Direct observation of band-gap closure for a semiconducting carbon nanotube in a large parallel magnetic field
S. H. Jhang, M. Marganska, Y. Skourski, D. Preusche, M. Grifoni, J. Wosnitza, C. Strunk
DDirect observation of band-gap closure for a semiconducting carbon nanotube in alarge parallel magnetic field
S. H. Jhang, ∗ M. Marga´nska, Y. Skourski, D. Preusche, M. Grifoni, J. Wosnitza, and C. Strunk Institute of Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Dresden High Magnetic Field Laboratory, Forschungszentrum Dresden-Rossendorf, 01328 Dresden, Germany (Dated: November 3, 2018)We have investigated the magnetoconductance of semiconducting carbon nanotubes (CNTs) inpulsed, parallel magnetic fields up to 60 T, and report the direct observation of the predicted band-gap closure and the reopening of the gap under variation of the applied magnetic field. We alsohighlight the important influence of mechanical strain on the magnetoconductance of the CNTs.
PACS numbers: 73.63.Fg, 75.47.-m, 73.23.Ad
Carbon nanotubes (CNTs) are attractive buildingblocks for nanoelectronic devices. While electronic prop-erties of CNTs are determined to be either metallic orsemiconducting once they are grown, a magnetic field B (cid:107) parallel to the tube axis provides an elegant way to tunethe band structure of a CNT after its growth [1]. The ori-gin of the sensitivity to B (cid:107) lies in the contribution of theAharonov-Bohm (AB) phase to the orbital phases pickedup by electrons encircling the perimeter of the tube. TheAB phase tunes the periodic boundary condition alongthe tube circumference and results in a φ -periodic mod-ulation of the band gap [1–3], where φ = h/e is the fluxquantum. Recently, significant drops in conductance G were induced by B (cid:107) for initially metallic CNTs [3–5] asthe energy gap of metallic CNTs linearly opens with mag-netic flux φ for φ ≤ φ /2. For semiconducting CNTs,theory predicts that the initial energy gap linearly de-creases with φ to close the gap at φ = φ /3, and reopensreaching a local maximum at φ = φ /2. The gap thencloses again at φ = 2 φ /3 and recovers its original valueat φ = φ [1, 6]. However, as actual magnetic fields B equivalent to φ are about 5000 and 50 T for CNTswith diameters d of 1 and 10 nm, respectively, the ABeffect of semiconducting CNTs has only been partiallyinvestigated for φ (cid:28) φ [7–9], and the direct observa-tion of the predicted semiconductor-to-metal transitionat φ = φ /3 has so far remained elusive. Moreover,while CNTs of d ≥ φ /3within the accessible fields of about 60 T in a special-ized pulsed-magnet lab, the magnetoconductance (MC)in thick CNTs is often strongly affected by disorder andquantum interference effects [10, 11], making it difficultto solely identify the AB effects on the band structure.In this Letter, we report a magneto-transport study ona clean semiconducting CNT performed in pulsed mag-netic fields of up to 60 T. The MC of the tube showed aclear manifestation of the AB effect on the band structurewhen located near the charge neutrality point (CNP).The conductance changes with B (cid:107) by more than 100times showing a peak, then a dip close to B (cid:107) = B /2 before approaching the second peak. The position of thepeak is shifted from the expected B (cid:107) = B /3, which canbe explained by the effect of mechanical strain originat-ing from the tube bending.Our experiments have been performed on devices madeof individual CVD-grown CNTs on Si/SiO /Si N sub-strates [12]. The heavily p -doped Si was used as a backgate and the thickness of the insulating layer was 350 nm.Pd (50 nm) electrodes were defined on top of the tubes bye-beam lithography. The distance between two Pd elec- (a) (b) S ) B ll (a) (b) G ( μ / h ) μ m B (T) G ( e / B (T) ll G B ll (T) V ) (c) E g ( e V B / 3 ( )
00 1 E B / 3 B (T) ll ϕ / ϕ FIG. 1: (a) MC of a semiconducting CNT device near thecharge neutrality point, measured at 3.1 K. The inset showsthe MC in a semilog scale. (b) The scanning electron micro-scope image of the measured device. The tube was smoothlybent while growing over 10 µ m on the substrates (black dot-ted line below the tube given as a guide line to the eyes). TheMC was studied between the two Pd electrodes, where thetube is almost linear. Magnetic fields were applied parallel tothis section of the tube, and the accuracy of the alignmentwas ∼ ± ◦ . (c) Calculated energy gap of a (95,15) semicon-ducting CNT in a parallel magnetic field. Solid and dashedlines are without and with the Zeeman effect, respectively.While the observed MC peak at B (cid:107) = 22 T can be related tothe gap closure at φ = φ /3, the diameter d = 8 ± . ≤ B /3 ≤ a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov trodes was ∼
