Ambipolar transition voltage spectroscopy: analytical results and experimental agreement

0.24 MB·English

by  Ioan Baldea

Ambipolar transition voltage spectroscopy: analytical results and experimental
agreement
Ioan Bˆaldea∗
Theoretische Chemie, Universita¨t Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany
This work emphasizes that the transition voltages Vt± for both bias polarities (V<>0) should be
usedtoproperly determinetheenergy offset ε0 ofthemolecular orbital closest toelectrodes’ Fermi
level and the bias asymmetry γ in molecular junctions. Accurate analytical formulas are deduced
2 to estimate ε0 and γ solely in terms of Vt±. These estimates are validated against experiments, by
1 showing that full experimental I-V-curves measured by Beebe et al [Phys. Rev. Lett. 97, 026801
0 (2006)]andTanetal[Appl.Phsy.Lett.96,013110(2010)]forbothbiaspolaritiescanbeexcellently
2 reproduced.

n
a PACSnumbers: 85.65.+h,85.35.Gv,73.63.-b
J

5
I. INTRODUCTION II. THEORETICAL FRAMEWORK
2

] An applied bias V affects the energy offset ε
l Transition voltage (Vt) spectroscopy (TVS) has been 0 ≡
hal rbeectewnetelynptrhoepmoseetda1ltFoegrmetiinensiegrhgtyinatnodththeeencelorsgeystoffmseotleεc0- eεl0e(cVtr)o|Vch=e0m→istrεy0,(2V0–)2.2 tMhiossetffeeacstilcya,nabsecuacrrceonutnlyteddofnorebiny
- ular orbital (HOMO or LUMO), which has long been means of a voltagedivision factor2 γ, whichspecifies the
s
bias at the location of the orbital’s “center of gravity”
e recognized to be a key quantity for the charge trans-
m port in molecular devices2. Due to its simplicity, it soon
ε (V)=ε +γeV. (1)
0 0
. became very popular for interpreting molecular trans-
at port measurements.3–14 Within the initial proposal re- With a potential origin as in Fig. 1, γ ranges from 1/2
m lying upon a tunneling barrier picture,1 Vt, the mini- to +1/2. Eq. (1) corresponds to a potential that i−s flat
- mum of the Fowler-Nordheim(FN) plot ln(I/V2) versus across the molecule and entirely drops at contacts. The
d 1/V determined fromI-V measurements,has been asso- interfacial potential drops are
n
ciatedtothepointwherethebarriertiltedbytheapplied
o bias change from trapezoidal to triangular. This yields δVs = V/2 [ε0(V) ε0]/e=(1/2 γ)V,
c − − −
[ eVt =ε0.6 Later, TVS was interpretedwithin a coherent δVt = [ε0(V) ε0]/e+V/2=(1/2+γ)V. (2)
transport model based on a single level.15 In the (real- −
1 istic) cases where the energy offset is sufficiently larger A positive (negative) γ-value corresponds to a larger
v than the level broadening due to the couplings to elec- (smaller) potential drop at the positive electrode, or, al-
7
trodes, the relationship deduced within the latter model ternatively,to a molecular orbitalenergy shifted upward
9
2 (eVt =1.15ε0 for a symmetrical orbitallocationbetween (downward) by a positive bias.
