Modeling STM images of chiral metallic carbon nanotubes

Accepted Manuscript Modeling STM images of chiral metallic carbon nanotubes

Ahmed A. Maarouf PII: DOI: Reference:

S0925-9635(17)30753-7 doi:10.1016/j.diamond.2018.03.027 DIAMAT 7067

To appear in:

Diamond & Related Materials

Received date: Revised date: Accepted date:

28 December 2017 22 March 2018 22 March 2018

Please cite this article as: Ahmed A. Maarouf , Modeling STM images of chiral metallic carbon nanotubes. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Diamat(2018), doi:10.1016/ j.diamond.2018.03.027

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Modeling STM images of chiral metallic carbon nanotubes Ahmed A. Maaroufa,∗ Department of Physics, Institute for Research and Medical Consultations, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia

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Abstract

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Scanning tunneling microscopy is a powerful tool for the study of the electronic and structural properties of carbon nanotubes. Efforts have been made to image

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discrete states on short metallic nanotubes. Here we address two important issues pertinent to understanding such images. (1) Electronic states of nanotubes

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have two bands at the Fermi energy. Reflection at the end of the tube will mix the bands in a non-universal way to form two distinct families of eigenstates.

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(2) Images of graphite and nanotubes can be drastically altered by asymmetries

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in the STM tip. We consider these images for finite chiral metallic tubes and demonstrate that by using both classes of image the tip effect can be extracted from the image. In this way the tip asymmetry can be quantified, and the

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eigenstates on the tube can be seen.

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1. Introduction

Scanning tunneling microscopy (STM) offers a unique opportunity for real

space determination of electronic structure on the atomic scale [1–3]. Using ∗ Corresponding

author Email address: [email protected] (Ahmed A. Maarouf)

Preprint submitted to Diamond & Related Materials

March 23, 2018

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STM, the electronic density patters have been observed in carbon nanotubes (CNT’s) structures [4–9]. Computationally, STM images of doped carbon nanotubes structures have been studied. [10] The robust one dimensional electronic

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structure of metallic CNT’s make them an ideal system for the study of onedimensional (1D) quantum effects on a mesoscopic scale. In particular, when cut to a finite length, they resemble the simple model of a 1D particle-in-a-box

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exhibiting discrete standing wave states. Venema et al. have observed such states by measuring the differential conductance as a function of voltage and

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position along the axis of a 30 nm long armchair CNT[6]. They found oscillations characteristic of individual eigenstates whose period was longer than that

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of the graphene lattice. These 1D line scans open the way for obtaining 2D spatial images which, remarkably, would reflect the atomic scale character of individual electron wavefunctions.

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Interpretation of 2D images of discrete states obtained in this way will be

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complicated for two reasons. (1) Since the electronic states of a CNT are derived from two bands, the structure of the wavefunctions (and thus the image)

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depends on the nature of the reflection of the waves from the tube ends. Previous theoretical work on finite tubes has focused on armchair tubes with high

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symmetry caps[11, 12]. In this case, symmetry prevents mixing between the two bands, and two distinct classes of wavefunctions can be distinguished by their symmetry. If the CNT has low symmetry caps or is chiral, however, this symmetry is lost, and the two bands will be mixed by reflection in a non universal way. (2) Asymmetry in the STM tip will severely modify the images. Early

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work by Mizes et al. [13, 14] highlighted the caution which must be exercised when interpreting STM images: observed images of graphite surfaces are regularly asymmetric, but the asymmetry of the image depends on the tip. This

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is crucial for the imaging of discrete states. The STM image is not simply a picture of the wavefunction.

In this paper, we develop a framework for interpreting energy resolved STM

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images for chiral nanotubes, taking into account both the tube-ends and tip effects. By focusing on the longest wavelength Fourier components of the wave-

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function, we extract the dependence of the standing wave patterns on chirality and the reflection at the ends, and we identify constraints which restrict the

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form of the patterns. We then show that for the purposes of analyzing the images the tip is characterized by 5 parameters. Since there are two bands there are two distinct classes of wavefunctions on the tubes. The images of these

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two states contain sufficient information to determine 4 of the 5 tip parameters.

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With the tip calibrated in this manner, the wavefunctions of the eigenstates can be extracted up to an unknown phase. This leads to specific predictions regard-

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ing the correlations between the images of different states and the evolution of the images with distance down the tube. In this sense, the wavefunctions of the

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eigenstates can be measured.

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Figure 1: (a) Low energy linear dispersion of metallic tubes about the two K points. rij denote the components of the rˆL . (b) Boundary conditions enforced by reflection at tube ends. (c) Spectral structure of the eigenstates. (d) Lowest star in reciprocal space of the tube lattice.