500 nm. The dc two-probe magnetocon-ductance was studied in pulsed magnetic fields, appliedparallel to the tube axis. The data presented here wereobtained from a CNT ( d ≈ G (0) is greatly suppressed as the Fermi energy is locatednear the CNP by applying a gate voltage V g = 4 V.With the application of B (cid:107) , G ( B (cid:107) ) exponentially in-creases by two orders of magnitude to recover the level ofone conductance quantum ( e /h ) until it reaches a peakat B = 22 T. The conductance then drops back to a min-imum around B min = 37 T, before increasing towards theexpected second peak. Although the second peak was notreachable in our experiment, the negative curvature ofthe MC curve near 60 T, seen in the inset of Fig. 1(a), in-dicates that the second peak is located not far above 60 T.We also notice G (0) (cid:28) G ( B min ). For comparison, we cal-culated the energy gap E g in B (cid:107) for a (95,15) CNT (with d similar to our tube) presented in Fig. 1(c); the largeconsecutive change in the conductance agrees in generalwith the band-gap modulation due to the AB effect. Theconductance peak at B = 22 T and the minimum at B min = 37 T can be attributed to the band-gap closure at φ = φ /3 and a local E g maximum at φ = φ /2, respec-tively. The observation, G (0) (cid:28) G ( B min ), results fromthe fact that E g ( φ /2) = E g (0). However, we note twoexperimental observations not explained within the sim-ple model: 1) The height of the first peak is smaller thanthat of the second peak. 2) If B = 22 T corresponds tothe band-gap closure at φ = φ /3, then the second peakshould already appear at B = 44 T. Also, the diameter d = 8 ± . ≤ B /3 ≤ ∼ ±
100 meV per 1% strain, due tothe shift of the K and K (cid:48) Dirac points under the strain.This change of the band gap (∆ E g ) is independent ofdiameter, whereas E g ∝ d − for semiconducting CNTs.Therefore, the effect of strain becomes more importantfor larger diameter tubes.Fig. 2 illustrates the shift of the K and K (cid:48) Dirac pointsunder uniaxial strain, and the resulting effects on thepositions of the MC peaks. The shift of the K -pointsdepends on the uniaxial strain σ (= L − L L ) and the chiralangle θ , and is given by [13]∆ k ⊥ = τ a − (1 + ν ) σ cos(3 θ ) , ∆ k (cid:107) = − τ a − (1 + ν ) σ sin(3 θ ) , (1) (b) K / (a) p = -1 k || k ⊥ K θ θ p = +1 p = 0 (d) (c) p = -1 p = +1 σ > 0 p = -1 K K / E k ⊥ strain G p = +1 k AB E σ > 0 Ф / Ф k ⊥ E K K / with B ll FIG. 2: (a) Hexagonal Brillouin zone with lines of allowed k ⊥ . K (red) and K (cid:48) (blue) points shift with an angle of 3 θ fromthe k ⊥ axis in the presence of uniaxial strain. (b) A zoom intothe area bound by the green box shows the position of the K point relative to the lines of allowed k ⊥ , depending on thetype p . [Here p = ± p = 0 are for semiconducting andmetallic CNTs, respectively.] (c) Corresponding Dirac coneswith lines of allowed k ⊥ for the CNTs with p = − σ > K and K (cid:48) points underthe strain, lower cones explain the resulting effect with B (cid:107) .The allowed k ⊥ states shift to the right with increasing B (cid:107) by k AB = (2 /d )( φ AB /φ ) due to the AB effect and close thegap of the tube when crossing the K -points. With the strain,those quantized k ⊥ lines intercept the K (cid:48) point earlier thanat φ = φ /3, and the K point later than at φ = 2 φ /3. (d)Resulting shift of the MC peaks for CNTs at σ >
0. Solidand dashed lines are with and without the strain, respectively.The two peaks move either closer (for p = +1) or away fromeach other (for p = − k ⊥ values due to the diameter shrinkage under thestrain. where τ = ± K and K (cid:48) Dirac points, a is theC-C bond length, and ν being the Poisson ratio. It isdisplayed in Fig. 2 for the case of tensile strain ( σ > K -points under σ >
0, thepositions of the gap closure at φ = φ /3 and at φ =2 φ /3 are also shifted either closer (for p = +1) or awayfrom each other (for p = −
1) depending on the type p of the semiconducting CNT (Fig. 2(d)). Here p = ± n, m ) satisfy n − m = 3 q + p with q being an integer. For the case of compressivestrain ( σ < p .Supposing the type of our tube as p = − σ > p = +1 and σ < p = − σ >
0, the first peak at B (cid:107) = B /3 is shifted to the left, DOS (a u ) ϕ / ϕ
100 0 1 DOS (a.u.)100 (95,15) 0 150 (95,15) p = +1 m e V ) ϕ / ϕ
10 (96 14)100 E ( m e V ) E ( m e (a) (b) p = 10-100 804020 60 0 8040-100 (b) p = -1B (T) || || FIG. 3: Model calculation of the DOS for (a) (95,15) and(b) (96,14) semiconducting CNTs in a parallel magnetic fieldat +0 .