5 electrodes16) turned out to be not much different from In the wide-band limit, wherein the transmission is
. theoriginal“barriershape”conjecture. Withcertainlim- Lorentzian, the current through a single level (Newns-
1
0 itations,abinitiostudies17,18givemicroscopicsupportto Anderson model) can be expressed analytically as (see,
2 the single-level model. A significant aspect in the TVS e. g., Refs. 16,23,24)
1 analysis18 is to properly account for the potential pro-
2eΓ2 Λ Λ
v: file asymmetry. In the presence of this asymmetry, the I = N g arctan + arctan − . (3)
i FN-plots also become asymmetric, a fact which reflects h Γa (cid:18) Γa − Γa (cid:19)
X
itself in different magnitudes of the transition voltages
Here, N isthe (effective)numberofmoleculescontribut-
r V = V for both bias polarities (V>0), as pointed
a out+tr6ece−ntlty−.19Itisthemainaimofthispa<pertoconsider ingtothecurrent,Λ± ≡ε0(V)±eV/2. Γa ≡(Γs+Γt)/2
andΓ √Γ Γ , Γ being the levelbroadeningsdue to
the ambipolar TVS in detail. g ≡ s t s,t
molecule-electrode couplings. In usual cases of interest,
The remaining part of this paper is organized as fol- Γ ε andvoltagesnotmuchhigherthanV ,25thear-
a 0 t
lows. The general theoretical framework will be pre- gum≪en|ts|oftheinversetrigonometricfunctionsofEq.(3)
sented in Sect. II. In Sect. III, accurate analytical for- are large, and one can approximate
mulas will be given enabling one to directly extract the
2e eV
quantitiesofphysicalinterestfromthetransitionvoltages I N Γ2 . (4)
measured for positive and negative biases. The accu- ≃ h g(ε +γeV)2 (eV/2)2
0
−
racyand the usefulness ofthese theoreticalformulaswill
be illustrated in Sect. IV, by showing that applications The comparison with the results based on the exact
to experimental I-V-data yield an excellent agreement. Eq. (3) shows that Eq. (4) very accurately describes the
Conclusions will be presented in Sect. V. FN-transition; see Fig. 2.
2
tion voltages V as
t1,2
χ ε /V = 2γ+ γ2+3/4,
t1 0 t1
≡ −
χt2 ε0/Vt2 = 2γ pγ2+3/4. (5)
≡ − −
p
Because 1/2 γ 1/2, it is easy to show that
− ≤ ≤
χ > 0 and χ < 0. Therefore, the signs of V and
t1 t2 t1
V are opposite. Denoting by V (> 0) and V (< 0)
t2 t+ t−
the transition voltage for positive and negative polar-
ities, V V and V V for LUMO-mediated
t+ t1 t− t2
≡ ≡
transport(ε >0), while for HOMO-mediatedtransport
0
(ε ε <0) V V and V V . In the HOMO
0 h t+ t2 t− t1
≡− ≡ ≡
case, V < V for γ > 0, whereas V > V for
t+ t− t+ t−
| | | |
γ < 0. For γ = 0, the result V /ε = 2/√3 =1.15 for
t± 0
symmetric case16 is recovered|. |
FIG. 1: Schematicrepresentation of a HOMO-mediated con-
Concerningthesignsingeneral,itisworthnotingthat,
duction (the HOMO much closer to the Fermi level than the
according to Eq. (1), a redefinition of the bias polarity
LUMO,ε ε)inasetupcharacterizedbyasymmetricelec-
h l
≪ (V V) yields a sign change in the division poten-
trodes [asymmetric voltage division γ(> 0) and asymmetric
→ −
molecule-electrode couplings Γs and Γt]. See the main text tial factor (γ → −γ). Therefore, the discussion can be
for details. restricted to the range, e. g., 0 γ 1/2.
≤ ≤
ToillustratetheaccuracyofthetransitionvoltagesV
t±
expressed by the above analytical formulas, a compari-
son with the transition voltages deduced from the exact
Eq. (3) is presented in Fig. 3. As visible there, the ex-
] 2
2
V)
g
Γ
[I/( 1 γ=0.2
log γ=0.1 Vt+_ /εh V /ε , γ=0.1
0.5 1.5 t+ h
γ=0
1 +_V /ε , γ=0
γ=0.1 t+_ h
0
Eq. (3), Γ=0.1 eV 0.5
a
Eq. (4) µ =-eV
-0.5 γ=0.2 Eq. (3), Γ=0.05 eV R 0
a
-4 -3 -2 -1 0 1 2 3 4 1/V(V) -0.5
V /ε , γ=0.1
t- h
-1 µ =-eV
R
-1.5
FIG. 2: Fowler-Nordheim (FN) plots for a single-level coher-
ent transport obtained by means of the exact Eq. (3) and 10 20 30 40 εh/Γa
approximate Eq. (4) for symmetric (γ = 0) and asymmetric
(γ =0)biasprofiles. TheyshowthatforlevelbroadeningsΓa
6
sufficientlysmallerthanitsenergyoffset(ε0=1eV),theFN- FIG.3: ThereducedtransitionvoltagesVt±/εhplottedversus
transition is accurately described by Eq. (4). See the main theinverseoftherelativeHOMO-level(ε0 = εh)broadening
text for details. Γa/εhforgivenpotentialdivisionfactors(γ =−0,0.1)deduced
exactly[Eq.(3),thicklines]andapproximately[Eq.(4),thin
horizontal lines]. Notice the rapid saturation for sufficiently
small level broadenings Γa. See themain text for details.
