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Kryvchenkova, O., Cobley, R. & Kalna, K. (2015). The Current Crowding Effect in ZnO Nanowires with a Metal

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Abstract

A simulation methodology to model tunnelling spectroscopy measurementsbased on the Price and Radcliffe formalism has been implemented withina finite element device simulator. The tip-sample system is modelled self-consistently including tip-induced bending and arbitrary realistic tip shapes.The resulting spectra of III-V semiconductors is compared against other ex-perimental results and a model based on the Bardeen tunnelling approach.Image force induced barrier lowering is found to increase the tunnelling cur-rent by three orders of magnitude when tunnelling to the sample valenceband, and by six orders of magnitude when tunnelling to the sample conduc-tion band, unlike previous models which assume the effect is equal in bothdirections.

1. Introduction

WHYS THIS WORK IS IMPORTANT:As the active regions of electronic materials and devices reduce in size to

the nanoscale, the analysis tools applied to them need to be able to char-acterise with a spatial resolution in the same length scale. Scanning probemicroscopy (SPM) is one such technique and can be used to make surface elec-tronic, morphological, optical, chemical and magnetic measurements down tothe atomic scale [1]. Unlike most electron microscopy techniques, SPM canbe applied to biased devices under operation [2, 3, 4, 5]. The main limitationof the method is that with all SPM techniques the probe can interact electro-statically and physically with the sample, changing the measured propertiesof the device under test [6]. Modelling the electrostatic probe interaction canbe used to quantify the measurement error, to match experimental resultsto device properties, or potentially to remove the effects of probe interac-tion [7, 8, 9, 10, 11].

OUR PROGRESS IN THE FIELD:In this work, we have developed a simulation methodology to reproduce

the scanning tunnelling spectroscopy (STS) and microscopy (STM) processusing the simulation tool ATLAS by Silvaco [12]. The developed methodologyallows a correct modelling of the bandgap for the variety of semiconductormaterials, and includes the effect of the tip induced band bending. It is

Preprint submitted to Elsevier July 31, 2013

known, that STM can be used to study a range of semiconductor devicesincluding quantum well lasers [13], photovoltaic devices [14], resonant tun-nelling diodes [15] and semiconductor sensors [16]. Our method allows mod-elling SPM on devices, including a complete treatment of the whole devicestructure, with simultaneous self-consistent solutions of the device operationwith the probe interaction across the whole device. The developed method-ology also allows realistic tip geometries including contaminated tips, and isreadily adapted to study the probe-sample interaction in all SPM techniques.The simulation of the quantum tunnelling between the metal probe and thesurface of the semiconductor is performed self-consistently, i.e. the electro-static potential is obtained in an iterative process in which the solution ofPoisson’s equation in 2D follows the calculation of the tunnelling currentuntil convergence is achieved. In the present work we study the bandgap ofthe III-V materials like GaAs, InP, Al0.3Ga0.7As, In53Ga0.47As and GaP, thathave a flat defect free (110) surface, but the developed methodology allowsallows a full treatment of the surface states. We include the effect of thebarrier lowering due to the image force, which we confirm in this paper tobe significant, but find the way it is accounted for previously may not becorrect.

PREVIOUS WORK:There exist a number of approaches and computational techniques to esti-

mate quantitatively the tip induced band bending on semiconductor surfacesby solving Poisson’s equation in 1D [24, 25, 26, 27], 2D [28] and 3D [29]. Mostcommonly, the computation of the tunnelling current is based on the transfer-Hamiltonian formalism [30, 31] and is used for quantitative and qualitativetunnelling current evaluation [32, 33, 34, 35].

ALTERNATIVE APPROACH:First-principles calculations such as density functional theory (DFT) [17,

18, 19] often used to model the STS spectra. [SAY WE DO NOT USE IT:]This approach can be used to model small systems of atoms, but it is notsuitable for the tasks like ours, where the whole device structure needs to beincluded in the calculations. In addition to the limitation to a small numberof atoms, DFT is based on a single electron approximation [20, 21] and theassumption that there is a link between density and the exact ground stateenergy. Consequently, the calculation of excited states is cumbersome andcomputationally expensive when a flexible basis is needed to simultaneouslydescribe the ground and excited states. DFT can model atomic scale band-structure, but when the system size is above 10 nm, as typical for studying

2

device surfaces, atomic level perturbation becomes negligible. Instead, thebehaviour of large areas of the surface which are electrostatically affectedhave to be taken into account. Therefore, Coloumb interactions have to beincluded within a model of the whole device structure and with the simulta-neous tip-sample tunnelling [22, 23]. The effect of the barrier lowering due tothe image force is of the many body nature of the tip-sample system, and itcan be included in DFT via an exchange-correlation potential [17], but onlyapproximatively at large computational costs.

