Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

Подождите немного. Документ загружается.

146 A. S. Alexandrov

() †

() () () ()

tcttat

αµ µ α

µµ µ

−

≠≠...

∑

∏

(15)

These correlation functions should be calculated up to the first order with

respect to the hopping integrals

. In this order, they satisfy the set of

coupled equations as

()

[() ( )] ()

dA t

itnt

αµ

αµ µ α µ

µµ

−

≠...

=−

∑

∏

(1) ()

() [ ( 1) ] ()

kk k

UA t N U A t

αµ α αµ

++−− .

(16)

When we have a finite number of molecular states, the set is finite. One

readily solves the set in the stationary case, when

()

and

()

αµ

do not

depend on time. For the

d -fold degenerate energy level, the one-particle

correlation function is found as

(1)

()

[()]

kkk

Anf t

rU i

αµ α αµ

−

=− .

−+

∑

(17)

Substituting it into Eq. (14), we obtain the stationary rate equation for the

electron density on the molecule as

[()]()0

nfrUZn

−

∑∑

(18)

Switching effect

Now using Eqs. (13, 8, 9) the current is found as

[( ) ( )] ()

rU Z n

−

=− ,

∑

(19)

where

()

IdI=/ . Let us consider two-fold, four-fold, and six-fold

degenerate molecular level. For

the kinetic equation is linear in n ,

and there is only one solution,

Negative U Molecular Quantum Dot 147

(0)

2[(0)()]

ffU

αα

+−

∑

(20)

The current through a two-fold degenerate molecular dot is found as

1221

(0)[1 ( )] (0)[1 ( )]

2[(0)()]

ffUf fU

ffU

αα

−−−

+−

∑

(21)

21(1)

−

−− .

(22)

For

4d = it has two physical roots, 0n

and

(3 5 ) 2 0 38n

−/≈.. In

this voltage range

(0) ( ) 0ffrU==, but

11 1

() (2) (3)1fU f U f U

== at

0T =. Using the sum rule

() 1

−

∑

and the rate equation (18), the

current is simplified in this voltage range as

. Hence we obtain two

stationary states of the molecule with low (zero at

) and high current,

076II≈. for the same voltage in the range 2( ) 2UeV

∆

−| | < < ∆, Fig. 3.

For

6d = , the kinetic equation has two physical roots in this voltage range,

0n = and 048n ≈. , which corresponds to 0I

and

096II≈. ,

respectively. Above the standard threshold,

2eV >∆, where

()1frU=

and

()0frU

there is only one solution, 05n

. with the current

II=.

Hence, different from the non-degenerate or double-degenerate MQD,

the rate equation (18) for

2d > has two physical roots in a certain voltage

range and the current-voltage characteristics show a hysteretic behavior,

Fig. 3. Ermakov [9] also calculated the I-V curves of the 4-fold degenerate

dot with the Coulomb and electron-phonon interactions, and found a

switching effect in the numerical I-V curves. However, Ref. [9] obtained an

unphysical population of each molecular state,

. Moreover the exact

theory [8] showed that the averaging procedure over phonons in the

There is no current bistability in this case. Moreover, if the temperature is

low (

∆

,| | ) there is practically no effect of correlations on the

current, (2)

VeV V

≈

Θ||−∆/||. Remarkably, four-fold or higher-

degenerate negative

U dots reveal a switching effect. In this case, the

kinetic equation is nonlinear, allowing for a few solutions. If

2( )eV U<∆−||, the only physically allowed solution of Eq. (18) for 4d =

and 6d = at zero temperature is 0n

. If 2( ) 2UeV

∆

−| | < < ∆, and

0T = , the rate equation is reduced to

148 A. S. Alexandrov

transformed Hamiltonian used in Ref. [9] cannot be applied in MQD since

it misses the vibron-side bands.

Fig. 3. The current-voltage hysteresis loop in the degenerate negative-U model of

a molecular quantum dot for two temperatures.

Mean-field approximation and switching

ˆˆ ˆ

UUnUnn

nn n

µµ

µµ µ

′

′′ ′

≠≠ ≠

≈−

∑∑ ∑

Then the MFA DOS is given by

() [ ( 1)]Ud n

ωδω

−−. Using the

Fermi-Dirac Golden rule the rate equation for

n becomes

We can better understand the origin of the switching phenomenon by taking

the limit

1d >> . The physical roots to Eq. (22) are 0n

and 05n =. in

this limit with the current

and

, respectively. This is precisely

the solution of the problem in the mean-field approximation (MFA), which

is a reasonable approximation for

 Indeed, using MFA one replaces

the exact two-body interaction in the Hamiltonian for a mean-field potential

Negative U Molecular Quantum Dot 149

2[(1)]

nfnUd

=− Γ +Γ − .

