Magnetoconductance of carbon nanotube p-n junctions
MMagnetoconductance of carbon nanotube p-n junctions
A. V. Andreev
Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA (Dated: June 5, 2007)The magnetoconductance of p-n junctions formed in clean single wall carbon nanotubes is studiedin the noninteracting electron approximation and perturbatively in electron-electron interaction, inthe geometry where a magnetic field is along the tube axis. For long junctions the low temper-ature magnetoconductance is anomalously large: the relative change in the conductance becomesof order unity even when the flux through the tube is much smaller than the flux quantum. Themagnetoconductance is negative for metallic tubes. For semiconducting and small gap tubes themagnetoconductance is nonmonotonic; positive at small and negative at large fields.
PACS numbers: 75.47.Jn, 73.23.Ad, 73.63.Fg
Magnetoconductance arises from the orbital and Zee-man coupling of electrons to the external magnetic field, H . As long as the flux through the crystalline unitcell is much smaller than the flux quantum Φ = hc/e magnetotransport may be described in the semiclassicalapproximation [1] ignoring the band structure changesdue to the presence of H . In most crystals this con-dition holds at all experimentally realizable fields. Re-cently much attention was focused on magnetotrans-port properties of carbon nanotube devices. Becauseof their large radius the magnetic field affects the one-dimensional electron spectrum [2] even at relatively weakfields. This leads to interesting magnetotransport phe-nomena [3, 4, 5, 6, 7, 8, 9, 10].In this paper we show that the magnetic field depen-dence of the one-dimensional band structure results in apeculiar mechanism of magnetoconductance of p-n junc-tions in carbon nanotubes. This mechanism is relevant tothe magnetoresistance of nominally undoped metallic andsmall gap nanotubes placed on an insulating substrate.In this case the long range disorder potential caused bycharged impurities in the substrate creates p- and n- re-gions in the tube, and backscattering of electrons arisesmainly from the gaps between p- and n- regions, wherethe semiclassical description of electron transport fails.We study the magnetoconductance of a p-n junctionformed in a clean single wall carbon nanotube for mag-netic fields parallel to the tube axis. The device is de-picted in Fig. 1 a ). The p- and n- regions can be formedby appropriately biasing the top gates. Such devices wererecently used to measure [11] thermodynamic propertiesof electron liquid in carbon nanotubes. It is shown be-low that for realistic device parameters similar to thoseof Ref. 11 the low temperature magnetoconductance be-comes of order unity while the flux Φ through the tubecross-section is much smaller than Φ .Before proceeding to detailed calculations let us qual-itatively discuss the origin of strong magnetoconduc-tance. We choose the x - and y - axes to be respec-tively along the tube axis and along its circumference.The one-dimensional electron sub-band spectrum is de- FIG. 1: a) Device sketch: a nanotube rests on an insulatingsubstrate. The n- and p- regions are created by biasing thetop gates. b) Band diagram of the device. Tilted solid linesrepresent the bottom of the conduction and the top of thevalence bands. The width of the center region is 2 L . Thewidth of the classically forbidden region is 2 x o = 2∆ /eE .Inset: one dimensional spectrum is obtained by dissectingthe graphene Brillouin zone by the p y = 0 line (dashed line).The low energy states lie in the vicinity of K and K (cid:48) points. termined by intersections of the graphene Brillouin zonewith the k y = (Φ / Φ + m ) /R lines, see the inset inFig. 