Nodal adsorbate bound states in armchair graphene nanoribbons: Fano resonances and adsorbate
recognition in weak disorder
C. H. Chiu and C. S. Chu
Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan
(Received 18 September 2014; revised manuscript received 4 November 2014; published 21 November 2014) We consider adsorbates on metallic armchair-graphene nanoribbons (AGNRs). An adsorbed atom on a nodal site of the gapless subband in the AGNR is found to induce a bound state. The nodal-adsorbate bound state alone does not cause reflection in the AGNR quantum transport in the one propagating-channel regime. Yet its manifestation, as a Fano resonance in G, is brought forth by the presence of a non-nodal adsorbate (or a scatterer) located on a longitudinally separated site. Both the adsorbate-dependent nature and the significantly enhanced energy resolution of the Fano structure in G invite adsorbate recognition. The Fano peak is shown to be robust against weak disorder such as from edge vacancies, due to the localized nature of the Fano resonant state. Adsorbates F and OH, described within an extended H¨uckel model, are considered as examples.
DOI:10.1103/PhysRevB.90.195436 PACS number(s): 72.80.Vp, 73.22.Pr, 73.20.Hb, 73.63.−b I. INTRODUCTION
The novel Dirac properties of graphene [1,2] has prompted much recent research on the quantum transport of graphene nanoribbons (GNRs), both experimentally [3–9] and theoret-ically [10–23]. One of the spectacular manifestations of the Dirac physics is in the metallic armchair GNRs (AGNRs), where a subband intercepts the Dirac point to become gapless [24]. Disorder effects on the GNR transport have been studied, with disorder due to edge vacancies [11,12,16], edge adsorbates [19], and adsorbates [20,21]. On the other hand, the motivation to tune the GNR energy gap from a microscopic point of view has led to researches on the effects of individual microscopic objects on the GNR. There have been studies of individual doping atoms [13], lattice defects [14,15], vacancies [17,18,22], and adsorbates [19,23] on the quantum transport of GNRs. These single-scatterer studies have found dip structures in G [13–15,17,19,22], whose dip energies are associated with the resonant-state energies due to the scatterer in the GNRs.
Interestingly, for the case of AGNRs, a single dopant [13] or a singe vacancy [17] are found to cause no reflection to the subband when they are located at certain sites of the AGNR. These sites are found to coincide with the nodal sites of the gapless subband in the AGNR [25]. The nodal sites are at transverse coordinate (see Fig.1) M= 3m − 1 for integer m. This is obtained from the transverse profile of the gapless-subband wave function, given by sin[knxa(M+ 1)]
(a complete form of the wave function will be presented in Sec.II AII), and from the quantized knx= 2π/(3a) where the
Dirac cone is intercepted. Further nodal-type behaviors are found in the AGNR current density profile [14,23], also at the nodal sites of the AGNR. There is no energy in these studies from which one can derive deeper insights about the nodal behavior. Thus far, the nodal behavior has evaded attentions. In this work, we will show that there is indeed a characteristic energy, namely, the nodal-adsorbate bound-state energy, for an adsorbate on a metallic AGNR. The intriguing features of this bound state are explored. Furthermore, its adsorbate-dependent energy is shown appropriate for adsor-bate recognition.
Using graphene for adsorbate recognition has been a goal of interest recently. An experimental observation of single gas-molecule adsorption/desorption events had been done on
graphene [26]. The basic quanta of charge transfer at the adsorbate site is obtained from the step changes in the Hall resistivity. This charge quanta is used for adsorbate recognition by comparing it with the theoretical calculation results for a number of target adsorbates [27]. Research on the graphene-based gas senors has since been intensive, in both the basic physical aspects [28,29], and the implementation aspects [30]. An alternate adsorbate-recognition scheme of single-adsorbate capability has been proposed recently [31]. This is by way of an optical measurement and with the introduction of a localized state called the localized plasmon resonance (LPR) [32], a collective mode on a patterned two-dimensional nanostructure. Incident lights will undergo diffractive coupling [33] with the LPR to give zero reflection at a wavelength λc for a
given incident angle [31]. At the zero reflection, the phase of the complex reflection coefficient encounters a singularity, meaning that it is highly sensitive and changes sharply with
λ≈ λc[34]. This sharp-change nature in the phase allows a
larger phase change to be measured when λcis shifted minutely
due to the presence of an adsorbate [31].
In this work, we consider quantum transport in metallic AGNR with nodal adsorbates. Our purpose is threefold here. First, we want to show that the nodal sites of the gapless subband have provided us a natural way to obtain localized states. Simply put, the localized state is a bound state induced by an adsorbate located on a nodal site of the gapless subband. As the localized state now carries information about the adsorbate, it plays an even more involved role for the adsorbate recognition here than the LPR states for the optical experiment above. Second, we want to show that the adsorbate bound state provides a significant boost in the energy resolution for the adsorbate recognition. The bound state will manifest as a Fano structure [35] in G, when another adsorbate is located on a longitudinally (along the ribbon) separated non-nodal site of the gapless subband. This Fano structure is robust against weak disorder from random edge vacancies. Third, quantum transport for mesoscopic systems is known to be sensitive to individual scatterers [36] and should thus be so to individual adsorbates.
