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A first principles scanning tunneling potentiometry study of an opaque graphene grain boundary in the ballistic transport regime

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Nanotechnology Nanotechnology 25 (2014) 415701 (11pp)

doi:10.1088/0957-4484/25/41/415701

A first principles scanning tunneling potentiometry study of an opaque graphene grain boundary in the ballistic transport regime Kirk H Bevan Division of Materials Engineering, Department of Mining & Materials Engineering, and Centre for the Physics of Materials, McGill University, Montréal, Québec, Canada H3A 0C5 E-mail: [email protected]. Received 12 May 2014, revised 14 July 2014 Accepted for publication 17 July 2014 Published 24 September 2014 Abstract

We report on a theoretical interpretation of scanning tunneling potentiometry (STP), formulated within the Keldysh non-equilibrium Greenʼs function description of quantum transport. By treating the probe tip as an electron point source/sink, it is shown that this approach provides an intuitive bridge between existing theoretical interpretations of scanning tunneling microscopy and STP. We illustrate this through ballistic transport simulations of the potential drop across an opaque graphene grain boundary, where atomistic features are predicted that might be imaged through high resolution STP measurements. The relationship between the electrochemical potential profile measured and the electrostatic potential drop across such a nanoscale defect is also explored in this model system. Keywords: scanning tunneling potentiometry, graphene, quantum transport, scanning tunneling microscopy (Some figures may appear in colour only in the online journal)

1. Introduction

resolution [5, 7]. The ultimate resolution limit of any scanning probe method lies at the scale of single atoms and even atomic orbitals [8]. Though this is now achievable with scanning tunneling microscopy (STM) [3, 8, 9], the conclusive observation of atomic scale features in STP imaging has remained elusive [2–7]. This is partly due to the fact, that it is not fully understood what unique potential drop features one might expect to observe at atomic length scales—if any at all1. Motivated by the desire to better chart this unknown, we present a theoretical study of atomic scale STP imaging in the ballistic transport regime.

Advanced measurement techniques possessing nanoscale and atomic scale resolution have played a pivotal role in the development of nanoelectronic device technology over the past decade [1]. Amongst all such characterization tools, scanning tunneling potentiometry (STP) provides indispensable insight into the nature of the potential drop across such devices [2–4]. As a surface sensitive technique, it is particularly amenable to emerging two dimensional (2D) devices constructed from materials such as graphene [5, 6]. Hence, over the past several years, there has been a push to characterize the voltage drop within graphene and related 2D systems via STP imaging [5–7]. These advances have been driven by a desire to better understand quantum/ballistic transport in short channel devices at ever increasing levels of 0957-4484/14/415701+11$33.00

1

It is important to note that molecular level potential measurements have been achieved with Kelvin probe force microscopy, a close cousin of scanning tunneling potentiometry (see [9]).

1

© 2014 IOP Publishing Ltd Printed in the UK

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2. STP model In this work we utilize the NEGF formalism to evaluate ballistic transport, within the ‘dual mode’ three probe STM and STP framework illustrated in figure 1. The essential properties of the NEGF method are well presented in [16, 17, 19–22]. However, to define those quantities relevant to our discussion, a brief overview is provided below. Beginning from the postulates of quantum mechanics [16], NEGF is primarily applied to compute the voltage drop and electrical current response of a nanoscale device/system. For the three probe system in figure 1, this is accomplished by partitioning the region of interest into an energy (E) dependent Greenʼs function of the form

Figure 1. Illustration of the three probe NEGF quantum transport

method applied to either scanning tunneling microscopy (where μ L = μ R , IS = 0, and IT ≠ 0 ) or scanning tunneling potentiometry (where μ L > μ R , IS ≠ 0 , and IT = 0).

−1

Chu and Sorbello conducted the initial pioneering STP theory work two decades ago, where they concluded that STP measurements probe non-equilibrium perturbations in the electrochemical potential [10]. This theoretical study also predicted the possible observation of sub-nanometer features, using a phenomenological scattering state approach. However, to the authorʼs knowledge, theories of STP imaging have thus far remained largely unexplored at the atomic scale [10, 11]. On the other hand, successful atomic scale STM theories exist in various forms [8, 12, 13]. The most widely adopted of these is the Tersoff–Hamann approximation [13]. By treating the probe tip as a point source/sink of electrons, it has provided a surprisingly robust framework for explaining many of the features observed in STM measurements [14, 15]. In this work we are able to intuitively bridge these two imaging modalities, by formulating tunneling probe measurements within the Keldysh non-equilibrium Greenʼs function (NEGF) formalism (as illustrated in figure 1) [16, 17]. In both cases, our analysis is built upon the same basic assumption: that the probe tip can be treated as a point source/sink of electrons. Thus we put forward the possibility that atomic features routinely achievable in STM measurements eventually might, in principle, also be observable in STP measurements. The theoretical framework of our STP model is outlined in section 2. Subsequently, in section 3 we present a first-principles application of the theory to graphene. In the first part of this analysis (section 3.1), we predict interesting atomic scale STP features that might be resolvable at an opaque graphene grain boundary [18]. As a followup to this study, in section 3.2, we delve deeper into the precise quantities that are probed in STP imaging and underscore the distinct separation of the measured electrochemical potential profile from the electrostatic potential profile—where the latter can be confusingly associated with the term ‘potential’ in the acronym STP. It is hoped that this theoretical study will provide impetus for further experimental studies aimed at achieving atomic scale STP imaging.