III. USEFUL ANALYTICAL RESULTS FOR
TRANSITION VOLTAGE SPECTROSCOPY
act results are rapidly approached for small Γ /ε 0.
a 0
→
Graphical results obtained by means of Eq. (5) are pre-
sented in Fig. 4 for values 0 γ 1/2. In the opposite
Eq. (4) can be used to deduce simple analytical ex-
≤ ≤
case ( 1/2 γ <0) they can be deduced by symmetry
pressions of the relevant quantities within the (realistic)
− ≤
V ( γ)= V (γ).
assumptionofalevelbroadeningsufficientlysmallerthan t± t∓
− ±
2
the energy offset, which are exact to (Γ /ε ) . By im- An important consequence of Eq. (5) is that both the
a 0
posing ∂ln(I/V2)/∂(1/V) = 0, oneOobtains the transi- voltage division factor γ and the energy offset ε can be
0
3
determined from V 26 noted,4,16 it is only a small range V > V (if at all28)
t± t
that can be sampled in experiments. The situation can
|ε0| = 2 Vt2++1e0||VVtt++VVtt−−||/3+Vt2−, (6) boGfeeanfuegrreatnhleearranliaimzlyepdtricoFavNled-epxblpoyrteulsnss(iinoIgn/Vsthvκea)lvmidsi.nf1oi/mrVuamr(b1iVt<ra=κry≤Vκt±2)c(.aκ2n9)
siqgnε0 Vt++Vt− also be deduced
γ = . (7)
2 V2 +10V V /3+V2
q t+ | t+ t−| t− χt1,2(κ)≡ Vt1ε,20(κ) = κ 1 1 −κγ±rγ2+ κ24−1!,
−
(8)
κ(κ+1) e V (κ)V (κ)
t+ t−
Vt1,V2t2 |ε0|= κ2−1 Vt2+(κ)+2κκ22|+−11|Vt+(κ)Vt−|(κ)|+Vt2−(κ),
1.8 q (9)
1.6 -ε /V
0 t2
1.4 ε0/Vt1 γ = signε0 Vt+(κ)+Vt−(κ) .
1.2 -V /V 2 V2 (κ)+2κ2+1 V (κ)V (κ) +V2 (κ)
t2 t1 t+ κ2−1| t+ t− | t−
1 q (10)
0.8 The assignment V (κ) V (κ) is the same as dis-
t1,2 t±
→
0.6 cussed above for κ = 2. The usage of κ-values smaller
0.4 than2resultsinlowertransitionvoltagesVt±(κ),29which
0.2 µR=-eV caneasierbe sampledexperimentally. Onthe otherside,
being smaller they are more affected by the relatively
0
0 0.1 0.2 0.3 0.4 γ large (>0.1V12,14) experimental errors.
Froma pragmatic standpoint, Eqs.(6) and (7) as well
as the more general Eqs. (9) and (10) represent the core
ofthispaper: theyenableonetodeterminethequantities
ε /V
h t γ<0 γ>0 ofphysicalinterestε0 andγ fromthe transitionvoltages
1.4 Vt± measured for both bias polarities.
1.2
IV. DISCUSSION OF EXPERIMENTAL DATA
1 25 ε /V
0.8 γ=0.25 γ=−0. -εhh/Vt+t- traTnhspeoarbtomveediaantaeldybtiycaalsrinesguleltlsevheollwdhionsegeenneerrgayl ofoffrsetthies
0.6 7
0.4 =0.067 γ=0 =−0.06 µR=-eV soufffiIRwceiieflsln.ntle1yxtlaanardpgepr1ly2th,thaoenbsetitasrieneseluedlcttsfrootrodett-hhienedeHuxpcOeeMdrimbOre-omnatedadelindaiatnetgda.
