Conductance through a one-atom point contact

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998

Conductance through a one-atom point contact F. Yamaguchi, Y. Yamamoto† ERATO Yamamoto Quantum Fluctuation Project, Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, U.S.A.

Linear conductance and current–voltage characteristics in a one-atom point contact are calculated. A one-atom point contact is modeled by a double-barrier tunnel junction with a central island which is either a single atom or an atomic cluster. A finite electron reflection and induced charge at both sides of the tapered constriction alter the conductance from the quantum unit of conductance in a perfectly smooth point contact. The conductance is expressed in terms of a difference 1W between the work function of the electrodes and the atomic energy level, a charging energy U and an energy level broadening h¯ 0 in our simple model of a one-atom point contact. c 1998 Academic Press Limited

Key words: conductance, atomic junction, charging energy, work function, energy level broadening.

1. Introduction In a quantum point contact in a semiconductor two-dimensional electron gas (2DEG) system, the electron momentum is quantized in the transverse direction if the constriction has a width comparable to the Fermi wavelength. Each of the transverse modes gives the quantum unit of conductance G Q = 2e2 / h as is interpreted by the Landauer formula [1] if the point contact reduces its transverse size adiabatically towards the constriction and the electrons propagate without reflection at the constriction. In quantum-point contacts fabricated in 2DEG systems [2], the adiabatic constrictions are indeed smooth enough over a distance larger than the Fermi wavelength (λF ∼ 40 nm) and the quantized conductance steps were observed with decreasing size of the constrictions. The studies about the quantized conductance in point contacts in semiconductor 2DEG systems were followed by atomic-scale point contacts with a mechanically controllable break (MCB) junction technique [3] and scanning tunneling microscopy (STM) [4]. Those point contacts, in principle, have constrictions as small as single atoms and such a point contact is not always adiabatically tapered towards the one-atom constriction in a length scale of the Fermi wavelength. For a three-dimensional metallic point contact, the ˚ is about the same order as the size of the constriction and electrons should Fermi wavelength (λF ∼ 5 A) suffer partial reflection. The finite reflection at both sides of the constriction confines the electrons between the two tapered structures and the energies of the confined electrons are quantized. Because of this small (atomic scale) confinement, the energy-level separation can be of the order of 1 eV or larger. The reflection of the electrons at the constriction and the discrete energy of the confined electrons increase induced charge at † Also at: NTT Basic Research Laboratories, 3-1 Morinosato-Wakamiya Atsugi, Kanagawa, 243-01, Japan.

0749–6036/98/030737 + 10 $25.00/0

sm970537

c 1998 Academic Press Limited

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Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 Reservoir

d

a Reservoir ˚ and Fig. 1. A one-atom point contact made of a crystal with bcc structure with (111) orientation. Interlayer distance d = 0.911 A ˚ for an one-atom contact of tungsten. nearest-neighbor distance within a layer a = 4.46 A

the constriction. This could correspond to a double-barrier tunnel junction with a central island (or quantum dot), in which a change in the electrostatic potential can be of the order of 1 eV or higher due to an extremely small system size. In this paper, the conductance through such a one-atom point contact is calculated by the following considerations. (1) The deviation from a perfectly smooth point contact is taken care of by a difference between a work function of an electrode and one of discrete energy levels in a central island. A one-atom point contact is modeled by a central island (either a single atom or an atomic cluster) and two electrodes (reservoirs) with identical work functions connecting to the central island. If the electrodes are made of different species from the central island, the difference between the work function of the electrodes and the energy level of the central island is treated in a similar way. (2) The reflection at the constriction, which determines a linewidth of a confined state, is expressed by dissipative couplings between the central island and the reservoir electrodes. (3) The change in the electrostatic potential is included as a change in the energy level of the central island.