[G (E ) ] = [ (E + iη)I − H − Σ L − Σ R − Σ T ] .

(1)

Here [H ] defines the device region Hamiltonian matrix, [I ] the identity matrix in an orthogonal representation2, and η represents a positive infinitesimal broadening constant. Device partitioning is made possible through the use of left/ right/tip self-energies [Σ L,R,T ]; the imaginary part of which contains information regarding the lifetime and coupling to the contacts in the form of [ΓL,R,T ] = i [Σ L,R,T − Σ L,† R, T ]. Under STM operation, again referring to the general model in figure 1, we assume that no current flows laterally through the sample such that IS = 0 and μ L = μ R = μS (with μS now describing the equivalent electrochemical potential of the left and right contacts). However, the STM tip is biased with respect to the sample through which a measurable tunneling current flows (IT ≠ 0 and μT ≠ μS). Under these conditions, the problem can be reduced to two probes by grouping the left and right contacts into a single sample selfenergy [Σ S] = [Σ L + Σ R ] such that [ΓS ] = [ΓL + ΓR ]. The tip current (IT) is then arrived at by integrating over the tip–sample transmission function through the bias window between μS and μT [23, 24] IT =

2q 2 h

∫μ

μT

Tr ⎡⎣ ΓT GΓS G †⎤⎦ dE ,

(2)

S

where we have assumed a low temperature spin degenerate integral (with the contact Fermi distributions well approximated by step functions) in units of eV [16]. Solving for the tip current during either constant current or constant height STM imaging, via equation (2), can easily become a very computationally demanding proposition [23, 24]. Hence, to simplify the problem, the transmission at each energy and scanning point may be solved more simply by approximating the tip as a point source/sink of electrons localized in real space at r , such that in a real space matrix/ grid representation [Σ T ] ≈ − iγT [δ (r0 − r, r2 − r)] 2 , which is non-zero only at the matrix/grid element (r0 = r, r2 = r). Here we treat the tip–sample coupling variable γT as a constant, further discussion on this approximation and its relation to more detailed formulations of the tip can be found in the appendix [12, 13]. 2 In a non-orthogonal representation the identity matrix [I ] becomes the overlap matrix [S ].

2

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The first consequence of this approach, is that we may neglect the perturbation introduced by the very small constant in the tunneling regime such that γT [G (E )] ≈ [(E + iη)I − H − Σ S]−1. Thus, the remainder of the transmission can be interpreted as a product of the spectral function of the substrate [AS ] = [GΓS G †]. Hence, the current at the tip becomes IT = =

2q 2γT h 2q 2γT



μT

∫μ

(μ L ⩾ μT ⩾ μ R ), such that the left and right contact tip currents cancel (IT = ITL + ITR = 0 ) during STP imaging. Within this interpretation, it is by measuring the variation in the tip electrochemical potential (required to obtain zero tip current) that one is able to build a profile of the ‘potential drop’ via μT (r). However, to establish what this measured μT (r) profile might be, it is to helpful make a few further approximations to the expression in equation (5). The vast majority of STP measurements are done on conductors, for which the potential drop at a scattering center is in the meV range [2–4, 26]. At these low lateral biases, transport can be treated within a linear response approach [16, 31] such that equation (5) reduces to

Tr ⎡⎣ [ δ ( r0 − r , r2 − r) ][ A S ( r2 , r1) ] ⎤⎦ dE

S

μT

∫μ

ρS (E , r)dE ,

(3)

S

where ρS (E , r) denotes the local density of states (LDOS) in the sample contained in the diagonal elements of [AS ] 2π [16]. The above reduction is essentially the Tersoff–Hamann STM approximation [13], arrived at through another route. This can be seen by decomposing the sample LDOS into the constituent contribution of each wave function ψj [16, 25] ρS (E , r) =

∑ψ j (r) ψ j* (r) j

ηπ

(

E − εj

2

)

.