2. Eigenstates of finite nanotubes

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In the effective mass model[15] wavefunctions of low energy states have the form

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





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 uK L (~r)   u−K R (~r)   + ψR (x)  , Ψ(~r) = ψL (x)      u−K L (~r) uK R (~r)

(1)

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where u±K L,R (~r) are the Bloch wavefunctions of the right and left moving bands at the ±K points in Fig. 1(a). ψL,R (x) are two component envelope functions

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which vary slowly with distance x along the tube axis. In terms of ψL,R (x) the

effective mass Hamiltonian is H = i~vF σ z ∂x , where vF is the Fermi velocity

and σ z is a Pauli matrix acting in the space of left and right movers. The eigenstates of a finite tube of length L depend on the reflection of waves

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from each end of the tube. Since there are two bands, each end is characterized by a two by two unitary matrix rˆ [16], which gives a boundary condition relating ψL and ψR as indicated in Fig. 1(b). Time reversal symmetry dictates that ˆr is

ψ0 eiqx and solving the eigenvalue problem,

(2)

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ψ0 = ψ0 rˆR rˆL e2iqL .

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symmetric. The eigenstates are determined by postulating ψR (x) = ψL (x)∗ =

The eigenvalues eiδγ of rˆR rˆL (δγ are two real numbers, γ = 1, 2) determine

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the energies of the discrete states, Eγm = ~vF qγm with qγm = (2πm − δγ )/2L. Thus, there are two distinct families of standing waves indexed by γ. The level

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spacing within a family is ΔE = π~vF /L, while the spacing between families is ~vF (δ1 − δ2 )/2L. For a 30nm tube ΔE ' 60meV . Eq. 2 defines the eigenvector

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ψ0 up to a phase which is specified by the boundary condition ψ0 = ψ0∗ rˆL . The

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eigenstates can thus be written in the general form

(3)

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γm γm ψR (x) = ψL (x)∗ = αγ , βγ eiqγm x ,

where αγ and βγ are complex numbers normalized by |αγ |2 +|βγ |2 = 1. Provided

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the reflection matrices vary slowly with energy on the scale of ΔE, αγ and βγ will not vary with m, and the two families of wavefunctions will be repeated in pairs, Fig. 1(c). Since the reflection matrices are unitary and symmetric, they each are parameterized by three real numbers. In the absence of specific knowledge about 5

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the caps, a nanotube is thus characterized by 6 unknown parameters. These determine δγ (which give the energies of the two families of states) as well as αγ and βγ - a total of 8 numbers. These are constrained by two real equations due

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to the orthogonality of eigenvectors: α1∗ α2 + β1∗ β2 = 0. Thus it is impossible to predict anything about the form of ψ0 for a given state - reflection matrices could be found to yield any value of α and β. However, due to the orthogonality

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of the eigenstates, the wavefunctions of the two families are correlated. This leads to non-trivial correlations in the images of the states.

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To visualize these wavefunctions, we consider a two dimensional projection of the wavefunction onto a cylinder a height h above the surface of the nanotube.

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The position on the surface of this cylinder is denoted by the two dimensional vector r. Due to the combined translational and rotational symmetries of the nanotube, the projected Bloch wavefunctions u±K,L,R (r, h) may be written as

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a sum of plane waves with wavevectors ±K + G, where G is a reciprocal lattice

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vector of the graphene lattice. Fourier components of the wavefunction decay exponentially away from the surface of the tube with an exponent proportional

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to the magnitude of their wavevector[13, 17]. Hence far from the surface the wavefunction is dominated by its lowest Fourier components: the lowest star of

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K shown in Fig. 1(d). While the full wavefunctions u±K,L,R (r) depends on the details of the atomic potential, the three Fourier components in the lowest star are determined up to a real multiplicative factor by symmetry. They can be

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determined within a tight binding calculation. We thus write

Ψγm (r, h) ∝

1 X

iKp ∙r φγm + c.c. p (x)e

(4)

p=−1

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where Kp are the Fermi point wavevectors indicated in Fig. 1(d). Using Eqs. (1,3) and the tight binding Bloch wavefunctions we find

(5)

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∗ iqγm x φγm cos ωp + iβγ e−iqγm x sin ωp , p (x) = αγ e

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where ωp = θ/2 − 2πp/3, and θ is the chiral angle. We have defined r = 0 to be the center of a hexagon.