OUTLINE OF THE PAPER:The developed simulation methodology for modelling STS in this work

uses a direct quantum tunnelling model based on Price and Radcliffe’s for-malism [36]. A detailed explanation of the model is given in Section 2.The developed methodology allows any arbitrary tip shape, as shown inSection 3. We have verified the developed methodology by modelling thetunnelling spectroscopy process to study the bandgap for a set of III-V tech-nologically important semiconductor bulk materials: GaAs (Section 3), InP,Al0.3Ga0.7As, In53Ga0.47As and GaP (Appendix A). To estimate the validityof the results of the 2D finite element solution described in this paper, inSection 4 we compare the computation results for p-GaAs with the experi-mental data and model implemented by Feenstra [35], as this model has agood agreement with experimental data and is widely accepted and used byother researchers [37, 2]. Simulations results to examine the role of the imageforce correction can be found in Section 5.

2. Simulation Methodology

The tip-sample system is described by a 2D model, and electrostatics aresolved fully in 2D real space. In self-consistent calculations, the current conti-nuity equations are solved iteratively in the semiconductor until convergenceis achieved to obtain the terminal currents. The tunnelling current can alsobe solved non self consistently (post processing) for comparison. In practice,the non self-consistent solution for a bulk semiconductor (like p-GaAs) will bevery close to the self-consistent solution of the current continuity equationsas shown later.

We use a direct quantum tunnelling model [12], in which the tunnellingcurrent calculation takes place along parallel slices through the gap. Thetunnelling model is based on a formula for elastic tunnelling proposed byTsu and Esaki [38], and further developed by Price and Radcliffe [36]. The

3

current density J through a potential barrier is obtained according to thenumber of generated carriers using the following formula [40]:

J =qkT

2π2~3m∗∫T (E) ·

· ln[

1 + exp(EFsamp − E)/kT

1 + exp(EFtip − E)/kT

](1)

where E is the charge carrier energy, m∗ =√mxmy (where mx and my are

carrier effective masses in the lateral directions), EFtip is the tip quasi-Fermilevel and EFsamp is the sample quasi-Fermi level. The integration term isdetermined with respect to the band edge position for every bias point. Inequilibrium, EFtip = EFsamp and the term J is equal to zero.

The transmission probability T (E), defined as the ratio of transmittedand incident currents, is calculated by solving Schrodinger’s equation in theeffective mass approximation [39]. It was demonstrated in the past thatthese approximations are very accurate to describe the sample-tip system inSTS [28] because electrons do not tunnel in to or from one specific energylevel, instead using a broad band of energies decaying away exponentiallyfrom the Fermi level. This approach was found to be more accurate than thecommonly used WentzelKramersBrillouin approximation for the case of thinbarriers for all energies [40].

The solution of the transmission probability T (E) is obtained accordingto the Gundlach model [42]:

T (E) =2

1 + g(E), (2)

with

g(E) =π2

2

mskmmmks

(Bi′dAio − Ai′dBio)2

+

+π2

2

mmksmskm

(BidAi′o − AidBi′o)

2+

+π2

2

mmms

λ20m2i kskm

(Bi′dAi′o − Ai′dBi′o)

2+

+π2

2

λ20m2i kskm

mmms

(BidAio − AidBio)2 (3)

4

where mm, ms and mi are effective electron masses in metal, semiconductorand insulator respectively, km and ks are the wavevectors in metal semicon-ductor respectively, λ0 = ~Θi/qFi, and Ai and Bi are the Airy functionsdefined as following:

Aio ≡ Ai

(ΦB − E~Θi

), Aid ≡ Ai

(ΦB + qFid− E

~Θi

), (4)

where ~Θi = (q2~2F 2i /2mi), ΦB is the barrier height for electrons or holes,

Fi is electric field in insulator, d is the insulator thickness [40, 12].The bandstructure is approximated in the anisotropic non-parabolic ap-

proximation and the complex active device system is fully included in themodel and solved self-consistently.

The tip-sample structure (including air surrounding the metal tip) is2.5 µm wide with the bulk semiconductor layer 0.9 µm deep. An exam-ple of the 2D circular tip structure with a regular triangular mesh is givenin Figure 1(e), where the projection of the cylindrical tip into 2D assumesthat the tip is uniform in the z-direction. We have checked, by numericalexperiments, that a sufficient number of grid points is used in the tip-sampleseparating region in the direction of the current flow. Homogeneous (reflect-ing) Neumann boundary conditions are used at all simulation cell boundariesexcept contacts where the Dirichlet boundary conditions are used. The metaltip is assumed to be a contact. An additional contact is added at the bottomof the semiconductor to allow for current to flow. The current flow from thesemiconductor through the insulator/air is via a tunnelling model only.

A model without image force correction assumes the abrupt change of thepotential at the interface between the gap and tip/sample surface. However,the image force correction model handles the non-physicality of a potentialdiscontinuity in two ways. Firstly, when the image force correction is added,the potential discontinuity is removed. When image force correction is notincluded, the finite element discretisation method fits a function to the pointsover the discontinuity and essentially produces a potential gradient depend-ing on the grid size. The wider method of reduction of the grid size untilstable convergence is achieved ensures no artifacts from this process affectthe results.