∑

(23)

For

0T = there are two stationary solutions of Eq. (23), 0n

and 05n =.

in the voltage range

2( ) 2UeV∆− | | < < ∆

, and only one solution, 05n =.

for

2eV >∆, where (1)2UUd=−/.

The MFA current is found as

(2 ) (2 )

nU=−.

(24)

Combining this equation and the rate equation (23) with

0dn dt/= , we

obtain the I-V characteristic equation as

()

(1 ) 1

(1 ) sinh 2

ln cosh 2 ,

ReVT

eV T

−=−×

∆∆



−/





where

[]

1coth(2)

sinh ( 2 )

RjeVT

eV T



=− / − .





(25)

The

IV/ curves are shown in Fig. 3 for different temperatures and

04U||=.∆.

Interestingly, the temperature narrows the voltage range of the

hysteresis loop, but the transition from the low (high)-current branch to the

high (low)-current branch remains discontinuous.

Let us examine the stability of each branch in the framework of the MFA

rate equation (23). Introducing small fluctuations of the electron density as

( ) exp( )nt n n t

=+ and linearizing Eq. (23) with respect to n

we find

the increment

2cosh

UnUeV

−



||Γ ∆− ||−/

=− Γ+





cosh

UnUeV

−



||Γ ∆− ||+/





(26)

One can see from this equation that at temperatures





the low-

current branch (

0)n = looses its stability at the threshold

2Ve

∆/ , while

150 A. S. Alexandrov

the high-current branch looses its stability at

()Ue

∆

−| | / .

In the

voltage range

VVV<< both branches are stable, 20

≈

−Γ< .

Finally, let us analyze the effect of a splitting of the degenerate molecular

level on the bi-stability. The degeneracy could be removed because of Jahn-

levels are evenly distributed in a band of a width

W. Then the MFA rate

equation (23) is modified as

2()

µα

=− Γ +Γ + ,

∑

(27)

where

′

∑

For 0T

in the stationary regime ( 0)dn dt

/= it has

two solutions,

0N = and 2Nd

/, in the voltage range

VWVV

with the current

0j = and 1

, respectively. We conclude that the level

splitting

WU<| | leads to a narrowing of the voltage range of the bi-

stability similar to the temperature narrowing shown in Fig. 3.

Conclusion

In conclusion, we have reviewed the negative Hubbard-U model of

tunneling through molecular quantum dots (MQD) with attractive

correlations. The degenerate MQD with effective attraction between

carriers has a hysteretic volatile memory if the degeneracy of the molecular

level is larger than two,

2d >. The origin of the bistability is illustrated in

Fig. 2. When the current flows through MQD, the "HOMO-LUMO" gap is

renormalized down to a lower value due to attractive correlations, so the

current-off voltage

V turns out smaller than the current-on voltage

V . The

hysteretic behavior strongly depends on temperature. The current bistability

vanishes above some critical temperature. Among potential candidates for

the negative

U quantum dot are single

C molecules ( 6d

), where the

electron-vibronic coupling proved to be particularly strong [18], or other

Teller distortions and/or the coupling with the leads. We assume that

1d 

carbon nano-structures including short nanotubes (

1d  ) connected to

metal electrodes. Other likely candidates are mixed-valence molecular

complexes [21].

Negative U Molecular Quantum Dot 151

References

1. Aviram A, Ratner M (eds) (1998) Molecular Electronics: Science and

Technology. Ann NY Acad Sci, New York; Nitzan A, Ratner MA (2003).