IT ≈

∫

(4)

+ η2

μ T (r ) =

≈

2q 2γT ⎛ ⎜  ⎝

μT

μT

L

R

ρL,R (E , r) =

⎞

Δμ ⎛ ρL ( μ S , r) − ρR ( μ S , r) ⎞ ⎜⎜ ⎟⎟ , 2 ⎝ ρL ( μ S , r ) + ρR ( μ S , r ) ⎠

(7)

1 (2π )n

∫

ψk L,R⊥ (r) ψk*L,R⊥ (r) dE ( k L,R) dk L,R∥

dk L , R ⊥

(8)

for a transport system of dimensionality n = (2 or 3). Here, k L,R is the lattice momentum vector of a given left/right scattering state, decomposed into k L,R ⊥ occurring in the direction(s) perpendicular to transport and k L,R∥ occurring in the direction of transport. Hence, within the NEGF prescription, the same physical assumption (namely, that of treating the scanning tip as a point electron source/sink) leads to both the Tersoff–Hamann STM approximation and the Chu–Sorbello STP approximation. However, NEGF is advantageous as it allows for the systematic inclusion of scattering effects through the application of additional self-energies [16, 31]. For example, one can explore the effect of electron–electron scattering on STM and STP measurements by including an additional dephasing self-energy Σ d = d p G determined by a dephasing parameter d p, such that the Greenʼs function becomes G (E ) = [(E + iη)I − H − Σ L − Σ R − d pG ]−1 3. The left and right dephased spectral functions needed for equations (3) and

⎞ Tr ⎡⎣ ΓT GΓR G †⎤⎦ dE ⎟ ⎠ R

∫ μ ρL (E , r)dE +∫ μ ρR (E , r)dE ⎟⎠.

(6)

where we have taken μS = 0 as our reference energy (i.e. ground). This relation has also been derived previously, though other means, and is the Chu–Sorbello approximation for STP measurements [10]. Again, this can be seen by further writing the left and right scattering LDOS terms in terms of their constituent wave function contributions [10, 16, 25], such that

μT

∫μ

( ( μ T − μ L ) ρL ( μ S , r )  + ( μ T − μ R ) ρR ( μ S , r ) ) ,

where μS is the sample electrochemical potential prior to and applying a lateral bias ( μ L = μS + Δμ 2 μ R = μS − Δμ 2). By further placing the STP imaging condition IT=0 on equation (6) we arrive at

In the limit of infinitesimal broadening η the Lorentzian becomes a delta function, which selectively ‘sifts’ out the contribution of each wave function (at its eigenenergy εj) to the current within the bias window as expressed by equation (3). Now using the picture developed above we can build a similar framework for interpreting or even predicting STP images. Under STP imaging, the sample potential profile is mapped by continuously adjusting the tip electrochemical potential ( μT ) to maintain zero tip current (IT = 0) whilst scanning across a latterly biased surface carrying a current (IS ≠ 0 and μ L > μ R ), as illustrated in figure 1 [2–4, 26]. There are other STP imaging approaches, but we take this simplest interpretation to avoid time dependent current complexities [2, 26–28]. Unlike STM imaging, STP imaging includes at least two current contributions: the current between the tip and the left contact (ITL) and the current between tip and the right contact (ITR), such that the total tip current in the ballistic regime [29, 30] is given by IT = ITL + ITR 2q 2 ⎛ μ T ⎡ Tr ⎣ ΓT GΓL G †⎤⎦ dE + = ⎜ h ⎝ μL

2q 2γT

(5)

Here we have started from the same above assumption: that the tip can be treated as a highly localized electron source/ sink, and have arrived at a total current that is a function of electrons ballistically injected from both the left and right contacts via [AL,R ] = [GΓL,R G †] = 2πρL,R . Importantly, we have placed the tip electrochemical potential ( μT ) between the left and right contact electrochemical potentials

3

Note that we have also dropped out small the tip self-energy in our dephasing expression, assuming the tip coupling is weak enough such that phase coherent Friedel oscillations are not perturbed by the tip.

3

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(7), can then be expressed as [31] ⎡ A d (E ) ⎤ = ⎣ L,R ⎦

∞

∑d pj−1[G ] j ⎡⎣ ΓL,R ⎤⎦ ⎡⎣ G †⎤⎦ . j

nitride (BN) [36]. Moreover, graphene can be tailored to incorporate topologically ‘flat’ grain boundaries and defects [18, 37]; and thereby, in principle, sidestep the tip jumping effects that have been problematic in STP measurements on rougher surfaces [2, 4, 32]. Importantly, this grain boundary has the unique property of being opaque to ballistic electron transport at low to moderate biases, due to the non-alignment of the left and right lead Dirac cones as shown in figure 2(g) (see [18]). Hence, it provides an excellent junction for inducing a sharp ‘atomistic’ electrochemical potential drop that can be used to both: (1) explore the resolution limits of STP measurements; and (2) demonstratively decouple and couple the electrostatic potential from the electrochemical potential (see footnote 4).