To examine the role of the image force, we implemented the tunnellingmodel by Schenk for ultrathin insulator layers [40]. The model accountsfor the image force effects using a pseudobarrier method which calculates

5

the transmission probability coefficient T (E) (Equation 2) of the modifiedtrapezoidal potential barrier. To evaluate the effective barrier height as afunction of electron incident energy, the image force potential is calculatedaccording to the formula by Kleefstra and Herman [41]:

Eim(x) =q16

16πεi

∞∑n=0

(k1k2)n

[k1

nd+ x+

k2d(n+ 1)− x

+2k1k2d(n+ 1)

], (5)

where

k1 = −1, k2 =εi − εsεi + εs

(6)

and ε is the relative dielectric permittivity of insulator (εi) and semiconductor(εs) regions, d is the insulator thickness, x is the position through the barrier.

The image force potential is added to the trapezoidal barrier potential.The model assumes the simple parabolic approximation of the barrier shapedue to the fact that for a very thin potential barrier the error is negligible [43].The barrier height is evaluated at three different energy levels as follows [12]:

φ(E) = φ(E1) +φ(E3)− φ(E1)

(E3 − E1)(E2 − E3)(E − E1) (E2 − E)

− φ(E2)− φ(E1)

(E2 − E1)(E2 − E3)(E − E1) (E3 − E) (7)

The Schenk tunnelling current model was implemented both self-consistentlyand in post processing calculations. For thin rectangular tips, and circulartips, there is little difference between a self-consistent solution and simplerpost processing calculations.

The material parameters used in simulations can be found in Table 1,where mc (×mo) is an effective mass in the conduction band, mhh (×mo) isa heavy hole effective mass, mlh (×mo) is a light hole effective mass and Eg

is a band gap.On the flat defect free (110) surface of III-V materials there are no sur-

face state distributions centred in the band gap and largely the surface canbe modelled as bulk [25]. Nevertheless, the developed methodology allows afull treatment of surface states very straightforwardly and can be includedin a wide range of ways from i) uniform distribution of defect states, to

6

Table 1: The material parameters used in simulations.

Material mc (×mo) mhh (×mo) mlh (×mo) Eg, [eV] Affinity, [eV]GaAs 0.067 0.49 0.16 1.42 4.07InP 0.0795 0.56 0.12 1.35 4.4GaP 0.13 0.79 0.14 2.75 4.4Al0.3Ga0.7As 0.092 0.571 0.157 1.8 3.75In0.53Ga0.47As 0.045 0.532 0.088 0.734 4.67

ii) realistic distribution either defined by iia) analytical functions (Gaussian,exponential) or even iib) a realistic distribution of surface states obtained ex-perimentally. In addition, the surface states can be included by iii) interfacetraps (need to define the energy levels of the traps, their nature [acceptor-like/donor-like], and a cross-section/trapping time).

3. Bandgap simulation for GaAs

To test the importance of the probe shape, several tip geometries weresimulated, as shown in Figure 1. The discretised geometry of the finiteelement model allows any realistic tip shape. The triangular tip shape Fig-ure 1(b) results in a slow convergence of calculations of tunnelling current athigh voltages, while Figure 1(c) and Figure 1(d) give similar results, as seenin Figure 1(f). For reduced computational time Figure 1(c) is used for theresults presented here, with a 70 nm tip radius, 1 nm tip-sample separationand a tip work function of 5 eV.

Tunnelling spectra for n-type GaAs (n-GaAs) with a doping concentra-tion of ND = 1018 cm−3 and ND = 1012 cm−3 are shown in Figure 2. ForND = 1012 cm−3 n-GaAs, the red dashed vertical lines at sample voltages of+1.5 V and −1.1 V indicate the onset of higher tunnelling outside of thebandgap region. Similarly, for ND = 1018 cm−3 n-GaAs, the vertical bluesolid lines at sample voltages of +0.65 V and −0.78 V indicate the onset oflarger tunnelling.

The change of the apparent band gap with doping concentration in Fig-ure 2 demonstrates the effect of tip-induced band bending. For the higherdoped case the onset of larger current corresponds to the tip crossing the va-lence and conduction band edges. In case of the low doping, the origin of theobserved gap is more complicated and the energy band alignments are shown

7

0T 100T 0 100

0T 35T 0 35T

50

0

10

0

50

0

10

0

Distance,T[nm]

Dis

tanc

e,T[n

m]

(a) (b)

(d)(с)

0TTTTTTTTTTTTTTTTT20Distance,T[nm]

3

0

0

Dis

tanc

e,T[n

m]

−3 0 33SampleTvoltage,T[nm]

log(

dI/d

V),

T[5Td

ecad

es/d

iv]

Triangular

CircularComplex(e)

(f)

Figure 1: Probe tip in 2D of rectangular (a), triangular (b), circular (c)shapes, and realistic complex (d) shape with a large radius of 70 nm with asmaller tip on the top with a radius of 10 nm. (e) 2D circular tip structurewith mesh.