2. Transition from quantum-point contacts to double-barrier tunnel junctions—localized states in one-atom point contacts In a one-atom point contact, electrons are either localized in a single atom between two tapered structures or completely delocalized in the entire structure. An electronic state is determined by the spatial configuration of constituent atoms. Figure 1 shows a one-atom point contact which has the possible sharpest constriction made from crystals with bcc structures. For example, such a one-atom point contact made from a tungsten ˚ and nearest-neighbor distance in a layer a = 4.46 A. ˚ The crystal has an interlayer distance d = 0.911 A electronic state of this one-atom point contact is calculated by separating the atomic clusters including the central single atom (system) from the rest of the structure (reservoir electrodes) as shown in Fig. 1. Figures 2A and 2B show the tight binding calculation results for the electronic state of a pyramidal atomic cluster structure consisting of one atom in the first layer, three atoms in the second layer and six atoms in the third layer which is followed by a reservoir electrode. When the interlayer distance d is relatively small, there is no localized state near the Fermi level due to strong coupling between atoms (Fig. 2A). However, if the interlayer

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 1.0 1st layer 2nd layer 3rd layer

Density of states (eV–1)

0.8

739 A

0.6

0.4

0.2

0.0 –6

–4 –2 0 2 4 Energy relative to Fermi level (eV)

6

1.0 1st layer 2nd layer 3rd layer

Density of states (eV–1)

0.8

B

0.6

0.4

0.2

0.0 –6

–4 –2 0 2 4 Energy relative to Fermi level (eV)

6

Fig. 2. Local density of states per atom in the first, second and third layer in the atomic cluster which has one tungsten atom in the first layer, three in the second layer and six (connected to a reservoir) in the third layer (the lower half of the one-atom point contact shown in ˚ and B, 2.5 A, ˚ and the nearest-neighbor distance Fig. 1), calculated with the tight-binding calculation. Interlayer distance d is A, 0.911 A ˚ The Fermi level is determined from charge neutrality in the atomic cluster. within a layer a = 4.46 A.

distance d becomes relatively large, there emerges a narrow peak in the local density of states in the first layer with the reduced densities of states in the second and third layers, which suggests the existence of a strongly localized state in the topmost single atom (Fig. 2B). A finite energy broadening h¯ 0 of the localized

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 Charging energy, energy broadening (eV)

740

Charging energy Energy broadening

1.2

1.0

0.8

0.6

0.4 1.0

1.5 2.0 2.5 Interlayer distance d (Å)

Fig. 3. Charging energy and energy broadening of the atomic energy level of a central tungsten atom at the constriction of the one-atom point contact of tungsten as the interlayer distance changes. The one-atom point contact is assumed to consist of a tungsten atom and two reservoirs with two atomic buffer layers as shown in Fig. 1. The decay rate of six atoms in the lower buffer layer is calculated by ˚ assuming each of the six atoms is connected to a one-dimensional wire with interatomic distance at 2.73 A.

state is due to tunnel coupling of the topmost atom to the reservoir electrodes. We also note that there is a difference 1W between the Fermi level of the reservoir electrode and the discrete energy level of the localized state. In order to estimate the charging energy U associated with the localized state, we modeled the atomic cluster by capacitively coupled distributed metal spheres. For example, tungsten atoms in the atomic cluster ˚ which have the same ionization energies as a real can be replaced with metal spheres with radius 1.12 A tungsten atom [5]. The central tungsten atom at the constriction has a finite energy broadening h¯ 0 due to tunneling to the two tapered structures and also a charging energy U due to electrostatic coupling to them. If the interlayer distance d increases, the charging energy U increases while the energy broadening h¯ 0 decreases because electrons are localized at the central tungsten atom, as shown in Fig. 3. As is evident from this result (‘spatial-isolation-induced localization’), the charging energy U is no longer negligible compared with the energy broadening h¯ 0 for a large d value and a double-barrier tunnel diode model [6] is more appropriate for a one-atom point contact rather than a standard quantum-point contact model [4]. The localization of electrons to a single atom occurs also when one applies an electric field across a one-atom point contact (‘field-induced localization’), because the electric field alters the atomic energy levels from layer to layer in the atomic cluster. The difference in the atomic energy levels induces the localization of electrons to a single atom. This was confirmed in the recent field emission spectrum measurement from a homogeneous single-atom tip [7]. If a central atom at the constriction is replaced with a foreign atom, it also induces a difference in atomic energy levels between the foreign atom and the rest of the constituent atoms. Electrons are localized at the central foreign atom because of this difference (‘foreign-atom-induced localization’) [8]. This was also demonstrated in the field emission spectrum measurement from a heterogeneous single-atom tip [9]. These mechanisms cause strong electron reflection at the constrictions and the localization electrons to

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single atoms, and the one-atom point contacts behave like a system consisting of two tunnel barriers and a central island between them.