γ γ
transportthroughmolecularjunctionsinCP-AFM(con-
1 2 3 4 -V /V
t- t+ ducting probe-atomic force microscopy)1,12 and CW
(crossed-wire)1 setups. For a concrete comparison with
FIG. 4: (a) Results for the transition voltages Vt± as a func- experiment,itisobviouslyimportanttocorrectlyassign,
tion of potential division factor γ deduced from Eq. (5). (b)
out of the two transition voltages measured for opposite
Ratio between the energy offset ε0 = εh < 0 and the tran-
− bias polarities in experiment, which is V and which is
sitionvoltagesVt± ofbothbiaspolaritiesasafunctionofthe t+
transition voltage asymmetry Vt−/Vt+. Each point of this Vt−. Therefore, before entering into details, I note that
curve corresponds to a given bias asymmetry γ, and a few the positive and negative biases have been chosen as the
values important for the discussion in the main text are in- ones utilized in the experiments of Refs. 1,12 discussed
dicated: γ = 0.25 Vt,max/Vt,min = 3.5, γ = 0.067 below.
→ | | →
|ε0/Vt,max| = 0.40,|Vt−| = |ε0|, and |Vt,min/Vt,min| = 1.36. Table I collects the experimental data for Vt± from
See themain text for details. Refs. 1 and 12. They have been used to compute the
values of ε and γ from Eqs. (6) and (7), which are also
h
TVS’s proof of value for molecular electronics is the given in Table I.
fact that the FN-minimum occurs at voltages below the Further, I will use these values of ε and γ to re-
h
valuescorrespondingtoresonanttunnelingwherethedif- produce the available I-V-data for anthracenethiol- and
ferentialconductanceismaximum. Thisisimportantbe- terphenylthiol-based junctions measured in Refs. 1 and
cause, with seldom exceptions,14,27 molecular junctions 1230), respectively. To this aim, I will employ Eq. (4),
cannot withstand such high voltages. Still, as already whichrepresentsaverygoodapproximationofthe exact
4
Eq. (3) for biases not too much larger than the tran-
sition voltages. The prefactor NΓ2 in Eq. (4) can be
g
determined from the experimental linear conductance I(µA)
(N1/2Γ = 0.051eV and N1/2Γ = 0.124eV, respec- 0.4
g g
tively). The theoretical curves obtained in this manner expt. Beebe et al
areplottedagainsttheexperimentalonesinFigs.5aand 0.2 γ=0.068, ε=0.623eV
h
b. Theagreementisexcellent,andthisdemonstratesthe γ=0.25, εh=0.869eV
remarkable accuracy of the present approach based on 0
Eqs. (4) and (1).31
Table I shows that for the molecular junctions of -0.2
Refs. 1 and 12 the energy offset is, in fact, very close to
theestimateofthebarrier-shapeconjectureifthetransi- -0.4 V =-0.85V V =0.62V
tion voltage for positive biases is employed, eV = ε .6 t- t+
t+ h
As visible in Fig. 4b, this does not hold in general but -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 V(V)
only for potential division factors close to γ =0.067.
Molecule, platform Vt+(V) Vt−(V) εh(eV) γ
Anth-SH,CP-AFM a 0.62 -0.85 0.62 0.068
Anth-SH,CW a 0.57 -0.85 0.59 0.086 I(µA)
TP-SH, CP-AFM a 0.67 -0.82 0.64 0.044
TP-SH, CP-AFM b 0.69 -0.85 0.69 0.069 2 eγ=xp0t.0. 6T9a,n ε et= a0l.694eV
TP-SH, CW a 0.66 -0.92 0.67 0.071 γ=0.25, ε=h0.967eV
h
0
TABLE I: Experimental values of Vt± for anth(racene)- and
terphenyl (TP)-based molecules and platforms from Refs. 1
(a)and12(b),andvaluesofε andγ calculated viaEqs.(6) -2
h
and (7). Seethe main text for details.