−8 −6 −4 −2 0 2 4 6 8

10−15

10−10

10−5

|dI/d

V|ef

oreN

D=

1018

cm−

3 ,e[nA

/V]

Sampleevoltage,e[V]−8 −6 −4 −2 0 2 4 6 88

10−10

10−5

100

|dI/d

V|ef

oreN

D=

1012

cm−

3 ,e[nA

/V]

Figure 2: Tunnelling current spectra for the n-GaAs with a doping concen-tration of ND = 1018 cm−3 (blue solid line corresponds to the left scale) andND = 1012 cm−3 (red dashed line corresponds to the right scale). The onsetsof higher tunnelling in the CB and VB are marked with vertical lines.

in detail in Figure 3 for the marked onset voltages of +1.5 and −1.1 V. In Fig-

8

−4−2 0 2 4 6−0.5

0

0.5

1

1.5

2

Distance, [nm]

Ene

rgy,

[eV

]

−4−2 0 2 4 6−2

−1.5

−1

−0.5

0

0.5

Distance, [nm]

Ene

rgy,

[eV

]

0 10 20−0.01

0

0.01

EF-eVEC

EV

EF

EF-eV

EC

EV

EF

(a) (b)

Figure 3: Energy band structure of the n-GaAs, ND = 1012 cm−3 obtainedat a sample voltages of +1.5 V (a) and −1.1 V (b).

ure 3(a), the applied tip voltage has induced depletion and inverted the sur-face to make it appear p-type. The tip applied bias has to be beyond +1.5 Vbefore significant tip-to-sample tunnelling can take place. In Figure 3(b), thesample surface is in accumulation appearing more n-type than the bulk. Theinduced surface accumulation region presents filled states in the conductionband (see Figure 3(b) inset) from which sample-to-tip tunnelling can takeplace before the tip crosses the valence band edge. This effect is describedby Koenraad as Type I accumulation [25]. For ND = 1018 cm−3 n-GaAs,a bandgap of 1.43 eV is expected, 0.01 eV larger than the true bandgap of1.42 eV, which would be within the systematic error of experimental data [44].The screening effect of the high doping reduces the amount of tip-inducedband bending. For a doping of ND = 1012 cm−3 n-GaAs, tip-induced bandbending widens the measured band gap to 2.60 eV.

An onset in the tunnelling current spectra is visible in the conductionband around +2 V in the highly doped n-type GaAs marked by an arrowin Figure 2. Figure 4 shows the band edge in detailed before and after thisonset, at +1.9 eV and +2.5 eV. At +2.5 eV the surface inversion has bent thevalence band above the sample Fermi level, allowing electrons to also tunnelfrom the tip to these empty inversion states. This is described by Koenraadas Type II inversion [25]. This induced depletion region is spatially localisedunder the tip. In Figure 4(c) and (d) the tunnelling current contributions

9

−10 −5 0 5

10−10

10−4

Distance,F [nm]

J,F[A

/cm

2 ]

−10 −5 0 5

100

10−5

Distance,F [nm]

J,F[A

/cm

2 ]

−5 0 5 10 15−1

0

1

2

3

Distance,F [nm]

Ene

rgy,F

[eV

]−5 0 5 10 15−1

0

1

2

3

Distance,F [nm]

Ene

rgy,F

[eV

]

CBF tunnelling

VBF tunnelling

EF-eV

EC

EF

EV

EF-eV

EC

EF

EV

(a) (b)

(с) (d)

Figure 4: Energy band structure profiles at sample voltages of +1.9 V (a) and+2.5 V (b). Corresponding tunnelling current density from the conduction(blue stars) and valence (red open circles) bands at sample voltages of +1.9 V(c) and +2.5 V (d).

from the valence and conduction band are shown as a function of distance,with the tip at the origin, for the corresponding case shown in Figure 2.At +2.5 V, the valence band tunnelling increases in magnitude beyond theconduction band tunnelling. In all cases, the current is larger under the tipand drops away almost exponentially from it.

The bandgap simulations for other semiconductor materials like InP, Al-GaAs, InGaAs and Ga can be found in Appendix A.