1 b ). Here R is the tube radius and m an inte-ger. The electron spectrum near the K and K (cid:48) pointsbecomes ± (cid:112) ∆ + ( (cid:126) v p x ) , where v ≈ × m/s isthe electron velocity in graphene, and ∆ is half the en-ergy gap between the valence and conduction sub-bands.In the center region between the p- and n- banks theexternal potential is assumed to be approximately lin-ear, U ( x ) = eEx , where e is the electron charge and E the electric field. The tilting of electron energy bands inthe external potential produces a spatial region of width2 x = 2∆ /eE , where electron motion is classically for-bidden, see Fig. 1 b ). The device conductance is governedby the Landau-Zener tunneling across this region. Withexponential accuracy the tunneling probability may be a r X i v : . [ c ond - m a t . m e s - h a ll ] J un found in the WKB approximation. The electron momen-tum p x along the tube depends on the position as p x = (cid:112) ( eEx − (cid:15) ) − ∆ / v . This gives the transmission prob-ability T = exp (cid:0) (cid:126) Im (cid:82) p x dx (cid:1) = exp[ − π ∆ / ( (cid:126) v eE )].The rigorous calculation given below shows that the pre-exponential factor is equal to unity. The magnetocon-ductance arises from the flux dependence of the bandgap [2], ∆ = ∆ ± (cid:126) v Φ /R Φ , where ∆ is half-the bandgap at zero flux and the ± sign corresponds to the differ-ent valleys, K and K (cid:48) . Accounting for electron spin andthe two valleys and neglecting the Zeeman splitting oneobtains for the device conductance G = 2 e h (cid:88) j =1 , exp (cid:34) − π (cid:126) v eER (cid:18) ∆ R (cid:126) v + ( − j ΦΦ (cid:19) (cid:35) . (1)If the electric field is not too strong, eER (cid:28) (cid:126) v /R ,the magnetoconductance becomes of order unity whilethe flux is still small, Φ (cid:28) Φ . In the case of metal-lic tubes, ∆ = 0, the magnetoconductance is negative.For semiconducting and small gap tubes the magneto-conductance is nonmonotonic; positive at small fields andnegative at large ones. The conductance maximum is at-tained at Φ max ≈ Φ ∆ R/ (cid:126) v . For semiconducting tubes,∆ ≈ (cid:126) v / R , this gives Φ max ≈ Φ /
3. For small gaptubes the zero flux gap arises only due to curvature ef-fects and is rather small, ∆ (cid:28) (cid:126) v /R . In this case theconductance maximum is achieved at Φ max (cid:28) Φ . Anonmonotonic magnetoresistance was recently observedin Ref. 8 in nanotube devices with a different geometry.In the noninteracting electron approximation the mag-netoconductance only weakly depends on the tempera-ture T as long as the latter is smaller than the Fermienergy in the banks. In this regime the energies ofelectrons participating in transport lie in the narrowband of width T around the chemical potential. In thisenergy range deviations of electric potential from thelinear form U ( x ) ≈ eEx are negligible. This resultsin energy-independent transmission coefficient and thustemperature-independent conductance.In the presence of electron-electron and electron-phonon interactions electrons can be transferred betweenthe p- and n- regions at finite temperature by thermalactivation. At T (cid:29) | ∆ − ∆ | = (cid:126) v Φ /R Φ the rate ofinelastic processes is practically independent of the mag-netic field and magnetoconductance arises mainly fromthe tunneling mechanism discussed above. In this regimeinelastic transfers shunt the tunneling mechanism andsuppress magnetoconductance. A crude estimate of thecharacteristic temperature T ∗ , above which the magneto-conductance suppression becomes significant, can be ob-tained in the tunneling regime by equating the activationrate, ∼ (cid:82) ∞ dx(cid:96) in exp[ − (∆ + eEx ) /T ] = TeE(cid:96) in exp( − ∆ /T ),with (cid:96) in being the inelastic mean free path, to the tunnel-ing rate, ∼ exp (cid:16) − π ∆ eE (cid:126) v (cid:17) . According to this estimate the noninteracting electron result, Eq. (1) provides a gooddescription of the conductance for T < eER .At zero temperature the finite reflection amplitude atthe p-n contact leads to the appearance of Friedel oscil-lations in the electron density. The additional scatteringof electrons from the Friedel oscillations in the