V =-0.85V V =0.69V
t- t+
-4
Noteworthy,allthecasespresentedinTableIarechar- -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 V(V)
acterized by small positive γ-values, revealing that the
potential drop at the soft contact (e. g., AFM-tip) is
slightly larger, δVt & δVs [cf. Eq. (2)]. So, even a (very) FIG. 5: The experimental I-V-curves measured in CP-AFM
small difference in the interfacial potential drops causes setup by Beebe et al1 and Tan et al12,30 for anthracene- and
a significant polarity dependence of the transition volt- terphenyl-basedjunctions,respectivelyplottedalongwiththe
age. This suggests that, more than the active molecule theoretical curves computed via Eq. (4) with the εh- and γ-
(whichcandiffer,seeTableI),thecontactsareimportant values of Table I. Also shown are the theoretical curves de-
for the V -asymmetry and calls for a systematic exper- duced by using the value γ = 0.25 given in Ref. 18, which
t±
completely disagree with experiments for negative voltages.
imental investigation on the role of the contact groups
Seethe main text for details.
(thiol, amine, etc).
The γ-values of Table I are significantly smaller than
those estimated via DFT calculations (γ 0.2 0.3)18
and that of γ = 0.25 claimed29 to be a≃ppropr−iate for bias are Vt− = −2.17V and Vt− = −2.42V, and they
clearly disagree with the experimental data of Refs. 1
the experiments of Refs. 1 and 12. With the value
and 12, respectively.
γ = 0.25, Eq. (5) yields HOMO-offsets ε = 0.87eV
h
(for V = 0.62V) and ε = 0.97eV (for V = 0.69V) It is worth mentioning that, as visible in Fig. 4, the
t+ h t+
for the I-V-curves shown in Refs. 1 and 12, respectively. Vt+/Vt− asymmetryveryrapidlyvarieswiththe division
Theverysmalldifference(muchsmallerthanthe experi- potentialfactorγ. Itbecomesimportantevenforsmallγ,
mentalinaccuracies)betweenthe abovevaluesandthose asrevealedbyTableI.Althoughmoreorlessasymmetric
given in Ref. 29 (ε 0.85eV and 1.0eV), deduced by I-V-characteristics [I(V) = I( V)] are ubiquitous in
assuming a Lorentzhia≈ntransmission[the key assumption molecularelectronics,such6al−arge−asymmetry(afactor
≃
underlyingEq.(3)24], demonstratesagainthe highaccu- 3.5)betweenthepositiveandnegativetransitionvoltages
racy of the present estimates based on Eq. (4). (Vt+ = 0.62V and Vt+ = 0.69V versus Vt− = 2.17V
−
Whiletheεh-estimates(εh =0.62eVandεh =0.69eV and Vt− = −2.42V, respectively) has not been reported
deduced in the present paper versus ε = 0.85eV and so far.
h
ε =1.0eVofRef.29)cannotbedirectlycheckedagainst A curious aspect should still be noted at this point.