4. Comparison with experimental data and other models

Feenstra’s Semitip model [45] is based on the Bardeen formalism usingthe Tersoff and Hamman approximation. The classic Bardeen model [30]assumes that the tunnelling current can be obtained from the difference inelectron scattering rates of the tip and the semiconductor sample, which isequal to the number of sample or tip states weighted by their occupation

10

−2 −1 0 1 2−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Samplesvoltage,s[V]

log(

dI/d

V),

s[nA

/V]

PresentsworkSemitips6Experimentalsdata

Figure 5: Experimental data for p-GaAs with a doping concentration ofNA = 1018 cm−3 after Ref. [35] and simulation results for Semitip 6 modeland present work. Simulations were made with the same set of parametersto study the difference in the models. Both models can be fitted exactly tothe experimental data with another set of the parameters.

probabilities, multiplied by their charge [46].The experimental data in Figure 5 was fitted to the simulation results of

Semitip 5 software with the following parameters: a tip radius R = 30 nm, atip-sample separation s = 0.9 nm, a contact potential ∆ϕ = −1.4 eV, and anarea of the tunnel junction of 0.7 nm2 [35]. The tunnelling current was alsoincreased by three orders of magnitude due to the image force effects [35, 47,48], which will be explained in Section 5. For our model in Figure 5, withthese parameters, the same magnitude is found without any image forcecorrection. Plots in Figure 5 for the latest version of Semitip 6 software andour model were created with the same set of free parameters as Semitip 5,to explain the differences in the models. Both models can be fitted exactlyto the experimental data with another set of the parameters.

One of the differences between the models arises in the calculation of themagnitude of the tip-induced band bending. The surface potential energydirectly under the tip apex was compared with the potential energy far insidethe semiconductor for voltages from −2 V to +2 V. The values are shownin Figure 6 for both Semitip 6 and our model. Both reproduce the realisticsituation of large tip-induced band bending when the semiconductor is indepletion (negative sample voltage for p-type material) and a small influencedue to the screening effect of the surface charge density when the semicon-ductor is in accumulation (positive sample voltage for p-type material). The

11

−2 −1 0 1 2−1.5

−1

−0.5

0

SampleDvoltage,D[V]T

IBB

,D[eV

]

PresentDworkSemitipD6

0 20Distance,D[nm]

0

20

Dis

tanc

e,D[n

m]

(a)D (b)

Figure 6: (a) Comparison of tip-induced band-bending models for presentwork and Semitip 6 model. (b) Potential distribution directly under the tipapex when no bias voltage is applied, contours are displaced by 0.1 V.

difference in the potential computation for the valence band is one of thesources of the mismatch in the models in Figure 5.

The standard material parameters used in our simulations differ fromthose used by Feenstra. In Semitip, the following parameters of the effec-tive mass in the valence band are used: heavy hole effective mass mh =0.643m0, light hole effective mass mlp = 0.081m0, split-off effective massmso = 0.172m0, while the material parameters for GaAs used in the presentwork are different and can be found in Table 1. However, when the sameparameters as in present work were used in Semitip, no significant differencewas found.

5. Image force simulations

The image force rounds off the corners of the trapezoidal barrier andincreases the tunnelling current. One approximation to avoid performingcomputations with this complex barrier shape is to use an equivalent lowerheight triangular barrier, which results in this constant magnitude scalingfactor for the tunnelling current, as it was used for Semitip models in Sec-tion 4.

For a rectangular tip shape, similar to Figure 1(a) but with a width of0.7 nm, and circular tip shape with the radius 30 nm, similar to Figure 1(c),both separated 0.9 nm from the p-GaAs surface with a doping concentrationNA = 1018 cm−3, spectra with and without image force correction are shownin Figure 7. There is a consistent four orders of magnitude increase in tun-nelling current when the correction is included for the rectangular tip shape(see Figure 7(a)). Different amount of the tunnelling current through the

12

210-1-2Sampleuvoltage,u[V]

log|

dI/d

V|,u

[2ud

ecad

es/d

iv]

withuimageuforceucorrectionwithoutuimageuforceucorrection

Magnitudeudifference,u[nA/V]

log|

dI/d

V|,u

[2ud

ecad

es/d

iv]

210-1-2Sampleuvoltage,u[V]

withuimageuforceucorrectionwithoutuimageuforceucorrection

magnitudeudifferenceuofutwoumodels magnitudeudifferenceuofutwoumodels

8axuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu8bx

uu20-2

4

6

8

xu10-4

Sampleuvoltage,u[V]20-2

Sampleuvoltage,u[V]

106

103

Magnitudeudifference,u[nA/V]

Figure 7: dI/dV spectra obtained when no image force correction is includedin the model and when image force correction is included in the model for(a) rectangular tip shape and (b) circular tip shape. The onsets show themagnitude difference in the currents when image force correction is includedand not included in the model.

conduction band (CB) due to the image force influence is well known [47].Previous models use constant scaling factor for the tunnelling current, as theycalculate the correction for CB only. We include the image force correctionfor valence band (VB) into our calculations. This results in a very differentimpact on CB and VB tunnelling for every bias point. With the same modelapplied to a circular tip, conduction band tunnelling current is found to bethree orders of magnitude larger and the valence band tunnelling current sixorders larger (see Figure 7(b)). The magnitude difference will change withthe different structure parameters, tip shape and semiconductor materialsused in the simulations. This compares to a uniform increase of three ordersof magnitude used by Feenstra model in the Section 4. When this imageforce correction is included in our calculation, the models matches the dataonly when the tip separation is increased. The difference in the image force

13

correction for circular and rectangular tip shapes might have an implicationon models where a circular tip shape is approximated by a staircase, com-monly used in the modelling of atomic force microscopy and Kelvin probeforce microscopy measurements [9, 7].