presenceof electron-electron interactions gives a correction [12]to the noninteracting result for the device conductance,Eq. (1). This correction is evaluated below to first orderin electron-electron interaction. It is given by Eq. (15)and is plotted in Fig. 2 b). It remains small even if theinteraction constant, e / (cid:126) v is of order unity.The device conductance in the noninteracting electronapproximation, Eq. (1), immediately follows from the re-sults of Cheianov and Falko [13] for a graphene p-n junc-tion that were obtained using transfer matrices. Belowwe present a consideration in terms of wave functionsthat is more convenient for the treatment of the interac-tion correction to magnetoconductance. Electron eigen-states of energy ε obey the Dirac equation, which by anappropriate basis choice can be cast in the form[ U ( x ) − ε − i (cid:126) v σ z ∂ x + ∆ σ y ] ψ = 0 , where σ i are Pauli matrices. Introducing the dimension-less coordinate ξ = eEx/ √ eE (cid:126) v , energy (cid:15) = ε/ √ eE (cid:126) v ,and momenta q = ∆ / √ v (cid:126) eE and k ( ξ ) = ( U ( ξ ) − ε ) / √ v (cid:126) eE we rewrite this equation as (cid:18) k ( ξ ) − i∂ ξ − iqiq k ( ξ ) + i∂ ξ (cid:19) (cid:18) uv (cid:19) = 0 . (2)The dimensionless momentum k ( ξ ) changes from − k f at ξ → −∞ to + k f at ξ → + ∞ , where k f (cid:29) k ( ξ ) is linear in ξ , k ( ξ ) ≈ ξ − (cid:15) , and the spinoramplitudes u and v satisfy the differential equation,( ∂ z + a + z ) f = 0 . Here a = i − q for u , and a = − i − q for v , and weintroduced the difference coordinate, z = ξ − (cid:15) . The in-dependent solutions of this equation are parabolic cylin-der functions [14] that can be expressed in terms of theconfluent hypergeometric function F ( α, γ, z ), f e ( z ) = exp (cid:18) − i z (cid:19) F (cid:18)
14 + ia , , iz (cid:19) ,f o ( z ) = z exp (cid:18) − i z (cid:19) F (cid:18)
34 + ia , , iz (cid:19) . Two linearly independent solutions of Eq. (2) are ψ = e − πq (cid:18) u e − u ∗ o (cid:19) , ψ = e − πq (cid:18) − u o u ∗ e (cid:19) , (4)where u e ( z ) = exp (cid:18) − i z (cid:19) F (cid:18) − i q , , iz (cid:19) , (5a) u o ( z ) = q z exp (cid:18) − i z (cid:19) F (cid:18) − i q , , iz (cid:19) . (5b)Equation (2) conserves the current along the tube axis, I x = ψ † σ z ψ . From the form of the current operator itis clear that the top/bottom components of the pseudo-spinor ψ represent the amplitudes of the right-/left- mov-ing waves. Thus the scattering states incident from theleft, ψ L , and right, ψ R , can be found by requiring that thebottom/top component of the spinor vanish at ξ → ±∞ .Using Eqs. (4) and (5) and the large distance asymptoticsof the confluent hypergeometric function [14], F ( α, γ, z → ∞ ) ≈ Γ( γ )Γ( γ − α ) ( − z ) − α + Γ( γ )Γ( α ) e z z α − γ , (6)we obtain for the scattering states incident from the rightand left, ψ L = (cid:18) u L v L (cid:19) = ψ + α ψ (cid:112) | α | , (7a) ψ R = (cid:18) u R v R (cid:19) = ψ − α ∗ ψ (cid:112) | α | . (7b)Here α is given by α = e − i π q (cid:16) − i q (cid:17) Γ (cid:16) − i q (cid:17) , | α | = tanh πq . (8)The z → ±∞ asymptotics of the right-moving wave(the top spinor component) in the scattering states ψ L and ψ R are; u L ( z ) ≈ √ πe − πq [1 − sgn( z ) | α | ] | z | i q e − i z (cid:112) | α | Γ (cid:16) + i q (cid:17) , (9a) u R ( z ) ≈ − q √ iπe − πq [1 + sgn( z )] | z | i q e − i z (cid:112) | α | Γ (cid:16) i q (cid:17) . (9b)From Eq. (9a) one finds the transmission amplitude, t ≡ lim ξ → + ∞ u L ( ξ ) u L ( − ξ ) = 1 − | α | | α | = exp (cid:18) − πq (cid:19) . This gives the transmission coefficient is T = exp( − πq ),in agreement with Ref. 13. Expressing q in terms ofthe system parameters using expressions presented in thetext above Eq.(2), and taking into account the magneticfield dependence of the energy gap one obtains the deviceconductance, Eq. (1). -4 -2 2 4-0.1-0.050.050.1 FIG. 2: a) Dimensionless electron density as a function of ξ for q = 0 . q = 0 .