h
experiment, there is a conclusive test to decide which By using the values ε = 0.87eV and ε = 0.97eV (the
h h
potentialdivisionfactor(γ 0.07orγ =0.25)iscorrect. curvesareindistinguishablefromthechoiceε =0.85eV
h
≃
Namely,forγ =0.25,thetransitionvoltagesfornegative and ε = 1.0eV), and γ = 0.25, and by adjusting the
h
5
prefactorinEq.(4)to reproducethe experimentallinear ε (atwhichtheoriginalTVSaimed),butalsothepoten-
0
conductance,theagreementwithexperimentsforpositive tial division factor γ without a notably increased effort:
biases is also good; see Figs. 5a and b. However, as vis- the only quantities to be determined experimentally are
ible there, there is a complete disagreement for negative the transition voltages V for both bias polarities. For
t±
biases. quantitative estimates, simple, very accurate analytical
Although the DFT inability to solve level alignment formulashavebeendeduced,whichhavebeenexcellently
problemsiswellknown,32,33 ithasbeenclaimed18,29 that validatedagainstavailableexperimentaldata. Theexcel-
DFT-estimates of the ratio ε /V could be trusted. The lent agreementfound in the present paper gives a strong
0 t
DFT-values of γ( 0.2 0.3) are based just on the fact support to the correctness of the Lorentzian transmis-
∼ −
thatthe ratioε /V isdeterminedbythevalueofγ. The sion[thekeyassumptionunderlyingEqs.(3)and(4)]and
0 t
above analysis reveals a further limitation of the DFT- rules out possible higher-order contributions ( V2,V3,
∼
based approaches to molecular transport, raising serious etc) to the RHS of Eq. (1) for the molecular junctions
doubts on its reliability for TVS studies. investigated in Refs. 1 and 12.
One should finally note that the above quantity ε The present analysis has emphasized the need to in-
h
is to be understood as the HOMO-offset characteriz- clude both bias polarities in order to obtain correct ε -
0
ing a molecule linked to two (non-biased) electrodes. and γ-estimates. In this context, a specification for
Therefore, it is not surprising that the present values the discussion13,35 of experimental data is helpful. Ac-
(ε = 0.62eV and ε = 0.69eV for anthracenthiol- and cording to a result of previous work,18 the ratio be-
h h
terphenylthiol-based junctions) deduced from transport tweentheenergyoffsetandthetransitionvoltage ε /V
0 t
| |
data can differ from the ultraviolet photoelectron spec- varies from 0.86 to 2. The present paper reconfirms
troscopy (UPS) data.1 Indeed, they do differ: UPS mea- this result with an important specification: the afore-
surements on anthracene- and terphenyl-thiol on gold mentioned range refers to ratio ε /V , which cor-
0 t,min
yielded εUPS 1.7eV and 1.8eV.1 Partly, the differ- responds to the transition voltag|e o|f the smallest mag-
h ≃
ence between ε and εUPS can be attributed to image nitude V min(V , V ). The ratio ε /V ,
effects,29,34 buthI do nothaddress this issue here. whichcotr,rmeisnpo≡ndstothte+t−rants−itionvoltageof|t0h|elat,rmgeaxst
magnitude V max(V , V ), ranges from 0 to
t,max t+ t−
≡ −
0.866 [cf. Eq. (5) and Fig. 4a].
V. CONCLUSION
Although the role of the asymmetric interfacial po- Acknowledgments
tential drops has already been noted in the original
TVS paper1 and considered more recently,18,19,29 a di- The author thank Horst K¨oppel for valuable discus-
rect quantitative analysis of full I-V-curves measured in sions and to Pramod Reddy for providing him the raw
experiments has notbeen attempted in previousstudies. experimentaldataused inFig. 5b. Financialsupportfor
The presentpaper showsthat, by resortingto ambipolar this work provided by the Deutsche Forschungsgemein-
TVS,itispossibletodeterminenotonlytheenergyoffset schaft (DFG) is gratefully acknowledged.
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24 I.BˆaldeaandH.K¨oppel, Phys.Rev.B81,193401(2010). timate of the geometrical average Γg = √ΓsΓt bu≥t not of
25 As already noted,4,16 with few exceptions,14 molecular theindividual electrodes’ hybridizations Γs,t; presumably,
junctions cannot withstand too large voltages and only a the coupling to the soft contact is smaller Γt < Γs. The
small range V &Vt can be sampled in experiments. arithmetic average Γa =(Γs+Γt)/2 could be deduced by
26 Eq. (5) yields expressions different from those given with- usingtheexactEq.(3)insteadofEq.(4),butthelargeex-
out deduction in Ref. 19. For LUMO-mediated transport, perimental inaccuracies (δVt >0.1V1,12) preclude reliable
(ε0 > 0), Ref. 19 gives Vt− /Vt+ = (1 2γ)/(1+2γ), evaluations.
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