6. Conclusion

An STM and STS simulation methodology based on the Price and Rad-cliffe tunnelling formalism using image force correction has been developedusing the Silvaco ATLAS. This 2D finite element model was applied to sev-eral semiconductor materials, with the origin of features in the spectra ex-amined. The simulations confirmed that at larger doping concentrations thescreening effect of the semiconductor has reduced the tip-induced band bend-ing. For n-type GaAs, the modelled spectra band gap deviates from the bulkvalue by only 0.1 eV or 0.7 % which is within the experimental systematicerror [44]. At lower doping concentrations, the screening is not sufficient,and tip-induced band bending causes the apparent bandgap to either widenor reduce depending on the tunnelling mechanism.

Comparison with Feenstra’s Semitip showed several differences. Firstly,the tunnelling model based on the Price and Radcliffe formalism [36, 40]provides a larger tunnelling current by around three orders of magnitudecompared to the model based on the Bardeen formalism [49, 45] which isused in Semitip. The image force correction gives a conduction band tun-nelling current increase of three orders of magnitude, and a valence bandtunnelling current increase of six orders, compared to the ’artificial’ uniformincrease of three orders of magnitude used in Ref. [35]. With our model,the calculated current matches the same experimental data with either alarger tip separation or a reduced tunnel junction area compared to thatcalculated by Feenstra. Finally, the Poisson-Schrodinger solver in our modelpredicts more tip-induced bending when sample is in depletion, than thatfrom Semitip. The result is a shift of the band onset in the spectra.

The use of our STM model developed within a commercial simulation tooloffers several advantages over other STM models. These advantages includei) the ability to use any realistic tip shape, ii) to include full device transportmodels for SPM on devices, and iii) to model spectra from SPM on powereddevices. The model can also account for surface states, and can readily beextended to other SPM techniques.

14

Acknowledgements

One of the authors (OK) would like to thank Zienkiewicz Scholarship(Swansea University, UK) for the financial support. We would also like tothank Prof. R. M. Feenstra for careful reading of the manuscript.

Appendix A. Bandgap simulations for InP, AlGaAs, InGaAs andGaP

420-2-4Sample voltage, [V]

log(

dI/d

V),

[5 d

ecad

es/d

iv] n-AlGaAs

n-InGaAs

n-GaP

VB CB

Figure A.8: Tunnelling spectra of the Al0.3Ga0.7As, In0.53Ga0.47As and GaPwith ND = 1016 cm−3. Curves are displaced by three orders of magnitude.

The simulation methodology was applied to Al0.3Ga0.7As, In0.53Ga0.47Asand GaP, all with a doping concentration of ND = 1016 cm−3. All materialparameters used in the simulations can be found in Table 1. These simulationused a 70 nm tip radius, 1 nm tip-sample separation and a tip work functionof 4.7 eV, with results shown in Figure A.8. Modelled band gaps for thisintermediate doping are 2.6 eV for Al0.3Ga0.7As compared to the experimentalvalue of 1.8 eV, and 1.4 eV instead of 0.734 eV for In0.53Ga0.47As, both dueto tip-induced band bending delaying the onset of increased current. ForGaP the apparent bandgap was 1.7 eV instead of 2.75 eV due to Type Iaccumulation as seen earlier. For GaP a feature is seen in the spectrum at+3 V, when Type II depletion shifts to Type II inversion [25], and tunnellingin to the valence band dominates over tunnelling in to the conduction band.

15

To demonstrate the modelling for p-type material, p-type InP with adoping concentration of NA = 1012 cm−3 is presented in Figure A.9 using thesame tunnelling parameters. The observed band gap of 2.85 eV is larger thanthe experimental bandgap of 1.42 eV due to large amount of the tip-inducedband bending in the low doped material, when the screening effect of thedoping is weak.

−2 0 2

10−10

10−5

100

Sample voltage, [V]

Tu

nn

elli

ng

cu

rre

nt,

[n

A]

Figure A.9: Simulated tunnelling spectra for p-InP. The vertical dashed linescorrespond to the onsets of the larger tunnelling in the conduction and thevalence band.

16

References

[1] E. Meyer. Scanning probe microscopy and related methods. BeilsteinJ. Nanotechnol., 1:155–157, December 2010.

[2] R. J. Cobley, K. S. Teng, M. R. Brown, P. Rees, and S. P. Wilks. Lasersstudied with cross-sectional scanning tunneling microscopy. Appl. Surf.Sci., 256(19):5736 – 5739, 2010.