(9)

j=1

From the real space diagonal elements of [AL,d R (E )] we can extract the dephased left and right scattering LDOS ρL,d R . In section 3.1 we will utilize this approach to examine the impact of dephasing on ‘atomically sharp’ STP imaging, a capability envisioned by the point electron source/sink approximation applied to arrive at equation (7). However, the STP interpretation expressed by equation (7) also challenges us at another fundamental level. Namely, it leads one to ask: what exactly do STP measurements capture, the electrostatic or the electrochemical potential profile [16, 31, 32]? This is a question of importance when interpreting STP images, since quite often STP measurements are used to quantify local electric fields around nanoscale defects [2–7, 26]. If we are measuring the electrostatic potential, then the gradient of a measured profile can be interpreted as the electric field [2–7, 26]. On the other hand, if we are measuring the electrochemical potential then the gradient does not necessarily measure the local electric field. As discussed in [16, 31] and [33], at low bias the electrostatic potential profile ϕ (r) can be related to the electrochemical potential profile in equation (7) through the linear response expression of Poissonʼs equation ⎛ ⎞ q2 −q 2 ⎜ ⃗ · εr ⃗ − ρS ( μ S ) ⎟ ϕ = ρS ( μ S ) μ T , ε0 ε0 ⎝ ⎠

( )

3.1. What atomistic features might be resolvable in STP measurements?

In order to correlate voltage drop measurements with ‘topographic’ features, STP images are typically compared against STM images [2–7, 26, 32]. A simulated constant height STM image of our chosen graphene grain boundary is presented in figure 2(b)—computed via equation (3) at a 1 V bias stretching from 10 meV to 1.01 eV above the Dirac cone (ED). Note, we are assuming a near intrinsic n-type graphene sheet with the Fermi level placed at E F = μS = ED + 10 meV (see figure 2(g)) [36]. As expected, the atomistic hexagonal patterning of graphene shows up clearly in the simulated STM image (in blue, figure 2(b)) [15]; whereas, the class II grain boundary is identifiable through a slightly higher LDOS region (in red, figure 2(b)). Interestingly, the increased LDOS in the defect region can be largely attributed to a ‘piling up’ of the left and right scattering states as shown in figures 2(e) and (f), rather than the strict localization of states near the scattering center [38]. With this ‘topographic’ information in hand, we can now begin to examine the voltage drop characteristics of this ‘atomically sharp’ system in detail. The electrochemical potential drop across this opaque graphene grain boundary, computed via equation (7) at bias of 1 meV, is shown in figure 2(c). We have extracted this simulated STP image at the same scanning height as the STM image in figure 2(b) (simulation details are provided at the end of this subsection). The most striking feature in this juxtaposition is the complete absence of atomistic features away from the graphene grain boundary in figure 2(c). Instead, we are primarily left with two extremes of solid red and blue (±0.5 meV) [7]. However, within the grain boundary scattering region a ‘mosaic’ type pattern emerges, containing tiled regions on the length scale of atomic bonds. Interestingly, these same general characteristics persist even upon the introduction of dephasing via equation (9), as displayed in figure 2(d). This indicates that such atomic scale features might be observable at both cryogenic and non-cryogenic temperatures. To understand the origin of the features in the simulated STP images, it is helpful to take a closer look at the averaged left (solid red) and right scattering (solid blue) LDOS presented in figures 2(e) (coherent) and (f) (dephased). At this

(10)

where εr is the relative static permittivity and ε0 is the vacuum permittivity. Importantly, only when the electrochemical potential varies on a length scale similar to that of the screening length (i.e. at sufficiently high carrier densities) can we assume that the electrostatic potential closely follows the electrochemical potential [16, 31]. Clearly, our theoretical interpretation is that one measures the electrochemical potential (see equation (7)). Therefore, our secondary goal in the next section (section 3.2) will be to illustrate the limits where the electrostatic potential maybe alternately either coupled or decoupled from the electrochemical potential about a nanoscale scattering center4.

3. Results To achieve these goals we have chosen to study the voltage drop characteristics of the class II graphene grain boundary illustrated in figure 2(a) [18]. Graphene represents an ideal ‘playground’ in which to address these problems, since transport in graphene is 2D and can be accurately probed through STP measurements [5, 6, 34, 35]. That is, transport through buried (non-surface) atoms can be ‘eliminated’ by placing graphene on a flat insulating substrate such as boron 4

A semiconductor p–n junction is a mesoscopic example of where the electrostatic potential is decoupled from the electrochemical potential (even at equilibrium). 4

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Figure 2. (a) Opaque graphene grain boundary. (b) Simulated STM Image. Simulated coherent and dephased STP images at a bias of 1 meV, (c) and (d) respectively. Simulated coherent and dephased in-plane averaged LDOS, (e) and (f) respectively—left scattering LDOS (red), right scattering LDOS (blue), and total LDOS (dashed black), the insets show a zoomed in view of the same data. (g) Non-overlapping left (red) and right (blue) transmission channels of the contact states at an opaque grain boundary, the channel transmission magnitude is indicated by the right axis color scale. For all results EF is set 10 meV above the Dirac point (ED).