3. Model of one-atom point contacts The one-atom point contacts discussed in the previous section are modeled by an atom and a pair of metal electrodes (L: left, R: right) with the identical work function W . The total Hamiltonian of the system is X X X † † εLk cLkσ cLkσ + εRk cRkσ cRkσ + ε0 cσ† cσ + U (c↑† c↑ − 12 )(c↓† c↓ − 12 ) H= k,σ ={↑,↓}

+

X k,σ

† (VLk cLkσ cσ

σ

k,σ

X † + h.c.) + (VRk cRkσ cσ + h.c.),

(1)

k,σ

† † and where cLkσ PcRkσ create electrons with energies εLk and εRk in the electrodes L and R and spins† {σ =↑, ↓} respectively, k,σ stands for summation over electron wavenumbers {k} and spins {σ }, and cσ creates an atomic electron with energy ε0 (= −E 0 measured from the vacuum) and spin σ . The atom between the two electrodes is assumed to have only one discrete energy level which accommodates two electrons (with spins ↑ and ↓), in the energy scales such as an applied bias voltage across the atom, energy broadening due to the dissipative couplings of the atom to the electrodes, the charging energy and thermal fluctuation. For an atom which has one electron when it is neutral, the fourth term in eqn (1) describes the change in the electrostatic energy due to the induced charge at the boundaries of the constriction when the atom deviates from its neutral state. This electrostatic energy U is calculated in terms of capacitances (CL and CR ) of the atom to the electrodes as e2 /{2(CL + CR )}. The actual value of CL and CR depend on the self-capacitance of an isolated atom and the configuration of the constituent atoms in the one-atom point contact. The last two terms in eqn (1) stand for tunneling of electrons through the constriction. The electrochemical potentials µL and µR of the electrodes L and R are located below the vacuum by the work function W , µ = −W , when an applied bias voltage across the atom is zero. A finite bias voltage V induces a difference between the electrochemical potentials, V = (µL − µR )/e. The difference between W and E 0 , 1W ≡ W − E 0 , describes ‘abruptness’ of the constriction, i.e., 1W is zero if the one-atom contact is made of identical atoms and the constriction is adiabatically tapered, but is finite if the tapered constriction is not gradual enough or a central atom is not identical to the rest of the atoms of the constriction.

4. Conductance through one-atom point contacts The conductance through the one-atom point contact is calculated with the T -matrix [10]. The linear conductance G through the constriction is written as Z 2e2 1 dεF(ε) f (ε){1 − f (ε)} , (2) G= h kB T µL ,µR →µ=−W where f η (ε) is the Fermi distribution function of the electrode η (L or R), defined as {exp[(ε−µη )/kB T ]+1}−1 for a temperature T , and both f L and f R approach f (ε) = {exp[(ε − µ)/kB T ] + 1}−1 in the limit of zero-bias voltages. The transmission coefficient F(ε) has a form of [11] 2 hni hni 1 − 2 2 2 , (3) + F(ε) = (h¯ 0) U 61 U 62 U U ε − ε0 − 2 + i h¯ 0 − ε−ε + U +i¯h 0−6 ε − ε0 + 2 + i h¯ 0 + ε−ε0 − U +i¯h 0−63 0 3 2

2

where hni is an average number of atomic electrons. The decay rates of the atomic electrons to the electrodes are related to the matrix elements VLk and VRk in eqn (1) and the density of states of the electrodes ρL and ρR

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Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 VL

tR

VR

ts

tR

Fig. 4. An atomic point contact which consists of two tapered electrodes (outside the dashed lines) and an atomic wire between them. If the electrodes are tapered adiabatically, the whole system can be regarded as an one-dimensional chain (as indicated by white and black circles). The electron-hopping matrix element between atoms in the electrodes is tR and that between atoms in the atomic wire is tS . The atomic wire is weakly coupled to the electrodes as described by VL and VR . The couplings VL and VR are smaller than tR and tS and treated as perturbations [10].