6, and q = 0 .
9. b) The ratio of the correctionto the transmission coefficient to the interaction constant λ ,Eq. (15), as a function of q . Next we evaluate the first interaction correction toEq. (1) at zero temperature. To first order in interac-tion the correction to the device conductance can be ob-tained by considering the change in the transmission am-plitude for a particle at the Fermi level that arises fromthe additional scattering from the Hartree-Fock poten-tial induced by the electron density [12]. The inducedHartree-Fock potential has two qualitatively different ef-fects on the transmission amplitude: i) By enhancing theeffective electric field inside the classically forbidden re-gion it increases the tunneling amplitude, and ii) It causesadditional backscattering from the Friedel oscillations inthe classically allowed region. The analysis below showsthat repulsive interaction increases the transmission am-plitude.Using Eqs. (5) and (6) it is easy to show that the spinorwave functions in Eq. (4) are normalized to a δ -functionof the dimensionless energy, (cid:90) ∞−∞ dξψ † i ( ξ − (cid:15) ) ψ j ( ξ − (cid:15) (cid:48) ) = 2 πδ ( (cid:15) − (cid:15) (cid:48) ) δ ij . (10)Therefore the electron density, upon subtraction of theuniform ion background, is n ( ξ ) = (cid:88) i =1 , (cid:90) ∞−∞ sign( − (cid:15) ) d(cid:15) π ψ † i ( ξ − (cid:15) ) ψ i ( ξ − (cid:15) ) . (11)In this equation the energy is measured from the Fermilevel and the electron hole symmetry of the problem wasused. Plots of electron density, Eq. (11), for differentvalues of q are presented in Fig. 2 a). The Friedel oscil-lations in charge density appear only at finite refectionamplitude and fall off as 1 /ξ . The extra power of 1 /ξ in comparison with the usual one-dimensional Fermi gasarises from the linearly growing external potential. Be-cause of the fast decay of the oscillations the correctionto the transmission amplitude is free from infrared diver-gences and arises from distances x ∼ (cid:112) (cid:126) v /eE .Below we consider the case of a short range interaction.This should be a reasonable approximation because theFourier transform of the Coulomb interaction depends onthe transferred momentum only logarithmically. Sincethe characteristic scattering momentum is (cid:112) (cid:126) eE/ v thedimensionless interaction constant can be estimated as λ ∼ κ e (cid:126) v ln eER (cid:126) v , where κ is the dielectric constant ofthe substrate. We restrict the consideration to metal-lic tubes, for which electron spectra in the presence ofthe flux remain degenerate in the two valleys. Then theHartree-Fock potential is V ( ξ ) = 3 λ n ( ξ ), where the fac-tor 3 = 4 − (cid:15) = 0, can bewritten as ψ L ( ξ ) + χ ( ξ ). The correction, χ ( ξ ), to thewave function satisfies the equation [ ˆ H + V ( ξ )] χ ( ξ ) = − V ( ξ ) ψ L ( ξ ) with ˆ H = ξ − i∂ ξ σ z + qσ y being the unper-turbed Hamiltonian. To first order in perturbation thesolution of this equation is χ ( ξ ) = (cid:90) dξ (cid:48) G R ( ξ, ξ (cid:48) ) V ( ξ (cid:48) ) ψ L ( ξ (cid:48) ) , (12)where G R ( ξ, ξ (cid:48) ) = ( ˆ H + iη ) − is the Green’s functionthat can be expressed in terms of the spinors in Eq. (4), G R ( ξ, ξ (cid:48) ) = (cid:88) i =1 , (cid:90) ∞−∞ d(cid:15) π ψ i ( ξ − (cid:15) ) ψ † i ( ξ (cid:48) − (cid:15) ) (cid:15) + iη . (13)In order to find the correction to the transmission am-plitude one needs only the large distance asymptotics of χ ( ξ ). Therefore only the on-shell part of the Green’sfunction will contribute to