Science 300:1384

2. Wu W, Jung G-Y, Olynick DL, Straznicky J, Li Z, Li X, Ohlberg DAA, Chen

Y, Wang S-Y, Tong WM, Williams RS (2004). Applied Physics A, accepted

3. Heath JR, Stoddart JF, Williams RS, Chandross EA, Weiss PS, Servic R

(2004). Science 303:1136

4. Alexandrov AS, Demsar J, Yanson IK (eds) (2004) Molecular Nanowires and

Other Quantum Objects. NATO Science Series, Kluwer Acad. Pub., Dordrecht,

the Netherlands

5. Collier CP et al. (1999). Science 285:391; Chen J et al. (1999). Science

286:1550; Gittins DI et al. (2000). Nature (London) 408:677

6. Stewart D, Chen Y, Williams RS, unpublished

7. Alexandrov AS, Bratkovsky AM, Williams RS (2003). Phys Rev B 67:075301

8. Alexandrov AS, Bratkovsky AM (2003). Phys Rev B 67:235312

9. Ermakov VN (2000). Physica E 8:99

10. Alexandrov AS (1983). Zh Fiz Khim 57:273 [Russ J Phys Chem 57:167];

(1998) Models and Phenomenology for Conventional and High-temperature

Superconductivity. Course CXXXVI of the Intenational School of Physics

‘Enrico Fermi’, G Iadonisi, JR Schrieffer, ML Chiofalo (eds), Amsterdam: IOS

Press, p 309

11. Su WP, Schrieffer JR (1980). Proc Natl Acad Sci 77:5626

12. Feldblum A et al (1982). Phys Rev B 26:815

13. Chance RR, Bredas JL, Silbey R (1984). Phys Rev B 29:4491

14. Ramsey MG et al (1990). Phys Rev B 42:5902

15. Steinmuller D, Ramsey MG, and Netzer FP (1993). Phys Rev B 47:13323

16. Swanson LS et al (1993). Synth Metals 55:241

17. Zhitenev NB, Meng H, Bao Z (2002) Phys Rev Lett 88:226801

18. Park J, Pasupathy AN, Goldsmith JI, Chang C, Yaish Y, Retta JR, Rinkoski M,

Sethna JP, Abruña HD, McEuen PL, Ralph DC (2000). Nature (London)

417:722

19. Alexandrov AS, Kornilovich PE (1999). Phys Rev Lett 82:807; (2002) J Phys:

Condens Matter 14:5337

20. Wingreen NS, Meir Y (1994). Phys Rev B 49:11040, and references therein

21. Wilson JA (2001). J Phys: Cond Matter 13:R945

22. Meir Y, Wingreen NS (1992). Phys Rev Lett 68:2512

23. Anderson PW (1975). Phys Rev Lett 34P:953

24. Street RA, Mott NF (1975). Phys Rev Lett 35: 1293

25. Micnas R, Ranninger J, Robaszkiewicz S (1990). Rev Mod Phys 62:113, and

references therein

26. Alexandrov AS, Kabanov VV (1996). Phys Rev B 54:3655

Configuring a Bistable Atomic Switch by

Repeated Electrochemical Cycling

F.-Q. Xie

, Ch. Obermair

, Th. Schimmel

1,2

Institute for Applied Physics, University of Karlsruhe, D-76128 Karlsruhe,

Germany

Institute of Nanotechnology, Forschungszentrum Karlsruhe, D-76021

Karlsruhe, Germany

Abstract: Recently, we have reported the stable and reproducible operation of

atomic-scale switches, which allow us to open and close an electrical

circuit by the controlled reconfiguration of silver atoms within an

atomic-scale junction. Here, we investigate the operation of such

atomic quantum switches, and we study in more detail the process

during which these switches are formed by repeated electrochemical

deposition and dissolution. We find that only after repeated

deposition/dissolution cycles, a bistable contact is formed on the

atomic scale, which allows to switch between a configuration where

the contact is closed, the conducting state or “on”-state, and a

configuration where the contact is open, the non-conducting state or

“off”-state. We demonstrate that before a bistable contact is formed,

irregular changes of the contact conductance are observed as a

function of the electrochemical cycling process. A sudden transition

to regular switching of the contact between a well-defined “on”-state

and the non-conducting “off”-state is observed, which indicates the

formation of a bistable contact configuration. Conductance

quantization at integer multiples of the conductance quantum

= 2e

/h (≈ 1/12.9 kΩ) is found at room temperature for the “on”-

state conductance.

Keywords: Quantized Conductance, Atomic-Scale Point Contact, Electro-

chemical Deposition, Silver, Atomic Quantum Switch, Nano-

Electronics, Molecular Electronics

153

R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 153–162.

154 F.-Q. Xie, Ch. Obermair, Th. Schimmel

Introduction

Due to their interesting physical properties and technological perspectives,

atomic-scale metallic point contacts are an object of intensive investigations

by numerous groups [1-12]. As the size of these constrictions is smaller

than the scattering length of the conduction electrons, transport through

such contacts is ballistic and as the width of the contacts is on the length

scale of the electron wavelength, the quantum nature of the electron is

directly observable. The conductance is quantized in multiples of 2e

/h,

where e is the charge of an electron and h is Planck´s constant.

Experimentally, two different approaches are available for the fabrication of

these metallic quantum point contacts: mechanically controlled deformation

of thin metallic junctions [1-5] and electrochemical fabrication

techniques [6-12]. In both cases, conductance quantization was observed

experimentally even at room temperature.