as (2π/h¯ )ρL |VLk |2 and (2π/h¯ )ρR |VRk |2 , when the energy dependence of the matrix elements and the density of states can be neglected. For simplicity, both decay rates are set to be 0 here. In eqn (3), 6i (i = 1, 2, 3) is defined as   X X 1 1 Ai |Vηk |2 + , 6i = (ε − ε0 ) − (εηk − ε0 ) (ε − ε0 ) + (εηk − ε0 ) η={L,R} k where A1 = f η (εηk ), A2 = 1 − f η (εηk ) and A3 = 1. For zero temperatures, the conductance eqn (2) becomes G= where F(ε)|ε=µ

= (h¯ 0)2 −1W +

U 2

1 − hni 2 + i h¯ 0 1 −

2e2 F(ε)|ε=µ(=−W ) , h

U −1W − U2 +3i¯h 0

+

(4)

hni 2

−1W −

U 2

 + i h¯ 0 1 −

U −1W + U2 +3i¯h 0

2  ,

(5) since f (ε = µ) = 12 , 61 = 62 = 12 63 = −i h¯ 0. Equation (5) describes the conductance in terms of by 1W , h¯ 0, hni and U . 4.1. One-atom point contact with 1W = 0 If the constriction is tapered adiabatically towards the central island in which electrons are confined, the difference between the work function W of the electrode and the energy level of the atomic electrons in the central island, 1W , is negligibly small compared with other energy scales (h¯ 0, kB T and U ) and the average number of atomic electrons is hni =1. The term F(ε) in eqn (5) is reduced to !2 3(h¯ 0)2 . (6) F(ε)|ε=µ(=−W ) =
U 2 + 3(h¯ 0)2 2 4.1.1. U/h ¯0  1 In sufficiently smoothly tapered constrictions, where the potential change due to the induced charge at the boundaries is negligible compared with the energy level broadening h¯ 0, eqn (6) is reduced to one so the conductance (eqn (2)) is identically equal to the quantum unit of conductance G Q . Each mode carries G Q when both 1W and U are negligible compared with the energy level broadening h¯ 0. As an extension of this discussion, the conductance through an atomic wire consisting of more than one

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Conductance (units of e2 h–1)

1.0 T = 4.2 K T = 300 K

0.8 0.6 0.4 0.2 0.0 5

10 15 Number of atoms

20

˚ and Fig. 5. Even–odd conduction oscillation as a function of the number of atoms in the atomic wire. The interatomic distance is 4 A ˚ [10]. the system-reservoir spacing is 6 A

atom connected to the two adiabatically tapered electrodes shown in Fig. 4 is calculated [10]. Under the assumption that both electrodes are adiabatically tapered towards the atomic wire and the whole system is made from identical atoms, 1W can be set zero. If the separation between the tapered electrodes and the atomic wire is sufficiently small, the charging energy U can be also neglected compared to the energy level broadening h¯ 0. The case where the atomic wire consists of one atom corresponds to a one-atom point contact which gives the quantum unit of conductance G Q at low temperatures as shown in Fig. 5. When the number N of constituent atoms of the atomic wire changes, the transmission coefficient F(ε) has a peak at the Fermi level (which gives a large conductance) for odd numbers of N and a valley at the Fermi level (which gives a small conductance) for even numbers of N . This conductance oscillation in the number of the atoms N originates from the mutual coupling between two adjacent constituent atoms in the atomic wire and the resulting modulation of the density of states of the atomic wire. The conductance oscillation is suppressed when the thermal energy kB T exceeds the widths of the peaks in the density of states. 4.1.2. U/h ¯0  1 If the tapered constriction is smooth enough to eliminate the difference between the work function of the electrodes and the atomic energy level, but the separation is not small enough to eliminate the induced charge at the boundaries of the constriction, U remains finite. In this case of U/h¯ 0  1, the transmission coefficient (eqn (6)) is suppressed as 144(h¯ 0/U )4 . 4.2. One-atom point contact with 1W 6= 0 The average number of atomic electrons hni at zero bias voltage V = 0 which appears in eqn (6) now deviated from one and given by 2f− , (7) hni = 1+ f− − f+ where f ± is defined as {exp[(1W ±U/2)/kB T ]+1}−1 as shown in Fig. 6 [6]. Figure 7 shows the conductance through one-atom point contacts with 1W/h¯ 0 = 0, 1, 2 and 3 as functions of U/h¯ 0 at zero temperatures. The conductance for U = 0 is suppressed as G Q × (h¯ 0)2 /{(1W )2 + (h¯ 0)2 }. This explains that an insufficient

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Number of atomic electrons hni

2.0

1.5

1.0

0.5

0.0 -2

-1

0 1W / U

1

2

Fig. 6. The average number of atomic electrons hni at zero bias voltage as a function of 1W/U for temperature kB T = 0.05U .