This paper is organized as follows. In Sec.II, we present our theoretical method for the analysis of the nodal-adsorbate bound state, the nodal-and-non-nodal adsorbate pair, and the
Green’s function for disordered AGNRs. Numerical results and discussions are presented in Sec.III. A conclusion is given in Sec.IV.
II. THEORY
In this section, we consider adsorbate-induced bound and resonant states in metallic AGNRs. Interplay between these states gives rise to adsorbate-dependent Fano resonance in G. Our focus is upon the one-propagating-channel (the gapless subband) regime. The energy of our interest is indicated in Fig.1by the G= 2e2/ hregion for a pristine AGNR (dotted
curve). This energy range, unless explicitly stated otherwise, is the energy for all our following discussions.
A. Nodal-adsorbate bound state
We show here that an adsorbate located on a nodal site of the gapless subband induces a bound state. For W -AGNR of large width W (the number of rows of carbon atoms across the AGNR) the bound-state energy is pinned at the band edge of the AGNR first-gapped subband (FGSB), and is adsorbate independent. For sufficiently narrow W -AGNRs, the bound-state energy moves away from the FGSB’s band edge, and becomes adsorbate dependent. Figure1 is shown first to put things into perspective. A bound state alone does not contribute to the scattering, as is shown in Fig.1 (the A1-only curve). On the other hand, an adsorbate located on a non-nodal site of the gapless subband induces a resonant state [14,20–22], which alone gives rise to a broad dip structure in G, as is shown in Fig.1(the A2-only curve). In the presence of both A1 and A2, a Fano structure is found in G, as is denoted by the black curve in Fig.1. The Fano peak coincides nicely with the adsorbate-dependent bound-state energy.
The physical picture for the above finding follows naturally from the Lippmann-Schwinger scattering description of the AGNR. The total wave function , projected onto lattice site
FIG. 1. (Color online) Fano resonances in conductance G of an AGNR with an adsorbate pair A1 and A2. A1 (A2) is a nodal (non-nodal) adsorbate. G’s are shown for a pristine AGNR (dotted curve), individual A1 (red curve), A2 (grey dashed curve), and a pair (black curve). Inset is the AGNR (width W= 20, along ˆx, and length along
ˆ
y) with the scatterers separated by L= 5 unit cells along ˆy. Also indicated are Bravais lattice vectors A1(blue arrows), A2(red arrows),
transverse site-position M, and M-spacing a. The adsorbates are F atoms. |j,s, given by js= φsj+ i Gssi jji Vji si ji, (1)
is expressed in terms of the lattice-site representation of φ, ˆ
G, and ˆV, which are the incident wave function, the AGNR Green’s function, and the scatterer on-site potentials, respec-tively. The lattice sites|j,s denote the unit-cell coordinates (Mj,Nj), and the sites s∈ {A,B} within a unit cell. The
Bravais lattice point is thus Rj = MjA1+ NjA2, with the
Bravais lattice vectors given by A1= a( ˆx −
√
3 ˆy) and A2 =
2√3a ˆy. For an AGNR of width W , we choose 0 Mj W− 1. The horizontal distance between two neighboring rows
of carbon atoms is a= 1.23 ˚A. In this work, we consider
W-AGNRs with W = 3p + 2 for integer p, which support a gapless subband [24,37,38], when the quantized kxintercepts
the Dirac point.
The standard nearest-neighbor tight-binding model is used for the AGNR, and the adsorbates, located on sites|ji,si,
are represented by on-site potentials Vji. The on-site potential
Vji = Vaof the adsorbate on the site, whose specific form is
derived from an extended H¨uckel model [20]. The extended H¨uckel model, as described in Ref. [20], has included the extended molecular orbitals α that are formed from the adsorbate-valence orbitals and the carbon atom-valence orbitals (pz excluded) to which the adsorbate is attached. It
results in an energy-dependent on-site potential Va, of the form
Va= t α |γα|2 E− εα , (2)
where t= 2.66 eV is the hopping coefficient of graphene. The values of the adsorbate parameters γα and εα have been
obtained by Ihnatsenka et al. [20]. For completeness, we list the parameters for adsorbates F and OH in TableI
For the case of a single nodal adsorbate, the condition for the induction of a bound state is derived from Eq. (1), by setting
φ= 0, to give 1− Gs1s1 j1j1(E) Va s1 j1 = 0. (3)
That the bound state can be found is facilitated by Gs1s1 j1j1(E)
becoming a real function of E when j1 is a nodal site of the
gapless subband. This is clearly shown in a subband-index (n) summation form of Gss
jj(E).
From Gssjj(E)= js| ˆG(E)|js, where ˆG(E) = (E −
ˆ
H+ iη)−1 is the retarded Green’s function, and Hˆ the
TABLE I. Parameters for adsorbate F and OH obtained by Ihnatsenka et al. in Ref. [20].