2% strain. For comparison, dashed lines indicate theband-edge position without strain. and the second peak at B (cid:107) = 2 B /3 to the right by theamount of ∆ B = ( d/ B | ∆ k ⊥ | . Assuming the shiftedpeaks at B = 22 T and B ∼
60 T, a simple calculation[17] leads to B ≈
82 T and ∆ B ≈ B corresponds to σ cos(3 θ ) ≈ · − from Eq. (1),supposing ν ≈ σ = 1 . · − for zigzag tubes ( θ = 0 ◦ ) would explain theshift of MC peaks observed in our data [19]. Without thestrain-induced shift, the model suggests the MC peaksoccur at 27 and 55 T with the φ -periodicity of 82 T.This value of B , equivalent to d = 8 . d = d σ ν ≈ · − nm) is negligible. Theexistence of a small strain is likely in our device, as ourtube was mechanically deformed during the growth onthe SiO substrates [Fig. 1(b)] [20].Taking the (95,15) tube ( d = 8.1 nm and θ = 7 . ◦ ) asa model CNT with p = −
1, we calculated the density ofstates (DOS) in a parallel magnetic field at +0 .
2% strain.For comparison, we present the DOS of a (96,14) tube,which has almost the same d and θ , but with p = +1. TheDOS was calculated from the dispersion relation, withmomenta close to the Fermi points modified according toEq. (1). We used periodic boundary conditions in the ax-ial direction, suitable for very long nanotubes. Displayedin Fig. 3, the dashed lines indicate the positions of theband edges without strain. At zero field, the axial straineither reduces (for p = −
1) or increases (for p = +1) theband gap of the CNTs, as demonstrated by previous ex-periments [15, 16]. With the application of B (cid:107) , the bandedges evolve, reflecting the orbital and Zeeman splitting.While the band gap is closed for both tubes at 27 and54 T without strain, the positions of the band-gap closureshift under +0 .
2% strain, resulting in the gap closure at22 and 58 T for the (95,15) tube, and at 32 and 48 T for the (96,14) CNT. The relation E g ( φ /
2) = E g (0)becomes under strain E g ( φ / > E g (0) (for p = − E g ( φ / < E g (0) (for p = +1)].The DOS calculated for the (95,15) tube at +0 . B ∼
60 T can be due to theband-gap closure (at φ = φ /3) from the inner shell,while the first peak at B = 22 T originates from theouter shell of the tube. However, corresponding d of 5.4and 9 nm for the inner and the outer shell, calculatedfrom this model, differ significantly from the known inter-shell distance in multi-walled CNTs ( ∼ G ( B (cid:107) ) versus V g . InFig. 4(a), MC traces at 3.5 K are displayed mainly forthe hole side of the CNP, where the CNT/Pd interfaceis most transparent [25]. At B (cid:107) = 0, the hole conduc-tance at V g = 2 V is high, almost 2 e /h , and decreasesrapidly as the Fermi energy is tuned towards the CNP G ( e / h ) g = 2 V5 V3.5 V G ( e / h ) B (T) ll B
22 T (a) (b) B CNP V g (V) FIG. 4: (a) G ( B (cid:107) ) traces at 3.5 K for various values of V g shown for the hole side of the CNP, except for V g = 5 V (dot-ted line) on the electron side. Near the CNP ( V ∗ g ≈ G ( B (cid:107) ) data exhibit a peak at B = 22 T. For B (cid:107) ≥
45 T,the extrapolated curves seem to merge at one point, imply-ing the complete gap closure at B ∼
60 T (see the extrap-olated dashed lines). (b) The gate characteristics G ( V g ) at B (cid:107) = 0 and 22 T, deduced from the G ( B (cid:107) ) traces. Datapoints at V g = 5.5 and 6 V are added from MC traces notshown in Fig. 4(a). G ( V g ) indicates that a small gap remainsat B = 22 T. Note the conductance for the electron side ofthe CNP is lower, compared to that of the hole side, due tothe larger Schottky barriers at the CNT/Pd interface. Solidlines are guides to the eyes. ( V ∗ g ∼ V g ≤ G ( V g ) at0 and 22 T, deduced from the MC traces in Fig. 4(a). G ( V g ) at B = 22 T shows that a small gap ∆ gap still re-mains at B , whereas the AB effect predicts a completegap closure at φ = φ /3. On the other hand, the ex-trapolated MC curves converge at B (cid:107) ≥