1.0 1W / ~ 0 = 0 1W / ~ 0 = 1 1W / ~ 0 = 2 1W / ~ 0 = 3

Conductance (units of 2e2 h–1)

0.8

0.6

0.4

0.2

0.0 0

2

4

U / ~0

6

8

10

Fig. 7. Conductance though a one-atom point contact as a function of the charging energy U . At points where ±U/2 = 1W , the conductance has maximum values because the difference 1W between the work function of the electrodes W and energy level E 0 happens to be absorbed because of an atomic-energy shift due to the charging energy U .

smoothness at the tapered constriction reduces the conductance from the quantum unit of conductance G Q . The conductance has a maximum value at U/2 = 1W , where the Fermi level of the electrodes coincides with one of the two quasi-energy levels of the atom, −E 0 − U/2. (To add one electron to the central island

Superlattices and Microstructures, Vol. 23, No. 3/4, 1998 1.0

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A

Current (units of e0)

0.5

0.0

–0.5 |1W| / U = 0 |1W| / U = 0.5 |1W| / U = 1.5 –1.0 –4

–2 0 Bias voltage eV/U

2

4

2.0

Number of atomic electrons hni

1W / U = –0.5 1W / U = –1.5 1.5

1.0

0.5 1W / U = 0 1W / U = 0.5 1W / U = 1.5 0.0 –4

–2 0 Bias voltage eV/U

2

4

Fig. 8. A, Current through a one-atom point contact and B, the average number of atomic electrons vs. bias voltage [6]. For |1W | < U/2, the number of atomic electrons is 1 at V = 0. Electron tunneling is blocked until the Fermi level of one of the electrodes coincides with one of the quasi atomic energy level, −E 0 ± U/2, at a bias voltage e|V | = 2(U/2 − |1W |) where the first step in the current is seen and the number of atomic electrons changes. Next steps in the current and the number of atomic electrons are seen at e|V | = 2(U/2 + |1W |). For |1W | > U/2, steps are located at e|V | = 2(±U/2 + |1W |).

when it is positively charged, the energy −E 0 − U/2 is required.) We note that even though electron transfer occurs resonantly through this energy level at V = 0, the partial reflection of electrons still exists and the overall conductance is smaller than G Q . The conductance has a peak also at 1W = −U/2, where the Fermi

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level of the electrodes coincides with the other of the quasi-energy levels of the atom, −E 0 + U/2. (To add one electron to the central island when it is neutral, the energy −E 0 + U/2 is required.) If one applies a bias voltage V across such a one-atom point contact, a current increases stepwise as the bias voltage is increased as shown in Fig. 8. As discussed above, the conductance at V ∼ 0 is large for 1W/U = ± 12 . Except for these singular cases, the conductance is suppressed for small bias voltages and current does not flow (‘Coulomb blockade’) until the voltage source supplies a sufficient energy to compensate for the energy shift due to the finite 1W and U .

5. Conclusion We presented a theory for the conductance through a one-atom point contact where a central island (single atom or atom cluster) located between two electrodes has a single discrete atomic energy level within the energy scales such as an applied bias voltage eV, energy level broadening h¯ 0, a charging energy U and the thermal energy kB T . The tapered constriction in such one-atom point contacts are not necessarily adiabatically smooth which causes electron reflection and induces electron localization and charging at the constriction. In addition to these effects originated from a geometrical shape of the constriction, an actual atomic structure has other sources of reducing the conductance to below the quantum unit of conductance. Our formula allows us to calculate such a reduced conductance in terms of a difference between a work function of an electrode and an atomic energy level 1W , a charging energy U and energy level broadening h¯ 0.

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