Adsorbate α(t) γα(t) F −10.862 4.363 −2.460 1.645 −0.914 1.180 OH −8.536 3.203 −1.802 1.779 −0.709 1.540
Hamiltonian of a pristine AGNR, we have Gssjj(E)= kn, ν js|kn,ν 1 E− νEkn+ iη kn,ν|js, (4)
of which νEkn is the graphene dispersion relation, ν= ±1, kn= (knx,ky), and η is a positive quantity of infinitesimal
magnitude. The subband wave function is given by js|kn, ν = 2 (W+ 1)Ny eiky √ 3a(2Nj−Mj) × sinknxa(Mj + 1) Cks nν, (5)
where Ny denotes the total number of unit cells
along the AGNR, ( CA
knν, C B knν) T = √1 2(1,νH ∗ kn/|Hkn|) T is
the normalized pseudospinor [39], with Hkn = −t [1 +
2e−i√3kyacos(knxa)], and Ek= |H(k)|.
Simplifying Eq. (4) leads to a useful subband-index summation form of Gss jj, given by Gssjj(E)= √ 3a 2π (W + 1) n Fjjngnjj,ss(E), (6)
where gjj,ssn (E) is the reduced Green’s function, gjjn,ss(E)= ν π/√3a −π/√3a dky eiky √ 3a Ljj Cs∗ knνC s knν E− νEkn+ iη . (7) The factor in the summation of Eq. (6),
Fjjn = 4 sin[knx(Mj + 1)a] sin[knx(Mj+ 1)a], (8)
is associated with the nth subband’s transverse profile, and is zero when j or j are at a nodal site. In Eq. (7), Ljj =
(Rjy− Rjy)/
√
3a is the longitudinal distance between the Bravais lattice points j and j. The subband index ranges 1 n W/2 such that knx= a(Wπ+1)( W/2 + n), where
x ( x) denotes the largest (smallest) integer not greater (less) than x.
The same-site reduced Green’s function gn
jj,ss(E) is a real
(complex) function of E when E lies outside (inside) of the
nth subband. This is seen from applying the Cauchy principal-value analysis to Eq. (7). Consequently, within our energy range of interest, a nodal adsorbate will shunt the only term of
Gss
jj(E) in Eq. (6) that is complex—the term from the gapless
subband. The nodal-bound state is hence orthogonal to the gapless subband, and does not cause reflection to the gapless subband, leaving G= 2e2/ hin our energy of interest. Its wave function is given by
js = VaGssjj11 s1
j1, (9)
with the energy fixed at the bound-state energy EB, and j1s1
fixed by the normalization.
A non-nodal adsorbate, following a similar argument, will have retained in Gss
jj(E) the term from the gapless subband
such that Gss
jj(E) becomes complex. The non-nodal adsorbate
cannot then form a bound state, since the condition Eq. (3) does not hold, but forms a resonant state instead. The resonant condition is given by 1− VaRe Gs1s1 j1j1(E) = 0. (10)
B. N Nodal adsorbates and one non-nodal adsorbate
Coherent interplay between the bound and resonant states gives rise to a Fano resonance in G. Taking the point of view that a non-nodal adsorbate plays the role of revealing the adsorbate-induced bound states in the AGNR, we consider in this subsection first a nodal and a non-nodal adsorbate pair. Then we consider N nodal adsorbates and one non-nodal adsorbate.
For an adsorbate pair consisting of a nodal (A1) and a non-nodal (A2) adsorbate, the Fano peak in G occurs exactly at the bound-state energy, and with G= 2e2/ h. This is most
easily seen from projecting Eq. (1) onto the lattice sites of A1 and A2, to give
1− VaGsj1j11s1 −VaG s1s2 j1j2 −VaGs2s1j2j1 1− VaG s2s2 j2j2 js1 1 js2 2 = φs1 j1 φs2 j2 . (11) The φs1
j1 = 0 for the incident wave function at A1 leads to the
important result
s2
j2 = 0, (12)
when the incident energy is at the A1’s bound-state energy, where Eq. (2) holds. That the total scattering wave function
j2s2vanishes (note: not the total scattering wave function j1s1 at the site of A1) is a result of the no-damping nature of the A1-induced bound state. The total wave function js, according to
Eq. (1), becomes
js = φjs+ VaGssjj11 s1
j1. (13)
The second term (scattered wave) in Eq. (13) is shown, in the following, to be evanescent waves, and does not contribute to the current. Thus we have perfect transmission (G= 2e2/ h)
at the Fano peak whose energy is at EB.
The reduced Green’s function gjjn,ss(E) is calculated by
both numerical and analytical methods. Excellent consistent results are obtained. Analytical results shed light on our un-derstanding of the physics. The characteristics of the reduced Green’s function are best illustrated by the analytic expression (s= s case) we obtained below via explicit calculations of the integral in Eq. (7) [40]. The calculation is facilitated by the canceling of a branch-cut integral through the ν summation in Eq. (7). We get gjjn,ss(E)= −ξ n E π 4√3 at2 E βn e−ξEn √ 3 jj κnya sinh(√3 κnya) , (14) for a non-flat-band subband, where jj = |Ljj|. For a
flat-band subflat-band (the n= 1 subband when W is odd), we have
g1jj,ss(E)= 2π √ 3 a E E2− t2+ 2iEηδjj. (15) Here κny = 1 √ 3 a ln(n+ 2 n− 1), (16)
where n= βn+ [1 − (E/t)2]/(4βn), and βn= − cos(knxa).