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In Fig. 2, we display images of the density n(r) = |Ψ(r, h)|2 . Apparent in this figure is the pattern of nodal lines, which pass through the center of every

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hexagon. Fig. 2(a,b) shows density maps of the two states for the special case

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of an armchair tube with ideal high symmetry caps as previously considered by Rubio et al.[11]. In this case the reflection matrices ˆrL,R are diagonal, and the eigenstates can be indexed as bonding or antibonding states according to

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their symmetry under reflection through a mirror plane. If the tube ends are not ideal, or the nanotube is chiral, there is no symmetry to prevent mixing between

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the two bands. This has a strong effect on the density map. Fig. 2(c,d) shows the two states of an armchair tube in the presence of mixing, while Fig. 2(e,f) show the states of a chiral tube. In Fig. 2(a-f) the plots are for a 30nm tube with m = 0, the phase exp(iqγm x) does not vary significantly over the image window since the length of the window lw  1/q. However along the tube the 7

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image evolves predictably on length scales ∼ 1/q according to (5). Fig. 2(g) is a longer image window for a 30nm tube with m = 6(corresponding to E ' 0.3eV as might be expected experimentally [6]). The change in phase over the length

is clear.

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of the window is about π, and the corresponding evolution of the wavefunction

To determine the constraints on the density maps, we project the lowest

nγm (r) =

1 X

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Fourier components of the density and write

iKp ∙r nγm + c.c. p e

(6)

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p=−1

The Fourier components of the density for each state are related to the wavefunc-

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tion via np = (φp+1 φp−1 )∗ , where we treat the indices p cyclically mod 3. This is easily inverted, φp = [(n−1 n0 n1 )1/2 /np ]∗ . Thus, the lowest Fourier compo-

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nents of the density contain all of the universal information about the structure

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of the wavefunctions. The three Fourier components obey

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p

φγm = 0. This p

“nodal constraint” reflects the fact that each Bloch state at the K point has a node at the center of every hexagon. In addition, there are two “orthogonality

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constraints”, derived by solving (5) for αγ ,βγ ,

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P

p,q

h i 1m∗ 2m0 Re φ φ = 0, p p p

(7)

h i 0 Im φ1m∗ φ2m p q−p cos (θ − 2πq/3) = 0.

Note the dependence on θ in (7). The correlation between the images depends on the chiral angle. 8

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Figure 2: Density maps of the two families of electronic states on finite tubes for (a,b) an armchair tube with no interband mixing, (c,d) with mixing, and (e,f) a chiral tube. (g) shows the evolution of the density along the tube. Black dots represent the underlying lattice.

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Measurement of nγm p would yield 3 complex numbers for each state. Since the overall magnitude is unknown, this leaves 5 measured real parameters for each state. The nodal constraint,

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p

1/nγm = 0 removes two, and the remaining p

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γm three variables determine the wavefunction ψR (x). Note that measuring a

single state gives no constraint on nγm p . The nodal constraint serves only to identify the center of the hexagons. Measuring both families of states, however,

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leads to falsifiable predictions: the centers of the hexagons determined by the nodal constraints must coincide, and the wavefunctions must be orthogonal.

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Moreover, as the phase eiqx advances along the tube, the images evolve in a predictable way.

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3. STM tip effects

In practice, it is to be expected that measurement of n(r) will be compli-

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cated in two ways: (1) the geometrical distortion due to tip tunneling onto a

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cylindrical surface; (2) artifacts arising from tip asymmetries. The effect of (1) was previously discussed by Venema et al.[18]. (2) is crucial in systems where

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the image is dominated by interference between a small number of standing waves. STM images of graphite are dominated by three standing waves with

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the period of the unit cell. While this leads to enhanced contrast in the images, Mizes et al.[13] argued that asymmetry due to multiple contacts between the tip and sample alters the phases and amplitudes of the three standing waves. The resulting images do not resemble the actual density and lack the threefold symmetry of the graphite lattice. Thus, an asymmetric tip makes it impossible

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to measure the Fourier components nγm characterizing a given state. Nonethep less, by exploiting the constraints outlined above, we can partially reconstruct the wavefunctions for both families of states.

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In their original theory of the STM, Tersoff and Hamann [19, 20] considered an ideal, spherical tip. They showed that the differential conductance is proportional to the local density of states evaluated at the center of curvature

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of the tip. To address the effect of tip asymmetry we consider a generalization of the Tersoff-Hamann theory. Suppose the STM tip is at a position R

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and height h above the surface. Following Ref. [20], we compute the tunneling matrix elements between sample and tip states by evaluating the Bardeen for-

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mula on an arbitrary surface S lying in the vacuum region between the tip and the sample. This leads to a general expression for the differential conductance dI/dV = G(R) of the form,

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drdr0 f (r, r0 )ρ(R + r, R + r0 , E),

(8)

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G(R) =

where ρ(r, r0 , E) =

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n

S

Ψn (r)∗ Ψn (r0 )δ(E − En ) is the nonlocal density matrix.