right incident states follow the right electrochemical potential. In a strongly scattering ballistic system, such as this, one should expect to see a flat potential profile beyond the scattering region. Whereas, the ‘mosaic’ features present in/near the grain boundary can be attributed to the competing decay of the left and right wave functions (as shown in figures 2(e) and (f)). In this boundary region neither contact electrochemical potential dominates and ‘atomistic’ features arise due to the competing propagation of wave functions, which bear the orbital and wave function symmetries of their source

short length scale [36], the dephased and coherent scattering electron density results are nearly identical. Both show electrons incident from the contacts quickly decaying within a ∼2 nm region about the grain boundary. Far to the left/right of the grain boundary the left/right scattering LDOS is the only quantity present in equation (7); hence, we have an LDOS quantity divided into itself giving the plateaued extremes (i.e. the absence of features away from the defect) which appear in figures 2(c) and (d). Or more physically stated, the left incident states follow the left electrochemical potential and the 5

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contacts. Hence, atoms themselves are not necessarily resolvable in STP measurements but the transport associated decay of atomic orbital features (derived from the scattering electronic states) could be resolvable. Nevertheless, it is difficult to ascertain whether such an ‘atomistic mosaic’ might be observed experimentally [8]. The point source/sink derived STM approximation, expressed by equation (3), has had remarkable success in the interpretation of atomic features in STM measurements (see articles citing [13]). It remains to be seen whether the same approximation, expressed by equation (7), can provide an accurate interpretation of eventual high resolution STP images. In principle, a tip capable of resolving atomistic STM features would also be capable of resolving atomistic STP features, so long as fine enough voltage stability could be achieved at both the tip and laterally across the nanoscale defect—not to mention the challenge of tip drift [2]. In this regard, the opaque characteristics of this transport system hold out particular promise. At cryogenic temperatures, in the absence of inter-valley graphene scattering, it is possible that biases greater than 10 meV might be applied across this opaque defect without inducing significant current densities, that typically result in destructive joule heating and electromigration [33, 39, 40]. This would enable one to observe fine features such as those plotted in figure 2(c) at voltage resolution tolerances as low as 0.01 meV (which have been repeatedly achieved in the literature) [3, 4]. However, it is important to note that the linear response assumptions made in arriving at equation (7) are less accurate at higher biases. Moreover, in the event that a near perfect opaque junction cannot be formed, due to the ‘patchwork quilting’ of grain boundaries for example [37], it would be favorable to conduct such a transport study at the low electron densities proposed here so that any leakage currents around an opaque barrier would not induce significant joule heating at grain boundary biases in excess of ∼1 meV [5]. Additionally, placing the Fermi level at such a low position relative to the cross over between the two Dirac cones (as shown in figure 2(g)), also places a higher bound on the phonon energies (and temperatures) needed to induce a significant inter-valley leakage current though such a grain boundary [41] (rendering it no longer opaque as discussed in [18]). However, the reported ∼1 μm long mean free path of graphene [29, 30], suggests that leakage currents arising from thermalization should not be problematic in samples with channel lengths much less than one micron (such as that in figure 2).

computed at sampling of 300 energy points and 128 k-points. The STP simulations were completing using 512 k-points, to ensure sufficient sampling accuracy at E F = ED + 10 meV (see figure 2(g)). In the dephased STP simulation (figure 2(d)), a dephasing value of 1 × 10−4 Htr2 was employed resulting in a ‘screening conservation value’ of 1.6 (see the discussion in [31]). In all presented simulations, a single-ζ polarized basis set with a cut off radius of 4.2 Bohr was employed [43], to allow execution of these simulations containing a large number of atoms (404 total) in a reasonable time frame (e.g. across 128 cores over a few days). The STP and STM images were therefore computed at a constant imaging height of 3 Bohr (below the 4.2 Bohr cutoff radius). We have performed similar simulations with a longer 8 Bohr basis set for a shorter channel region (approximately 3 nm in width), and found that nearly identical atomistic patterning still appeared at constant imaging heights up to the radial extent of the basis (approximately 7 Bohr). Lastly, it is important to note that though density functional theory (DFT) is utilized in this work, all atomistic results presented here could have also been obtained through an alternate singleparticle approach (e.g. Hückel theory) [44–47].

3.2. When do we also measure the electrostatic potential profile?

With this detailed scattering analysis in hand, let us move on to the second goal of this study: to illustrate the extremes of electrochemical–electrostatic coupling that can exist in STP measurements. Thus far, we have established that the grain boundary in figure 2(a) is capable of producing a relatively sharp electrochemical potential drop in the range of ∼2 nm (and possibly even atomistic STP features) –in the absence of tip jumping effects [2, 4]. So, it remains to be shown that if one studies transport through this system at near intrinsic doping levels (i.e. E F = μS = ED + 10 meV), that a much longer (decoupled) electrostatic potential drop on the order of ∼20 nm should result [36]. At the conclusion of this analysis, we will argue that a much higher doping level is often needed to safely assume that STP measurements map both the electrostatic and the electrochemical potential at nanoscale defects. To estimate how long the electrostatic potential drop should be at such low doping levels, we need to move to a more manageable long range description of electron scattering in graphene. The atomistic NEGF approach utilized in figure 2 is simply too computationally intensive to be applied over extended length scales. However, the effective Hamiltonian of the Dirac cone in graphene provides a compact framework for such studies [36] and is given by