When E lies outside of the nth subband, ξEn= 1 and κnyis
real, we have an evanescent nature in gn
jj,ss(E). When E lies
inside the nth subband, ξn
then gn
jj,ss(E) is of a propagating-wave nature. These features
also lead to our finding that the coupling between the bound and the resonant states, represented by the off-diagonal terms in Eq. (11), decreases with increasing j1j2exponentially.
Remarkably, for configurations that involve N nodal adsor-bates and a non-nodal adsorbate, the Fano-peak conductances still manage to maintain their G= 2e2/ h value. Following
our analysis in Eq. (11) and Eq. (13), but for N nodal adsorbates, the vanishing of the total scattering wave function
N+1 ≡ sN+1
jN+1 = 0 at the (N + 1)th adsorbate, that is, the
non-nodal adsorbate, can be achieved. It is given by the condition
det Q(E)= 0. (17)
Matrix elements of Q are given by
Qii =
1− VaGii, i= i,
−VaGii, i= i. (18)
Here subscripts i and idenote the nodal adsorbates. To keep notations simple, we define Gii ≡ Gsjisi
iji.
The useful relation Gii = G∗ii is obtained from the
trans-formation properties of E(k) and Cs
knνwhen ky changes sign.
That Qii is, in fact, real results from CkAnμbeing real. These
properties of Q lead to a real equation in Eq. (17); the energies it gives correspond to the Fano peak energies, and should also correspond to the bound-state energies of the N nodal-adsorbate configurations. The exponential behavior of
Giifor i= ihelps to suppress the energy shift of these Fano
peaks from the adsorbate bound-state energies of the individual nodal adsorbate.
C. Green’s function ˆ˜G for disordered AGNRs
The AGNR Green’s function ˆ˜Gin the presence of vacancies is given by the relation
ˆ˜
G= ˆG + ˆG ˆV ˆ˜G = ˆG + V0G ˆˆ IvG,ˆ˜ (19)
where ˆGis the Green’s function for the pristine AGNR. The vacancy potential ˆV ≡ V0 Iˆv= V0
i |i i|, where ˆIvis the
projection operator into {|i}, the subspace spanned by the vacancy lattice sites. Here V0is the vacancy on-site potential,
which will be set to infinity at the end of the derivation. The derivation in this subsection becomes that for edge vacancies when their lattice sites are at the edge sites of the AGNR.
Applying the projection operator ˆIv from the left to
Eq. (19), we get ˆ
IvGˆ˜ = ˆIvGˆ + V0IˆvG ˆˆ Iv G.ˆ˜
After rearranging terms, we obtain ˆ
Iv Gˆ˜ = [ ˆIv− V0Iˆv G ˆˆ Iv]−1IˆvG.ˆ (20)
We substitute Eq. (20) into Eq. (19) to get ˆ˜
G= ˆG + V0G ˆˆ Iv[ ˆIv− V0IˆvG ˆˆ Iv]−1IˆvG.ˆ (21)
Finally, taking the V0→ ∞ limit, we arrive at
ˆ˜
G= ˆG − ˆG ˆIv[ ˆIvG ˆˆ Iv]−1IˆvG.ˆ (22)
The expression for the one-vacancy case of Eq. (22) is the same as in Ref. [41].
Our results also check with the expectation that the eigenstates ˜φin the presence of the vacancies should vanish at their respective lattice sites. This expected condition ˆIvφ˜ =
˜
φ†Iˆv= 0 is satisfied by applying ˆIvto Eq. (22) to get
ˆ
Iv Gˆ˜ = ˆ˜G ˆIv= 0.
The eigenstates ˜φand their counterparts φ for the pristine AGNR are connected by the relation ˜φ= φ + ˆG V0Iˆvφ˜. The
relation is reexpressed in the form ˜
φ= φ − ˆG ˆIv[ ˆIvG ˆˆ Iv]−1 Iˆvφ. (23)
Building on the above relations for the treatment of the edge vacancies, we consider the case of an adsorbate pair with the edge vacancies in the background. The adsorbate pair consists of a nodal (A1) and a non-nodal (A2) adsorbates. The total scattering wave function ˜ resulted from an incident wave ˜φ
is given by ˜ js = ˜φjs+ VaG˜ ss1 jj1˜ s1 j1+ VaG˜ ss2 jj2˜ s2 j2. (24)
Projecting ˜at the adsorbate lattice sites, we obtain a similar equation as in Eq. (11), given by
1− VaG˜s1s1j1j1 −Va ˜ Gs1s2j 1j2 −VaG˜js22sj11 1− VaG˜ s2s2 j2j2 ˜ js1 1 ˜ s2 j2 = ˜ φjs1 1 ˜ φs2 j2 . (25) In the weak disorder regime, when the density of the edge vacancy is low, the nodal physics and its evanescent effects will play a crucial role. The evanescent effects are exerted via
G1i or Gi1, where i denotes the edge-vacancy lattice sites.