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The tip convolution function f (r, r0 ) contains all information about the tip. For the spherical tip of radius R considered in Ref. [20] f (r, r0 ) extends over a range

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R + h, but due to its special analytic form integration over S “reconstructs” the local density at the center of curvature of the tip. For a double tip, f (r, r0 ) is

the sum of two terms. For an arbitrary asymmetric tip the convolution function could be very com-

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Figure 3: STM images of Fig. 2(a) projected into the lowest star taken with (a) an ideal tip and (b) an asymmetric tip.

plicated. However, the problem at hand is simplified by the fact that the density

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is dominated by just three standing waves which “pick” a few Fourier compo-

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nents of f (r, r0 ). When biased to energy Eγm , the differential conductance at constant height is given by,

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G

γm

(R) =

1 X

iKp ∙R fp nγm + c.c. + ... p e

(9)

p=−1

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where fp = f (−Kp+1 , −Kp−1 ) and nγm are the Fourier coefficients of the denp

sity. Hence extracting the Fourier coefficients from a real image gives not nγm p but gpγm = fp nγm p . Figure 3 compares images projected into the lowest star taken

with an ideal tip (fp = 1) and an arbitrary “non-ideal” tip. For images in the lowest star, an imperfect tip is described by three com-

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plex numbers fp . Extracting an overall phase (ζ0 ) and magnitude, we write fp = |f0 |eiζ0 fˉp , where fˉ0 ≡ 1. |f0 | affects only the overall magnitude of the current, and the phases of fˉ±1 reflect the arbitrariness of the origin of R. This

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leaves three intrinsic parameters characterizing the tip: |fˉ±1 |, which describe the rotational asymmetry, and the phase ζ0 .

To interpret an STM image in terms of the wavefunctions of the eigenstates, the tip must be calibrated. This may be partially accomplished by imposing the P1

fˉp /gpγm = 0. For an individual

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nodal constraints, which take the form

p=−1

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state this represents two equations in four unknowns and cannot be solved, but by using two eigenstates, fˉ±1 can be determined. The tip calibration holds

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provided the height and orientation relative to the tube are both constant, hence images taken at different places on the same tube should have the same tip calibration. The phase ζ0 can not be determined.

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Knowledge of fˉ±1 allows the determination of the Fourier components of the

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density, nγm p , up to an overall phase which depends on the unknown ζ0 . Without knowing ζ0 , the true density nγm (r) can not be fully recovered. Nonetheless, ζ0

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does not affect the orthogonality relations in Eq. (7), and they yield two non trivial constraints relating the real images of the states. Moreover, the evo-

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lution of the images as the phase exp(iqγm x) advances along the tube can be predicted using Eq. (5,6). The wavefunctions (αγ , βγ ) can be partially recon-

1m structed given our knowledge of nγm p . Using the parameterization ψR (x) =

√ √ 2m ( 1 − λ2 ei(ξ1m +η) , λei(ξ1m −η) ), ψR (x) = (−λei(ξ2m +η) , 1 − λ2 ei(ξ2m −η) ), it is possible with (5) to extract ξγm and λ from the images. The phases ξγm ad-

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vance with qγm x, while λ describes the mixing between the two bands due to reflection. The relative phase η is inextricable without knowledge of ζ0 .

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4. Conclusion In conclusion, the low energy discrete eigenstates of finite nanotubes belong to two distinct families. The densities of these two families are correlated by

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the orthogonality of their wavefunctions, and within each family the densities differ only due to the phase of the wavefunction. The density of a single state

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evolves predictably along the tube with the phase of the wavefunction at a rate proportional to the energy of the state relative to the K points. This may pro-

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vide a possible mechanism for a quantitative measure of doping. STM images of graphite and nanotubes are particularly susceptible to artifacts due to asymmetries in the tip which complicates interpretation of the images. However, in

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this special case, the symmetry of the honeycomb lattice, and the presence of

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multiple distinguishable states, can be used to retrieve five of the six parameters characterizing the two families of states from the STM images. In the process,

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partial knowledge is gained about the STM tip, which would probe the validity of the conventional assumption of a spherically symmetric tip. Therefore,

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successfully measuring 2D STM maps of discrete states on finite tubes would not only provide an elegant example of elementary quantum mechanics on a mesoscopic scale, but could also allow investigation of doping, band mixing at tube ends, and STM tips.

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Acknowledgment The author would like to acknowledge, with gratitude, the invaluable insights of Charles L. Kane throughout this work, and to express his thanks to Neil

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Wilson, University of Warwick for the fruitful discussions.

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Graphical abstract

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Highlights • Low energy discrete eigenstates of finite metallic nanotubes belong to two distinct families.

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• Densities of the families are correlated by the orthogonality of their wavefunctions.

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• STM Images of these families can be used to parameterize the STM tip..

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Graphics Abstract

Figure 1

Figure 2

Figure 3