3.1.1. Atomistic simulation details. The structure presented in

figure 2(a) was relaxed within VASP [42] to a force tolerance accuracy of 0.01 eV/Ang. 2D properties were enforced by placing a 1.5 nm vacuum buffer region between graphene sheets. All calculations were performed using the local density approximation [20]. The electronic structure of the transport region was computed at a sampling of 64 transverse k-points and 128 energy contour points, to a convergence accuracy of 1 × 10−6 Htr [20]. The STM simulation was

⎡ 0 e−iθ k ⎤ ⎥, [H (k) ] = vF k ⎢ iθ ⎣e k 0 ⎦

(11)

where vF is the Fermi velocity and θ k is the angle for the Dirac cone momentum vector k = k x x + k y y , such that θ k = arctan (k x k y ) [36, 48]. In a two probe system, the left 6

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scattering contribution in solid red and the right scattering contribution in solid blue. Note that the scattering properties are plotted over more than 1 μm , since LDOS oscillations are approximately 100 nm in length at the chosen low electron density (E F = ED + 10 meV). Dephasing can be included phenomenologically by introducing a randomly averaged phase shift 0 < α ⩽ 2π , such that rs → rs eiα , into each transverse scattering state. The dephased LDOS results are shown in dashed red and blue in figure 3(a); when compared against the coherent scattering results, significantly dampened Friedel oscillations are evident. Lastly, to physically incorporate the increased LDOS present near/inside the grain boundary (see the dashed results in figure 2(e) and (f)), a slightly offset fitted Gaussian distribution ∼2 nm in width has been phenomenologically added to both the left and right incident LDOS (this is represented as a scattering center peak in figure 3(a)). Importantly, by including this ‘Gaussian fit’ we are able to arrive at an electrochemical potential drop that is broadened over ∼2 nm (solid black in figure 3(b)) and thereby corresponds well with the first-principles results in figure 2. Note, that the same ‘Gaussian fits’ are utilized in both the coherent and dephased LDOS scattering results, resulting in closely matching electrochemical potential profiles (just as in figure 2). Without such a fit, one would instead arrive at an abrupt electrochemical potential drop by employing equation (12). As an aside, we note that the Friedel type oscillations present in the coherent scattering LDOS (see figure 3(a)) do not show up in the coherent electrochemical potential profile (see figure 3(b))–in fact the two overlap very closely, and are not distinguishable in this plot. This is because our defect is entirely opaque in the coherent limit, producing ‘extended plateaus’ for the same reason that ‘atomistic plateaus’ appear in figure 3(c) (see the earlier discussion). However, this is not to say that Friedel type oscillations should not be observable in an electrochemical potential profile, merely that they require a defect with an average transmission somewhat greater than zero (see the discussion in [31]). Though much more could be said, we shall leave an extended theoretical investigation of Friedel oscillations in STP measurements to future work. We now have all of the necessary quantities needed to estimate the electrostatic potential drop via equation (10). All results computed in this manner are shown in green and red in figure 3(b); the solid and dashed results grouped within each coloring, represent coherent and dephased screening results, respectively. The green electrostatic potential estimates were computed by assuming a sequence of infinitely stacked graphene sheets (i.e. bulk graphite), each with the same strongly scattering grain boundary in the center (as illustrated in the adjoining figure 3(b) inset) [49, 50]. At approximately 16 nm in width, these bulk screening results represent a lower bound on what the expected electrostatic potential profile might be for this system (under low doping conditions). Importantly, one obtains an even longer electrostatic ∼30 nm wide potential drop inside the graphene sheet if it is treated as an insulated 2D conductor sandwiched between BN and vacuum

Figure 3. (a) Effective Hamiltonian coherent (solid) and dephased

(dashed) electron scattering density at E F = ED + 10 meV. (b) A decoupled electrochemical potential drop (black) and electrostatic potential drop (green and red) at a opaque defect—dephased results are dashed and coherent scattering results are solid. The structural insets depict the screening environment assumed in calculating a given potential drop: bulk screening (green) or a single graphene sheet sandwiched between BN and vacuum (red). (c) The electrostatic potential (colored) follows the electrochemical potential ( μT ) more and more closely as the Fermi level is raised from E F = ED + 10 meV to E F = ED + 400 meV.