In the following, we will trace these evanescent effects in several physical quantities of interest for the elucidation of the evanescent effects on the Fano structures.
The incident wave function ˜φs1
1 is found to take on the
form ˜φs1
1 = δ ˜φ evan,s1
1 from projecting Eq. (23) to the nodal site
A1. The superscript in δ ˜φevan,s1
1 indicates that the term is of
evanescent nature (between A1 and the edge vacancies) and is small when evanescent suppression prevails. Since φ1s1= 0, the G1i factor in the scattering term of Eq. (23) brings about
the evanescent effects to ˜φs1
1.
For ˆ˜G, or its matrix element ˜Gj1, the second term in Eq. (22)
contains Gi1such that ˜Gj1should be cast in the form
˜
Gj1= Gj1+ δ ˜Gevanj1 , (26)
where j denotes a general lattice site. Furthermore, for the
j = 1 case, the second term of Eq. (26) is of even higher order
in the evanescent effects. Finally, for the Fano structure, we consider Eq. (25). At around the Fano peak energy EB, we
have the equation −Vaδ ˜Gevan11 ˜
s1
1 − VaG˜12˜2s2 = δ ˜φ evan,s1
1 , (27)
from which we conclude for ˜2s2 the form ˜2s2= δ ˜evan,s2
2 .
Putting all these together, the total scattering wave function in Eq. (24) becomes ˜ ls = ˜φls+ Va Gl1+ δ ˜Gevanl1 ˜ s1 1 + VaG˜l2δ ˜2evan,s2. (28)
The term involving Gl1 is an evanescent wave and does
not contribute to G. The last two terms in Eq. (28) consist of the propagating channel but suffer suppression from the evanescent effects. Thus, in the vicinity of EB, the conductance
Gis basically that determined by ˜φ. In other words, the Fano peak will rise to the conductance as is determined by the background edge vacancies. The shift in the Fano-peak energy is also small, being protected by the evanescent effects. These features are confirmed in our numerical results.
III. RESULTS
Numerical examples presented in this section aim at the illustration of three major points. The first point is the nodal-adsorbate bound states. The second is the adsorbate recognition from using the bound states. And the third is the robustness of the adsorbate recognition against weak disorder.
A. Nodal-adsorbate bound states
The nodal-adsorbate-induced bound state in metallic AG-NRs is presented in this subsection. Taking a 20-AGNR as an example, the nodal sites are at M = 2, 5, and 8, which are equivalent to nodal sites at M= 17, 14, and 11, respectively. For the case when the adsorbate is F, their bound-state energies are−0.209t, −0.171t, and −0.159t, corresponding sequentially to the nodal sites listed earlier. As a reference, the first-gapped subband band edge is at−0.25t.
Key features we present include the nodal-adsorbate bound-state wave functions, their Fano structures in G by a neighboring non-nodal adsorbate, the localized (or evanescent) nature of the bound state, and the bound states that are formed from a few nodal adsorbates.
Figure2presents (a) the wave function and (b) the||2.
Equation (9) is used for this purpose. The adsorbates F are located on nodal sites M= 17 (a); 5 (b); and 8 (c). The fact that
is real, according to the discussion in the paragraph just after Eq. (18), allows us to make the plot as it is. The positions of the nodal adsorbate are at the centers of the yellow triangular structures in the contours. Similar triangular structures have been obtained, but in the local density of states at zero energies around a vacancy in bulk graphene [41,42]. For the sake of our following discussions, we denote the nodal-adsorbate site to be an A site. Its nearest neighboring site along the ribbon direction is the B site in the same unit cell. The||2 contour in Fig.2 (lower figures) shows that||2weights more, in general, on
the B site. This trend is in agreement with that for a vacancy [41]. The very large on-site potential V0 for the vacancy has
pushed the wave function of its zero-energy state to the B (A) site entirely when the vacancy is at an A (B) site.
Another trend of interest that holds better for the more central-nodal adsorbate (M= 8) is related to the asymmetric, or symmetric, weighting of the||2 for the A, B sites in a
unit cell. This weighting feature depends on the longitudinal displacement of the unit cell of interest from the nodal adsorbate. In Fig. 2, asymmetric (symmetric) weighting of ||2 occurs when the unit cell is on the smaller (larger)
y-coordinate side of the nodal adsorbate. The origin of this is that||2for A sites exhibit a π/3 symmetry (with a y → −y
symmetry) while that for B sites exhibit a 2π/3 symmetry (without a y→ −y symmetry). The A-, B-site correlations in the above conclusion will switch accordingly if the nodal adsorbate is switched to a B site. It is because of the A, B site symmetry in the AGNR. When the nodal adsorbate is close to the AGNR edge (M= 17), the second trend is greatly disrupted. Similarly, looking at the central-nodal adsorbate in the -contour plot, Fig.2(c)(upper figure), we see that the three lines of A-B sites leading outward from
FIG. 2. (Color online) Nodal-adsorbate bound-state wave functions (upper figures) and||2(lower figures). Adsorbates at nodal sites
M= 17, 5, and 8 are presented in (a)–(c), respectively. The adsorbate is F. The sites in (b) marked by grey (red) open symbols are B sites to be attached upon by non-nodal adsorbates in Fig.3. For display purpose, s-like functions are used in place of the site orbitals.
