (= 2for CNTs and = 1 for GNRs ), L is the length of theCNT or GNR, f v , f c are the occupation functions of thevalence band states in the p-type side and the conductionband in the n-side respectively, and v g ( k x ) is the groupvelocity. This sum may be converted into the equivalentintegral I T = 2 g v eh (cid:90) eV [ f v ( E ) − f c ( E )] T W KB d E , (6)where f v ( E ) = 1 / (1 + exp [( E − eV ) /k B T ]) and f c ( E ) =1 / (1+exp [ E /k B T ]). Here k B is the Boltzmann constant.For the band diagram shown in Figure 1, the net tunnel-ing current evaluates to I T = 2 g v e h T W KB × V T ln [ 12 (1 + cosh VV T )] , (7)where V T = k B T /e . This expression captures the tem-perature and bias voltage dependence of the tunnel-ing current. If the applied bias is much greater thanthe thermal energy (
V >> V T ), the current reducesto a form similar to the Landauer expression I T ≈ (2 g v e /h ) T W KB ( V − V T ln 4). Thus, the tunneling cur-rent has a negative temperature coefficient at high biasconditions. The dependence of tunneling currents inGNRs on voltage and temperature are illustrated in Fig-ure 2 (a, b).The tunneling current per unit width of GNRs is max-imized for W = (cid:112) π (cid:126) v F / e F . For example, for F ∼ W ∼ . ∼ µ A/ µ m, comparable to traditional FETs. Forthinner ribbons at higher fields, the current densities canapproach 1000 µ A/ µ m. The dependence of the currentdensities on temperature, electric field, and GNR widthsare shown in Figure 2 (c, d).The BTBT probability in traditional parabolic-bandstructure semiconductors in the triangular barrier FIG. 2: Tunneling currents in GNR p-n junctions: (a) W =5nm device, voltage dependence at various temperatures, (b)temperature dependence at various voltages. Tunneling cur-rents per unit width for different GNR widths at (c) varioustemperatures at F = 1 MV/cm, and (d) for various F at 300K. To maximize the tunneling current density, an optimum( F , W ) combination exists, as illustrated by the dashed lines. approximation depends on the bandgap as T parabolic ∼ exp ( − √ m (cid:63) E / g / e (cid:126) F ), where m (cid:63) is a reduced carriereffective mass. The BTBT probability for GNRs andCNTs retains the same dependence on electric field, butdue to the difference in bandstructure there is a strongerdependence on the bandgap. The effective mass does ap-pear in the tunneling probability of the GNR or CNTdiodes since their bandstructure is not parabolic at anyenergy. If one compares the Zener tunneling probabili-ties in diodes made of CNTs or GNRs with other direct-bandgap semiconductors of the same bandgap, then theratio T carbon T parabolic ∼ exp (cid:2) − E / g e (cid:126) F ( π (cid:112) E g v F − √ m (cid:63) (cid:3) (8)indicates that the GNR or CNT p-n diode will have a higher interband tunneling probability if the relation E g < (16 v F / π ) × m (cid:63) is satisfied. From the k · p theoryfor traditional direct-bandgap semiconductors, the elec-tron effective mass (in the conduction band) is related tothe bandgap by the approximation m (cid:63)c ≈ ( E g / m ,where the bandgap is in eV and m is the free electronmass. This leads to the requirement m v F > ( 3 π
16 ) ×
10 eV , (9)which is always satisfied since the LHS is m v F ≈ . always have a higher reverse-bias Zener tunnelingprobability than a traditional semiconductor of the samebandgap.Two more facts tilt the tunneling probability decisivelyin favor of CNTs and GNRs. First, for bulk 3D p-njunctions, the transverse kinetic energy of carriers canbe large, and leads to a further exponential decrease ofcarrier tunneling probability , which is avoided in 1Dstructures. Second, if normal parabolic bandgap semi-conductors are shrunk to length scales comparable tothat of CNTs and GNRs, their bandgap and the effectivemasses increase further due to quantum confinement.Though Zener tunneling currents are detrimental intraditional devices such as rectifiers, field-effect and bipo-lar transistors, it is important to note that the funda-mental switching action in these devices is controlled bythermionic emission over barriers, which require a min-imum of ( k B T /e ) ln(10) ∼
60 mV per decade change ofcurrent at 300 K (the ‘classical’ limit). However, a newcrop of tunneling-FETs have been recently proposed anddemonstrated ), which rely on the very mechanismstudied in this work. These devices are capable of reach-ing far below the classical limit for switching by exploit-ing quantum mechanical tunneling. It is for such devicesthat high interband tunneling current drives in carbon-based 1D nanostructures hold a distinct advantage, andmuch promise in the future.We gratefully acknowledge financial support from NSFawards DMR-06545698 & ECCS-0802125 & from the Na-noelectronics Research Initiative (NRI) through the Mid-west Institute for Nanoelectronics Discovery (MIND),and acknowledge discussions with Joerg Appenzeller &Eric Pop for this work. P. Avouris, Z. Chen, and V. Perebeinos,
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