scattering states of this Hamiltonian can be represented as ⎧ ⎛ −iθ k 2 ⎞ ⎛ e−iθ k ′ 2 ⎞ ⎪⎜ e ⎟ ei k·r + rs ⎜ iθ 2 ⎟ ei k ′·r ; x ⩽ 0 i θ 2 k ⎪⎝ e ⎝ e k′ ⎠ ⎠ (12) ψk L ( r ) = ⎨ ⎪ ⎛ e−iθ k 2 ⎞ i k·r x ⩾ 0, ⎪ ts ⎜⎝ iθ k 2 ⎟⎠ e ; e ⎩

where r = x x + y y , θ k ′ = π − θ k , and k′ = − k x x + k y y . An equal and opposite expression applies to right scattering states ψk R (r) (incident from the −x -direction). The transmission and reflection coefficients (rs and ts) are k-vector dependent and their magnitude can be extracted from atomistic NEGF calculations. In particular, taking the transmission coefficients T in figure 2(g), we arrive at |ts | = T and |rs | = 1 − T . Since, this grain boundary is opaque in the ballistic limit we have the reduced Hamiltonian scattering parameters of |rs | = 1 and |ts | = 0 . Within the above framework, one can arrive at the total scattering LDOS by summing across all scattering states via equation (8). The coherent scattering LDOS results obtained through this approach are shown in figure 3(a), with the left 7

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regions each 1 nm thick (dashed and solid red in figure 3(b) indicated by the adjoining inset) [49, 50]. This 2D conductor limit is likely more representative of the decoupling between the electrostatic and electrochemical potential one might expect for near intrinsic graphene on an insulator such as BN —the system proposed at the onset of this study [36]. Hence, from the results in figures 2 and 3, it is evident that one should be cautious in the interpretation of STP imaging data. At low carrier concentrations, the electrostatic potential does not necessarily follow the STP measured electrochemical potential. When the two are decoupled, STP imaging is not able to reliably characterize the electrostatic voltage drop and associated non-equilibrium electric fields. This concept of electrostatic decoupling follows directly from Landauerʼs concept of resistivity dipole formation about defects in the residual (ballistic) resistivity limit [16, 31–33]. At what carrier concentration can one approximately assume that the STP potential also corresponds to the electrostatic voltage drop? Roughly speaking, this occurs when the electrochemical potential varies relatively slowly with respect to the screening length λ. The screening length is directly related to the electron density at the Fermi energy ρ (E F ) by λ = εr ε0 eρ (E F ) . However, this bulk metric can vary depending on the geometry of the chosen system. For our model grain boundary a more precise estimate can be obtained by raising the Fermi level until the two potentials are ‘strongly’ coupled as illustrated in figure 3(c). In these calculations we again assume a 2D conductor sandwiched between BN and vacuum regions each 1 nm thick, to arrive at a conservative doping level of E F = ED + 400 meV that is achievable experimentally [51]. Note that even when the Fermi level is raised to such an extent, the electrostatic potential never completely follows the electrochemical potential (see light blue and black curves in figure 3(c)); since there must always be at least some residual charge density that provides the voltage drop at a defect, as expressed compactly by equation (10) [16, 31]. Lastly, it is worth nothing that we have made use of the fact that the electrochemical profile across the defect does not vary much as the Fermi level is raised in this particular system. This is demonstrated in figure 4, where the length of the electrochemical potential drop remains ∼2 nm as the Fermi energy is raised from E F = ED + 10 meV to E F = ED + 400 meV. Here it can be seen that atomistic electrochemical features from 25 meV through to 100 meV are nearly identical. However, new features begin to appear in the range of E F = ED + 200 meV through to E F = ED + 400 meV, as the band structure dispersion begins to transition away from a linear Dirac like relation [52]. Indeed, in figure 4(f) we see that the electrochemical potential drop broadens to ∼3 nm in width at E F = ED + 400 meV (rather than shrinking with the screening length), suggesting that the electrochemical potential and electrostatic potential would follow each other even more closely at high doping levels than is indicated in figure 3(c).

3.2.1. Electrostatic simulation details. In all long range

electrostatic simulations, the graphene sheet was assumed to behave as a 0.34 nm thick 2D conductor [51, 52], with a uniform electron density transverse to the direction of transport. For the insulator simulations, where graphene was sandwiched between a 1 nm BN region and a 1 nm vacuum region, periodic boundary conditions were assumed transverse to the direction of transport—similar periodic boundary conditions were applied to the ‘bulk’ screening simulation. The contact electrostatic potential was solved iteratively through mixed Neumann and Dirichlet boundary conditions (see the discussion in [31]). The relative dielectric constant (εr) of BN and graphene were both taken to be ∼3, as reported in recent experimental studies [49, 50].

4. Conclusion In summary, we have presented a theory of STP measurements in the ballistic transport regime, whereby the surface electrochemical potential was derived to be the quantity imaged. Moreover, by treating the scanning probe tip as a point source/sink of electrons, within the NEGF prescription, the close similarity between the Tersoff–Hamann STM approximation and the Chu–Sorbello STP approximation was demonstrated. A first-principles implementation of this theory was then conducted to study the potential drop across an opaque graphene grain boundary. It was predicted that atomistic features might be observable in STP measurements within the grain boundary region, where scattering states incident from each contact possess competing magnitudes. Subsequently, in the same model graphene system, we examined the physical meaning of the potential measured via STP imaging. It was shown that one should only safely assume that STP imaging measures both the electrochemical and electrostatic potential in the limit where: electrochemical potential variations occur on the same length scale as the sample screening length. Moreover, we expect that this physical interpretation should also carry over to insulating and semiconducting systems (see footnote 4). Lastly, it is important to note that future work should focus on STP tip properties that are not included in the point source approximation applied herein, including: tip induced atomic relaxations in the sample, tip electronic structure, and tipsample electronic coupling (as discussed in the appendix) [12, 23, 46, 47, 53–55].