FIG. 3. (Color online) A-, B-site symmetry (asymmetry) in Fano structures for adsorbate pairs. Adsorbate pair in Fig.2(b) with the non-nodal adsorbate located on a site of the A-B unit cell denoted by (a) grey (red) open square, and (b) grey (red) open circle. Solid (black) curves are for non-nodal adsorbate located on the A site, and open (red) symbols are for non-nodal adsorbate located on the B site.
the three vertices of the yellow triangle have exhibiting monotonically decreasing characteristics. On the other hand, the three directions leading outward from the three bases of the yellow triangle have exhibiting oscillatory decaying characteristics. These characteristics should be of interest for exploring interactions between nodal adsorbates.
The asymmetric (symmetric) weighting of||2in a unit cell is reflected in Fig.3, via the Fano structure in G, when a non-nodal adsorbate is in the neighborhood. The non-nodal adsorbate is the one in Fig.2(b). The unit-cell locations of the non-nodal adsorbates for the adsorbate pair are denoted by grey (red) open symbols in Fig.2(b). The symmetric weighting case, Fig.3(a), occurs when the non-nodal adsorbate is located on the larger
y-coordinate side of the nodal adsorbate. It is illustrated by the almost complete overlapping in the Fano structures in Fig.3(a). The solid black curve and the red open symbols denote the case of an A- and B-site non-nodal adsorbate, respectively. On the other hand, the asymmetric weighting case, Fig.3(b), occurs when the non-nodal adsorbate is located on the smaller
y coordinate side of the nodal adsorbate. It is illustrated by the very different Fano-structure widths in Fig.3(b). Again, the solid black curve and the red open symbols denote the case of an A- and B-site non-nodal adsorbate, respectively. Note that both of the dip structures in G experience quite a significant relative energy shift (≈ 0.025t) between the two
G curves. In contrast, that the Fano-peak energy shows no relative energy shift, and the Fano-peak G remains at 2e2/ h
is in excellent accord with our analysis leading up to Eq. (17). Since these two key features of our focus in this work remain intact, regardless of whether the adsorbate pair belongs to AA type or AB type, we will assume only AA-type adsorbate-pair configurations in the following examples for the sake of focusing our presentations on the essentials.
The robustness in the Fano peak is due to the localized nature of the bound state. This is shown in Fig. 4, when the L dependencies of the Fano structure in G and the Fano width are presented in Figs.4(a) and 4(b), respectively. As
FIG. 4. (Color online) Fano structures and Fano widths vs sepa-ration L in an adsorbate pair. (a) Conductance G for L= 10, 5, and 2, in units of 2√3a. (b) Exponential-type dependence of the Fano width εFano, the peak-dip energy difference, on L for non-nodal adsorbate
at M= 4, 6, and 9. Inset (solid curve) is for non-nodal adsorbate at M= 3.
the longitudinal separation L decreases, from L= 10, 5, to 2, the Fano peaks in Fig. 4(a) show broadening peak-and-dip structures, while the Fano-peak energy and G remain unchanged. Here L is in units of 2√3a. A more systematic study is shown in Fig. 4(b), when the L dependence of the Fano width εFano is presented. The Fano-peak width
εFanois defined as the Fano peak-dip energy difference. For
the same nodal adsorbate, we present curves from different non-nodal adsorbates (M= 3,4,6,9). The cases for M = 4, 6, 9 show exponential dependence of εFano on L. The
case for M= 3, shown in the inset of Fig. 4(b), seems to need further exploration below. It turns out, as is shown in Fig. 1 (dotted curve), that the band edge of the first and second gapped subbands contribute comparably to the exponential behavior, because of their proximity in energy to the bound state. Our analysis shows that these two exponential contributions come to a cancellation at a certain L, leading to the seemingly nonexponential dependence on L. An explicit exposition of this understanding is by artificial keeping in ˆG
only the first-gapped subband (the ˜ε1 curve), or only the
second-gapped subband (the ˜ε2 curve), or only both (the
˜ε3 curve), and compare with the full numerical calculation
(full curve), as is nicely presented in Fig.4(b) (inset). The exponential or evanescent nature behind the Fano structures is thus established.
Fano characteristics for adsorbate configurations involving more than one nodal adsorbates are presented in Fig.5. The implication on the bound states is our focus here. Specifically, we consider the N = 3 case, where indicated in the inset are the nodal-adsorbate positions (red solid circles) and the
FIG. 5. (Color online) G for three nodal adsorbates and one non-nodal adsorbate. Lattice sites (M,N ) are (8,6), (5,−3), and (17,−6) for the nodal adsorbates, and (4,0) for the non-nodal adsorbate. All adsorbates are F in (a), and the non-nodal adsorbate is changed to OH in (b).