[3] A. Lochthofen, W. Mertin, G. Bacher, L. Hoeppel, S. Bader, J. Off, andB. Hahn. Electrical investigation of v-defects in gan using kelvin probeand conductive atomic force microscopy. Appl. Phys. Lett., 93(2):022107,2008.

[4] S. Saraf and Y. Rosenwaks. Local measurement of semiconductor bandbending and surface charge using kelvin probe force microscopy. Surf.Sci., 574(2-3):L35 – L39, 2005.

[5] R. J. Cobley, P. Rees, K. S. Teng, and S. P. Wilks. Analyzing real-timesurface modification of operating semiconductor laser diodes using cross-sectional scanning tunneling microscopy. J. Appl. Phys., 107(9):094507,2010.

[6] L. Koenders and A. Yacoot. Tip geometry and tip-sample interactionsin scanning probe microscopy (spm). In Wolfgang Osten, editor, Fringe2005, pages 456–463. Springer Berlin Heidelberg, 2006.

[7] S. Sadewasser, Th. Glatzel, R. Shikler, Y. Rosenwaks, and M.Ch. Lux-Steiner. Resolution of kelvin probe force microscopy in ultrahigh vac-uum: comparison of experiment and simulation. Appl. Surf. Sci., 210(1-2):32–36, March 2003.

[8] D. Ziegler, P. Gava, J. Guttinger, F. Molitor, M. Lazzeri L. Wirtz,A. M. Saitta, F. Mauri A. Stemmer, and C. Stampfer. Variations inthe work function of doped single- and few-layer graphene assessed bykelvin probe force microscopy and density functional theory. Phys. Rev.B, 83:235434, June 2011.

[9] E. Koren, N. Berkovitch, O. Azriel, A. Boag, Y. Rosenwaks, E. R. Heme-sath, and L. J. Lauhon. Direct measurement of nanowire schottky junc-tion depletion region. Appl. Phys. Lett., 99(22):223511, 2011.

17

[10] S. Fremy, S. Kawai, R. Pawlak, T. Glatzel, A. Baratoff, andE. Meyer. Three-dimensional dynamic force spectroscopy measurementson kbr(001): atomic deformations at small tipsample separations. Nan-otechnology, 23(5):055401, 2012.

[11] ITRS Reports, October 2012. http://www.itrs.net/reports.html.

[12] SILVACO. ATLAS User’s Manual, 2007. User manual.

[13] R. J. Cobley, K. S. Teng, M. R. Brown, S. P. Wilks, and P. Rees. Directreal-time observation of catastrophic optical degradation in operatingsemiconductor lasers using scanning tunneling microscopy. Appl. Phys.Lett., 91(8):081119, 2007.

[14] T. J. Kempa, B. Tian, D. R. Kim, J. Hu, X. Zheng, and C. M. Lieber.Single and tandem axial p-i-n nanowire photovoltaic devices. Nano Lett.,8(10):3456–3460, 2008.

[15] W. Wu, S. L. Skala, J. R. Tucker, J. W. Lyding, A. Seabaugh, E. A.Beam, and D. Jovanovic. Interface characterization of an inp/ingaasresonant tunneling diode by scanning tunneling microscopy. J. Vac.Sci. Technol., A, 13(3):602 –606, may 1995.

[16] A. Gurlo, M. Ivanovskaya, N. Brsan, M. Schweizer-Berberich,U. Weimar, W. Gpel, and A. Diguez. Grain size control in nanocrys-talline in2o3 semiconductor gas sensors. Sensor. Actuat. B: Chem.,44:327 – 333, 1997.

[17] Vienna Ab-initio Simulation Package User’s Manual, 2013.

[18] ACCELRYS User’s Manual, 2013. User manual.

[19] ATOMISTIX User’s Manual, 2013. User manual.

[20] P. Hohenberg and W. Kohn. Inhomogeneous Electron Gas. Phys. Rev.,136(3B):B864–B871, November 1964.

[21] W. Kohn and L. J. Sham. Self-Consistent Equations Including Exchangeand Correlation Effects. Phys. Rev., 140(4A):A1133–A1138, November1965.

18

[22] M. V. Fischetti and S. E. Laux. Monte carlo study of electron transportin silicon inversion layers. Phys. Rev. B, 48(4):2244–2274, July 1993.

[23] V. A. Ukraintsev. Data evaluation technique for electron-tunneling spec-troscopy. Phys. Rev. B, 53(16):11176–11185, April 1996.

[24] R. M. Feenstra and J. A. Stroscio. Tunneling spectroscopy of thegaas(110) surface. J. Vac. Sci. Technol. B. Microelectron. NanometerStruct., 5(4):923–929, 1987.