45 T [Fig. 4(a)],indicating a complete gap closure for the assumed sec-ond peak around 60 T. The remaining small gap at B is responsible for the smaller height of the first peak inFig. 1(a), compared to that of the second peak.The Zeeman effect splits antiparallel spin states andreduces the band gap by ∆ E Zeeman ≈ · B (cid:107) ,affecting the φ -periodic modulation of the band gap [2,26] as shown in Fig. 1(c). Including the Zeeman effect,the small gap observed at B is closed when the Zeemancontribution becomes larger than ∆ gap at higher B (cid:107) . Thecomplete gap closure for the assumed second peak around60 T suggests the size of ∆ gap < E Zeeman corresponds to ∼ gap at B , thetube bending can mix the states between the quantizedlines of allowed k ⊥ and open a gap for metallic CNTs[14]. Therefore, a bending-induced gap ∆ bend , competingwith ∆ E Zeeman , is present at φ = φ /3 and at φ = 2 φ /3for curved CNTs, partly contributing to ∆ gap . However,∆ bend ∝ ( d/D ) [14], with an estimated bending diam-eter D of ∼ µ m, is too small ( (cid:28) gap . On the other hand, the inter-shell interactioncan also lead to a gap, for example, when the symmetryis lowered by disorienting one shell axis with respect tothe other [21]. Therefore, the observed ∆ gap at B forour tube might originate from the inter-shell interaction,apart from the bending-induced gap.Finally, we discuss the possible effect of spin-orbit cou-pling [27, 28] on the MC of semiconducting CNTs. Thespin splitting induced by spin-orbit coupling results in apeculiar double-peak MC structure for a chiral metallicCNT, as reported in our previous work [5]. For semi-conducting CNTs, the MC peak at φ = φ /3 does notsplit into two, as the Zeeman contribution at φ = φ /3(∆ E Zeeman ≈ /d meV[nm − ]) is much larger thanthe spin-orbit energy splitting (∆ SO ≈ . /d meV[nm − ])[29].In conclusion, our experiment clearly shows that asemiconducting CNT can be converted into a metallicone with the application of large B (cid:107) , providing a consis-tent confirmation of the AB effect on the band structureof semiconducting CNTs. In addition, we reveal thatthe position of the band-gap closure at φ = φ /3 can betuned by mechanical strain. Combined control of boththe strain and the AB effect may open up new possi-bilities for magneto-electronic and magneto-optical CNTdevices. We acknowledge B. Witkamp and H. van der Zant forhelp in the growth of CNTs. This research was sup-ported by the Deutsche Forschungsgemeinschaft withinGRK 1570 and SFB 689 and by EuroMagNET under theEU contract No. 228043. ∗ e-mail:[email protected][1] H. Ajiki and T. Ando, J. Phys. Soc. Jpn. , 1255 (1993).[2] U. C. Coskun et al ., Science , 1132 (2004).[3] B. Lassagne et al. , Phys. Rev. Lett. , 176802 (2007).[4] G. Fedorov et al ., Nano Lett. , 960 (2007).[5] S. H. Jhang et al ., Phys. Rev. B , 041404 (2010).[6] J.-C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. , 677 (2007).[7] E. D. Minot et al ., Nature (London) , 536 (2004).[8] S. Zaric et al ., Science , 1129 (2004).[9] G. Fedorov et al ., Appl. Phys. Lett. , 132101 (2010).[10] G. Fedorov et al ., Phys. Rev. Lett. , 066801 (2005).[11] B. Stojetz et al. , Phys. Rev. Lett. , 186802 (2005).[12] J. Kong et al. , Nature (London) , 878 (1998).[13] L. Yang and J. Han, Phys. Rev. Lett. , 154 (2000).[14] L. F. Chibotaru, S. A. Bovin, and A. Ceulemans, Phys.Rev. B , 161401 (2002).[15] E. D. Minot et al ., Phys. Rev. Lett. , 156401 (2003).[16] M. Huang et al ., Phys. Rev. Lett. , 136803 (2008).[17] B /3 − ∆ B = 22 T and 2 B /3 + ∆ B = 60 T.[18] D. S´anchez-Portal et al ., Phys. Rev. B , 12678 (1999).[19] Eq. (1) was confirmed by experiment [16] with an ad-ditional prefactor of 0.57. This would result in twice aslarge strain to explain the shift of the MC peaks in ourdata.[20] As our tube is fixed to electrodes on the substrate, differ-ent thermal-expansion coefficients between Si and CNTcan also play a role. Using α Si ∼ . × − K − and α CNT ∼ − K − at room temperature [16], a roughestimate leads to σ = ( α Si − α CNT ) ∆ T ≈ − , with∆ T ≈ −
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