nodal adsorbate position (black solid circle). The non-nodal adsorbate plays the role of revealing the bound states. Figure5(a)shows that there are three Fano structures. The fact that these Fano peaks all have G= 2e2/ hshows, according
to our analysis following Eq. (17), that the Fano-peak energies correspond to bound states. These bound-state energies are at −0.216t, −0.169t, and −0.158t, which are pretty close to the bound-state energies for individual nodal adsorbate, at M = 17, 5, and 8, respectively, given in the beginning of this subsection. In terms of percentage-energy shifts, they are −3.2%, + 1.2%, and +0.6%. The signs of these percentage-energy shifts are consistent with level repulsion. The M= 17 bound-state level has a greatest percentage-energy shift due to the proximity of the M= 17 to the M = 5 adsorbates. The Fano width of the M= 17 is narrowest because of the farthest distance the associated nodal adsorbate is separated from the non-nodal adsorbate. To illustrate that the results are robust against the type of non-nodal adsorbate, we show in Fig.5(b) the case when the non-nodal adsorbate becomes OH. For refer-ence, we also plot in Fig.5(b)the G (dotted curve) for a single OH. In summary, the exponential nature of the nodal bound states has kept intact the Fano-peak energies and G values.
B. Adsorbate recognition
The key feature we present here is the adsorbate-recognition capability provided by the adsorbate-dependent bound states. For comparison, we use both F and OH as adsorbates. For illustration, we have included three metallic AGNR of different widths.
Presented in Figs.6(d)and6(e)are Fano peaks of adsorbate pairs from adsorbate species F and OH, respectively, and for 20-AGNR. The Fano-peak energy spectrum is clearly distinct for the two adsorbate species. In contrast to the broad dip of the adsorbate resonant state, the Fano profile has significantly enhanced the energy resolution. Adsorbate recognition is
FIG. 6. (Color online) Look-up figures for adsorbate recognition. (a)–(c) are look-up figures for AGNR of W= 17, 20, and 23, respectively. Va curves for adsorbate OH (F) are denoted by solid
left- (right-)pointing triangles with Vareferring to the right-ordinate
scale. 1/Re [G11] curves for W -AGNR at nodal sites MI are denoted
by open symbols. G curves in (d) and (e) denote adsorbate pairs (L, M1, and M2) for F and OH, respectively. Vertical dotted lines in
(a)–(c) are guides to adsorbate recognition for (d).
facilitated by look-up-figures, as in Figs. 6(a)–6(c). Here 1/[G11] (which is also 1/Re [G11] in the energy range of our
interest) intercepts Va to produce the adsorbate bound-state
energies. Each nodal site MI , the I th nodal site counted from a ribbon edge, contributes a 1/Re [G11] curve. These curves, as
shown in Figs.6(a)–6(c)for W = 17, 20, and 23, respectively, serve to identify the adsorbate represented by, for instance, the Fano peaks in Fig.6(d). Guided by the three vertical dotted lines, we see that OH is clearly out of the consideration, and F fits best for the W = 20 case in Fig.6(b). Furthermore, both the W= 23 and 17 cases can be ruled out when the entire Fano-peak spectrum is utilized. Thus our scheme could also provide us information about W . The boasting of the energy resolution by the Fano structures, in contrast to the energy resolution provided by the width of the resonant state, has made possible the above adsorbate recognition scheme.
C. Robustness of our adsorbate recognition against weak disorder
The robustness of the Fano structures as well as the adsorbate recognition capability against weak disorder is illustrated. The disorder is provided by edge vacancies. We have performed the ensemble average of G over disorder
FIG. 7. (Color online) Effects of disorder due to edge vacancies on the Fano resonance of an adsorbate pair from F (a)–(c) and OH (d)–(f). The adsorbates are at M1 and M2 with separation L= 5
along a 20-AGNR. Conductance curves Gp (dashed curves) are for
an adsorbate pair only,GBG (grey curves) are for edge-vacancy
disorder only, andG (solid curves) are for both the adsorbate pair and edge-vacancy disorder. The edge vacancies distribute over L0∼
1 μm with concentration Pv= 0.125%. Average conductance G is
obtained for 100 disorder samples.
configurations. We have also performed the average of G for a given disorder configuration over a small energy interval. Both averages have demonstrated the robustness of the nodal physics found in this work.
Effects of disorder on our Fano-peak results due to random edge vacancies are presented in Fig.7. The edge vacancies are distributed on the AGNR sites along its two edges with an occupancy Pv= 0.125%, and over a length L0∼ 1 μm. An
edge vacancy is described by an on-site potential V0→ ∞.
The ensemble average over 100 edge-vacancy configurations gives usG (solid curves) in the presence of an adsorbate pair, and the background conductanceGBG (grey curves) in the
absence of any adsorbate. The adsorbate pairs in Fig.7are the same as those in Figs6(d)and6(e), where their conductances are plotted again in Fig.7as Gp(dotted curves) for comparison.
Disorder effects of G on an adsorbate pair for F is presented in Figs.7(a)–7(c), and for OH is in Figs.7(d)–7(f).