[25] G. J. de Raad, D. M. Bruls, P. M. Koenraad, and J. H. Wolter. Inter-play between tip-induced band bending and voltage-dependent surfacecorrugation on gaas(110) surfaces. Phys. Rev. B, 66:195306, November2002.

[26] S. Loth, M. Wenderoth, R. G. Ulbrich, S. Malzer, and G. H. Dohler.Connection of anisotropic conductivity to tip-induced space-charge lay-ers in scanning tunneling spectroscopy of p-doped gaas. Phys. Rev. B,76:235318, Dec 2007.

[27] M. Hirayama, J. Nakamura, and A. Natori. Band-bending effects onscanning tunneling microscope images of subsurface dopants: First-principles calculations. J. Appl. Phys., 105(8):083720–083720–4, 2009.

[28] R. M. Feenstra. Electrostatic potential for a hyperbolic probe tip neara semiconductor. J. Vac. Sci. Technol., B, 21(5):2080–2088, 2003.

[29] R. M. Feenstra, J. Y. Lee, M. H. Kang, G. Meyer, and K. H. Rieder.Band gap of the Ge(111)c(2 × 8) surface by scanning tunneling spec-troscopy. Phys. Rev. B, 73:035310, January 2006.

[30] J. Bardeen. Tunnelling from a many-particle point of view. Phys. Rev.Lett., 6:57–59, January 1961.

[31] G.D. Mahan. Many-Particle Physics, 2nd ed. Physics of Solids andLiquids. Plenum Press, 1983.

[32] M. Tsukada, K. Kobayashi, and N. Isshiki. Effect of tip atomic and elec-tronic structure on scanning tunneling microscopy/spectroscopy. Surf.Sci., 242(13):12–17, February 1991.

19

[33] J. M. Bass and C. C. Matthai. Scanning-tunneling-microscopy and spec-troscopy simulation of the gaas(110) surface. Phys. Rev. B, 52:4712–4715, August 1995.

[34] B. W. Hoogenboom, C. Berthod, M. Peter, Ø. Fischer, and A. A. Ko-rdyuk. Modeling scanning tunneling spectra of bi2sr2cacu2o8+. Phys.Rev. B, 67:224502, June 2003.

[35] R. M. Feenstra, Y. Dong, M. P. Semtsiv, and W. T. Masselink. Influ-ence of tip-induced band bending on tunnelling spectra of semiconductorsurfaces. Nanotechnology, 18(4):044015, January 2007.

[36] P. J. Price and J. M. Radcliffe. Esaki tunneling. IBM J. Res. Dev.,3(4):364–371, October 1959.

[37] C.K. Egan, A. Choubey, and A.W. Brinkman. Scanning tunneling mi-croscopy and spectroscopy of the semi-insulating cdznte(110) surface.Surf. Sci., 604:1825 – 1831, 2010.

[38] R. Tsu and L. Esaki. Tunneling in a finite superlattice. Appl. Phys.Lett., 22(11):562–564, 1973.

[39] D. K. Ferry. Semiconductors. Macmillan, 1991.

[40] A. Schenk and G. Heiser. Modeling and simulation of tunneling throughultra-thin gate dielectrics. J. Appl. Phys., 81(12):7900–7908, 1997.

[41] M. Kleefstra and G. C. Herman. Influence of the image force on the bandgap in semiconductors and insulators. J. Appl. Phys., 51(9):4923–4926,1980.

[42] K.H. Gundlach. Zur berechnung des tunnelstroms durch eine trapezfr-mige potentialstufe. Solid-State Electronics, 9(10):949 – 957, 1966.

[43] R. Stratton. Volt-current characteristics for tunneling through insulatingfilms. J. Phys. Chem. Solids, 23(9):1177 – 1190, 1962.

[44] R. M. Feenstra. Tunneling spectroscopy of the (110) surface of direct-gapiii-v semiconductors. Phys. Rev. B, 50(7):4561–4570, August 1994.

20

[45] R. M. Feenstra and S. Gaan. Quantitative determination of nanoscaleelectronic properties of semiconductor surfaces by scanning tunnellingspectroscopy. J. Phys.: Conf. Ser., 326(1):012009, 2011.

[46] A. D. Gottlieb and L. Wesoloski. Bardeen’s tunnelling theory as appliedto scanning tunnelling microscopy: a technical guide to the traditionalinterpretation. Nanotechnology, 17(8):R57, April 2006.

[47] J. G. Simmons. Generalized formula for the electric tunnel effect betweensimilar electrodes separated by a thin insulating film. J. Appl. Phys.,34(6):1793–1803, 1963.

[48] G. Binnig, N. Garcia, H. Rohrer, J. M. Soler, and F. Flores. Electron-metal-surface interaction potential with vacuum tunneling: Observationof the image force. Phys. Rev. B, 30:4816–4818, October 1984.

[49] C. B. Duke. Tunneling in solids. Solid state physics: Supplement.Academic Press, 1969.

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