Acknowledgments R. Möller, C. Bobisch, W. Wang, M. R. Beasley, G. M. Stocks, and S. Datta are gratefully acknowledged for helpful discussions. NSERC of Canada and FQRNT of Québec are graciously thanked for financial support. Computational resources were generously provided by the US National Science Foundation Network for Computational Nanotechnology and by Compute Canada via Calcul Québec and the Canada Foundation for Innovation. 8

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Figure 4. Non-equilibrium electrochemical potential profile with increasing electron concentration (or rising Fermi energy). (a) Shows the

mapping of atomic positions onto the computed profiles for the Fermi energy placed (b) 25 meV, (c) 50 meV, (d) 100 meV, (e) 200 meV, and (f) 400 meV above the graphene Dirac point. The atomic structure in (a) is reduced size version of that in figure 2—provided again for visual comparison.

Appendix

given by

[ Σ T ] = [ τST ] [ GT ][ τST]† ,

It is useful to consider how one arrives at the point source approximation when starting from a ‘full’ tip self-energy [23, 46]. Suppose we describe the combined tip and sample system in the form ⎡ HT τTS⎤ ⎢ ⎥ ⎣ τST HS ⎦

ψT ψS

ψT ψS ,

{ } { } =E

(A.2)

where [GT ] = [(E + iηT )I − HT ]−1 is the tip Greenʼs function with broadening ηT [17, 56]. To simplify this self-energy let us begin by establishing our basis of representation. The full eigenstate at energy E in equation (A.1) can be represented by a linear combination of the orthogonal tip ψT and sample ψS eigenstates—a reasonable approximation when the tip and sample are independent electron sources [17]. This reduces equation (A.1) to a 2 × 2 Hamiltonian for each tip/sample eigenstate combination. We can further express these eigenstates in a real-space basis χj, such that ψT = ∑ j α j χ j and ψS = ∑ j β j χ j . Under the simplest interpretation each χj is orthogonal, located at a point rj ,

(A.1)

where [HT S ] are the tip/sample Hamiltonian sub-matrices and [τTS ] = [τST ]† is the tip–sample coupling [23]. The ‘full’ selfenergy of the tip acting on the device in equation (1) is then

9

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filling an infinitesimal volume of a3 with a normalizing height of a−3 2 . To approach the analytical regime within such a basis we need only move towards the limit a → 0 [17]. Thus in our representation, {ψT } and {ψS } are column matrices containing the coefficients αj and βj respectively. Within the tip–sample eigenbasis representation, the selfenergy coupling is given by

(

)

[ τST ] = { ψS } † [T ] + [U ] { ψT },

equation (3): [Σ T ] = −iζ 2υ = −iγT 2, which is non-zero only at real-space matrix element rj (the tip apex origin). The above reduction has made use of quite a few approximations, that result in considerable computational savings [12]. Improved tip models rely upon questioning the validity of each of these assumptions one by one, but usually come at a computational cost [12, 23, 46, 47, 53–55]. For example, improving upon the energy dependence of the tip Greenʼs function will likely result in a fuller description of bias dependent STM features [12]. Likewise, improving the tip–sampling coupling approximation will probably result in a ‘fuller’ description of bonding interactions between the tip and sample [23]. For further details on the wide range of tip models available the reader is referred to [12, 23, 46, 47, 53–55].

(A.3)

where [T ] and [U ] are the kinetic energy and potential energy matrices of the system Hamiltonian in the real-space basis. Now let us assume that: (1) the tip occupies a potential region [U ] ‘outside’ the surface that closely resembles a flat barrier; and (2) the tip eigenstate {ψT } closely resembles a ‘s-like’ atomic orbital insomuch as it interacts with the sample. The Greenʼs function solution to a point source δ (r) in a flat potential region has a solution that very much resembles a ‘slike’ atomic orbital of the form g (r) ∝ e−r 2 mU  r . Its analytical and real-space matrix representations are given by

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⎛ 2 ⎞ ⎜−  2+ U ⎟ g (r) = δ (r) ⇔ ([T ] + [U ]){g} = {δ},(A.4) ⎝ 2m ⎠

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where {δ} is a column matrix with value 1 at rj and 0 at all other points. By introducing an energy multiplier ζ we can normalize the Greenʼs function and convert it from inverse energy units, to arrive at the following matrix representation [U ] (ζ {g} ) = + ζ{δ} − [T ] (ζ {g} ) = [U ]{ ψT }.

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