Eq. (12) at ξ → ∞ , χ ( ξ ) = − i (cid:88) i ψ i ( ξ ) (cid:90) dξ (cid:48) ψ † i ( ξ (cid:48) ) V ( ξ (cid:48) ) ψ L ( ξ (cid:48) ) . (14)Since the wave functions ψ , are related to the scatteringstates ψ L,R by the unitary transformation Eq. (7) thesum over i in Eq. (14) can be understood to run over i = L, R . Next we notice that for i = L the integralin Eq. (14) is purely real. Therefore this term will onlychange the phase of the transmission amplitude, but notits modulus. Thus to compute the first correction to thetransmission amplitude we may replace χ ( ξ ) by˜ χ ( ξ ) = − i ψ R ( ξ ) (cid:90) dξ (cid:48) ψ † R ( ξ (cid:48) ) V ( ξ (cid:48) ) ψ L ( ξ (cid:48) ) . The transmission amplitude can be found from the equa-tion t = t +lim ξ → + ∞ (1 , · ˜ χ ( ξ ) u L ( − ξ ) . Using Eq. (9b) we obtainfor the correction δ T to the transmission coefficient, δ T λ = − e − πq | α | (cid:90) dξ (cid:48) n ( ξ (cid:48) )Im[ α ∗ ψ † R ( ξ (cid:48) ) ψ L ( ξ (cid:48) )] . (15)It may be evaluated using Eqs. (7), (4), (5) and (11), andis plotted as a function of q in Fig. 2 b). At weak reflec-tion, q <
1, the correction to transmission coefficient issmall even if the interaction constant is of order unity.This is because the Friedel oscillations amplitude is pro-portional to the reflection amplitude. Thus the noninter-acting result, Eq. (1), provides a good description for the low temperature conductance of metallic devices. In thetunneling regime, q (cid:29)
1, the relative correction to thetransmission amplitude is expected to be strong becauseof the exponential dependence of the tunneling ampli-tude on the effective electric field. Therefore the methoddescribed above becomes inapplicable.In summary, the low temperature magnetoconduc-tance of p-n junctions in clean single wall carbon nan-otubes was studied in the geometry where the magneticfield is along the tube axis. For weak band-tilting field E (cid:28) (cid:126) v /eR the magnetoconductance of long, L (cid:29) R ,junctions becomes of order unity while the flux throughthe tube is much smaller than the flux quantum. In thenoninteracting electron approximation the device con-ductance is given by Eq. (1). The magnetoconductanceis positive for metallic tubes and nonmonotonic for semi-conducting and small gap tubes. The interaction correc-tion to the zero temperature magnetoconductance wasstudied to first order in perturbation theory. It arisesdue to the change in the effective electric field in thegap between the p- and n- regions and due to the scat-tering from the Friedel oscillations. In contrast to theone-dimensional Fermi gas, the Friedel oscillations in thepresent geometry fall off as 1 /x , which leads to the ab-sence of infrared divergence in the correction to the tun-neling amplitude. The net correction to the tunnelingprobability is positive. It is given by Eq. (15) and isplotted in Fig. 2 b). If the reflection coefficient is nottoo strong the noninteracting electron result, Eq. (1), israther accurate even if the coupling constant is of orderunity. At finite temperatures, in addition to tunnelingacross the classically forbidden region electrons can betransferred between p- and n- regions by being promotedacross the band gap due to inelastic electron-electron andelectron phonon scattering. The estimate of activationtransfer rate shows that tunneling processes describedby Eq. (1) dominate the transport for T < eER .I would like to thank D. Cobden, E. Mishchenko, T. D.Son and B. Spivak and for useful discussions. [1] R. Peierls,