As the contact in such junctions is ultimately formed by only one or a

few individual atoms, it has already been suggested that the controlled

movement of the contacting atom(s) could lead to a switch or a relay on the

atomic scale. By moving only one atom in and out of position, a quantized

electrical current could be switched on and off. Such a development would

not only mean that the ultimate limit for the size of a switch would be

reached. It would also provide the basic functional unit for the potential

future field of quantum electronics. However, building such a device will

require reproducible control of the position of individual atoms within the

quantum point contact by means of a third electrode, the control electrode

or gate electrode.

In the past, first experiments demonstrating controlled switching of

atomic positions were reported. Eigler et al. [13] showed the switching of

the position of a xenon atom between a scanning tunnelling microscope

(STM) tip and a nickel sample surface by applying voltage pulses between

STM tip and sample. Fuchs and Schimmel [14] demonstrated the switching

of atomic positions in a solid surface at room temperatures by applying

voltage pulses with the tip of an STM. Another approach was performed

recently by Terabe et al. [15], who switched the conductance across a small

gap containing a solid electrolyte by voltage pulsing. In these experiments,

however, there was no way of controlling the conductance across the

atomic-scale junction by means of an independent third electrode, the

control electrode or “gate” electrode. Such an independent control

electrode, however, is necessary for the fabrication of transistors or relays

on the atomic scale.

Configuring a Bistable Atomic Switch 155

Recently, we could demonstrate the first implementation of a transistor

on the atomic scale [16, 17]. The atomic-scale transistor can be reversibly

switched between a quantized conducting on-state and an insulating off-

state by applying a control potential relative to a third, independent gate

electrode. For this purpose, an atomic-scale point contact is formed by

electrochemical deposition of silver within a nanoscale gap between two

gold electrodes, which subsequently can be dissolved and re-deposited, thus

allowing to open and close the gap. However, it turned out that only after

numerous such deposition/dissolution cycles, a contact is formed which

reproducibly opens and closes as a function of the voltage applied to the

gate electrode. Here, we study the effect of this electrochemical cycling

process, and we discuss the mechanisms of formation and operation of the

atomic-scale quantum transistor.

Experimental

The experimental setup is illustrated in Fig. 1. Atomic-scale silver quantum

point contacts were fabricated by electrochemical deposition from aqueous

solution. The electrolyte consisted of 1 mM AgNO

+ 0.1 M HNO

in bi-

distilled water. For the deposition of the point contacts, a gap was

fabricated between two gold electrodes on a glass substrate. On the ends of

these gold electrodes which form the gap, silver was deposited

electrochemically until the gap was closed. Three types of electrodes are

involved in this electrochemical experiment: a) the two gold electrodes,

which served both as leads for the atomic-scale point contact and as

electrochemical working electrodes, on which the metal is deposited from

the electrolyte, b) the counter electrode for the electrochemical deposition

current and c) the quasi-reference electrode providing the electrochemical

reference potential.

The potential difference between the working electrodes and the quasi-

reference electrode is controlled by a “gate voltage” applied to a

bipotentiostat.

Within the atomic-scale transistor described below, the two gold working

electrodes served as “source” and “drain” while the input of the

bipotentiostat which controls the potential of the working electrodes

relative to the quasi-reference electrode in the following is referred to as the

“gate” electrode. By variation of the voltage applied to this “gate”

electrode, the conductance between “source” and “drain” could be switched

(see below).

156 F.-Q. Xie, Ch. Obermair, Th. Schimmel

Fig. 1. Illustration of the experimental setup. Silver quantum point contacts are

electrochemically grown within a nanoscale gap between two electrodes deposited

on a substrate. After repeated electrochemical deposition/dissolution processes, a

bistable contact configuration is formed, and the reproducible switching of the

contact between the two Au working electrodes is achieved by means of an

The gold working electrodes (thickness approx. 100 nm), were covered

with an insulating polymer coating except for the immediate contact area

where the electrode ends were next to each other, forming a sub-micron or

nanometer-scale gap. Typical gap widths were of the order of 100 nm.

Silver wire of 0.25 mm in diameter with 99.9985% purity was used for the

counter electrode and the quasi-reference electrode.

For conductance measurements, an additional voltage of -12.9 mV was

applied between the two gold electrodes.

When applying a positive

potential at the gate electrode (i.e. gold working electrodes having a

negative bias relative to the electrochemical reference electrode), silver

islands formed on the two gold electrodes, two islands finally meeting each

other by forming an atomic-scale contact. During deposition, the

conductance between the two gold electrodes was continuously measured.

As soon as a predefined conductance value was reached, the computer-

controlled feedback stopped further deposition of silver on the working

electrodes.

independent gate electrode.

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

Подождите немного. Документ загружается.