Figure7shows that our Fano results remain significant in the presence of disorder. Equally remarkable is the robustness of the Fano-peak energies against the disorder. Even though the conductance at the Fano peak is no longer of value 2e2/ h, it
manages to rise to the same level asGBG. This is understood
from our analysis in Eq. (28). The monotonic decreasing of GBG to zero as E approaches the Dirac point (E = 0) is the
symmetric edge-vacancy-induced gap-opening feature [20]. Thus the Fano-peak conductances for F suffer less suppression from disorder than that for OH, due to the farther distance of their Fano-peak energies from the Dirac point. Yet the Fano structures of OH remain significant enough for our purposes here. On the same token, our results should hold for the case of hydrogen-passivated AGNR, due to its small energy-gap opening effect on the metallic AGNR (|E| 0.02t
FIG. 8. (Color online) Same plot as in Fig. 7 except that the number of edge-vacancy configurations is 3. The adsorbates are at M1and M2with separation L= 5 along a 20-AGNR. Adsorbate pair
is from F (a)–(c) and OH (d)–(f). Conductance curves Gp (dashed
curves) are for an adsorbate pair only, and GBG (grey curves)
are for edge-vacancy disorder only (100 configurations average, Pv= 0.125% over 1 μm). Conductance GEis averaged over both
an energy interval E and over three edge-vacancy configurations, where E= 2.5 × 10−3t.
for W = 20) [43,44]. The recent atomically precise bottom-up fabrication of AGNR [45–47] shows an encouraging trend of achieving low edge-disorder W -AGNR in the near future. The recent development of the STM tip-assisted nanopatterning [48] would provide the exciting adsorbate-maneuvering capa-bility for the experimental implementation of this study.
The ensemble averagedG in Fig.7has taken 100 random edge-vacancy configurations into consideration. It is legitimate to see how this large demand of configurations could be reduced for the realization in an experiment. Making use of the mesoscopic characteristics of the quantum transport [36], we introduce, for a given edge-vacancy configuration, an energy average of G over a small energy interval E. For our purpose here, we choose E= 2.5 × 10−3t. We show, in Fig.8, that this energy average procedure has dramatically reduced the demand on the configuration size by using only three randomly chosen configurations. With merely three configurations, we see that the Fano characteristics inGE
are very clear in Figs. 8(a), 8(b), 8(d), and 8(f). As for Figs. 8(c) and8(e), theGE already shows sharp-change
characteristics at the positions of the Fano peaks (guided by the
Gpcurves). This rapid converging to the Fano characteristics by energy averaging is an impressive demonstration of the mesoscopic physics [36].
We have established that the Fano structures of an adsorbate pair (a nodal and non-nodal pair) is robust against weak edge-vacancy disorder. The effect of other adsorbates on the adsorbate pair is addressed in the following. Figure 5 has shown that when these other adsorbates are nodal, the Fano structures of the adsorbate pair are not suppressed. On the other hand, when these other adsorbates are non-nodal, they could affect the Fano structures. In the low adsorbate
FIG. 9. (Color online) Effects of an additional random adsorbate A3 on the Fano structures of an adsorbate pair in the presence of edge-vacancy disorder. The adsorbate pair, located at M1and M2, are
separated by L= 3. Adsorbates are F in (a)–(c), and OH in (d)–(f). A3 locates randomly over M and A3-pair L separation is within a 0.5-μm range.GEis defined as in Fig.8, but for five configurations
of edge vacancies and A3.
concentration regime, we consider the effects of an additional random adsorbate on the Fano structures of the adsorbate pair in the presence of edge-vacancy disorder. This is presented in Fig.9. Adsorbates are F in (a)–(c), and OH in (d)–(f). For the GE, we have included five random configurations of the
additional adsorbate and the edge vacancies. The additional adsorbate locates randomly over all possible M sites and over a longitudinal distance of 1.0 μm. This is equivalent to an adsorbate concentration Pa≈ 3.1 × 10−3% for the adsorbate
system. The edge-vacancy concentration Pv≈ 0.125% is the
same as in Fig.8. Overall, Fig.9shows that the Fano structures remain evident even though there are signs of suppression. In particular, they are quite robust for Figs.9(a)– 9(d), and recognizable (guided by the Gpcurves) for Figs.9(e)and9(f).
IV. CONCLUSION
In conclusion, using AGNR as a platform, we have performed investigation on the concept of nodal physics and the possible physical objects that it can lead to. Specifically, we have pointed out that nodal adsorbate induces a bound state in metallic AGNRs, that the bound-state energy is adsorbate dependent, and that it is robust against weak disorder. It is revealed in Fano resonances via a non-nodal adsorbate, which greatly enhanced the energy resolution. It allows adsorbate recognition, when its H¨uckel parameters are known, and H¨uckel parameters probing, when the adsorbate is known. Finally, the generic nature of this work should encourage further research on nodal physics in other physical systems.
ACKNOWLEDGMENTS
This work was supported by Ministry of Science and Technology, Taiwan (Contract No. NSC 102-2112-M-009-005-MY3), NCTS Taiwan, and a MOE-ATU grant.
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![TABLE I. Parameters for adsorbate F and OH obtained by Ihnatsenka et al. in Ref. [ 20 ].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7692611.143665/2.911.470.838.985.1126/table-parameters-adsorbate-and-oh-obtained-ihnatsenka-ref.webp)
