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The effective mass of an electron when tunneling through a molecular wire
Chemical Physics 281 (2002) 347–352 www.elsevier.com/locate/chemphys
The effective mass of an electron when tunneling through a molecular wire C. Joachim a,*, M. Magoga b a
CEMES–CNRS, 29 rue J. Marvig, BP 4347, Toulouse Cedex 31055, France b Nanotimes, 29 rue J. Marvig, Toulouse Cedex 31400, France Received 5 October 2001
Abstract Observed from the electrodes, the electronic tunnel transport regime though a molecular wire is described by a generalized non-parabolic dispersion relationship. The calculation of the complex-band structure of a molecular wire leads to an analytical expression for the effective mass m of a tunneling electron through this wire. m is calculated for different molecular wires and compared to the ones of standard solid state tunnel junction. In the tunneling regime, the conductance optimization of a molecular wire depends equally on this m and on its homo–lumo gap. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction When connected between two metallic pads, a conjugated molecular wire forms a well-located conduit in space for the electrons to be exchanged between these pads through the wire. In term of the pad–molecular wire–pad junction conductance, the efficiency of such a molecular pathway is orders of magnitude better than the vacuum for an equivalent distance between the pads. At low bias voltage, the pad–molecular wire–pad conductance G is given by the well-known expression GðEF Þ ¼ G0 ðEF Þ ecðEF ÞL with G0 ðEF Þ the contact conductance, cðEF Þ the tunnel inverse damping
*
Corresponding author. Fax: +33-5-6225-7999. E-mail address: [email protected] (C. Joachim).
length of the tunnel process at the pads Fermi level EF and L the length of the molecular wire [1]. In the standard ballistic and diffusive transport regimes, cðEF Þ ¼ 0. The electrons are exchanged between the pads in a resonant regime via the electronic band structure of the material embedded between the pads. The electron effective mass m (and the average time between two collisions, if any) governs the transport in this regime. In a tunnel transport regime through a molecular wire i.e., for a bias voltage much lower that the homo– lumo gap of the molecule, cðEF Þ 6¼ 0. There is no resonance at EF between the pads and the molecular energy levels of the wire. But the super-exchange mechanism of the electrons between the pads via such a wire still depends on the distribution of the molecular levels opening and supporting the tunnel pathway. For example, cðEÞ depends on the highest occupied–lowest unoccu-
0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 3 7 2 - 5
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pied (homo–lumo) gap v of the wire and on a spectral function PðEÞ which takes into account the distribution of the molecular levels of the wire each side of the gap [1]. We demonstrate here that the dependence of cðEÞ on v and PðEÞ defines an effective mass m ðEÞ for the tunneling electrons. First, the calculation technique of the complex electronic band structure EðkÞ (with k the complex valued wave vector of the tunneling electron) for a molecular wire is presented and applied to a simple polyphenyl molecular wire. In a second part, the relationship between cðEÞ and the electron dispersion function kðEÞ is given. Then, kðEÞ is expended in power of the Kramers determinant to demonstrate how m ðEÞ depends on its parabolo€ıdal shape in the homo–lumo gap of the wire. The effective mass m for different wires are compared with those of solid state materials included in the junction and studied in the tunnel regime. In conclusion, we show how a tunnel process through a molecular wire must be described by a two band model with an energy dependent m ðEÞ rather than by a simple effective tunnel barrier whose height is adjusted to the observed cðEF Þ. In the following, it is supposed that the tunnel transport regime is elastic i.e., the electron tunneling time is much smaller than intramolecular relaxation times.
2. The molecular wire complex valued band structure Through a molecular wire with v 6¼ 0, the tunnel electrons are guided by the electronic evanescent waves coming from each pad and over-lapping on the molecular wire. Those waves are not special solutions of the electronic eigenvalue Schr€ odinger equation [2] but are parts (together with the ordinary propagative electronic states) of the electronic states associated with the complex-band structure of the pad–molecule–pad junction. In this formalism, EðkÞ is an analytic function of the complex valued k vector of the tunneling electrons. A molecular wire is usually made of a regular but finite repetition of the same monomer of unit length L0 . Therefore, in a mono-electronic approximation, the electronic complex-band struc-
ture EðkÞ of the periodic extension of this finite chain is obtained by solving the electronic H ðrÞ/k ðrÞ ¼ EðkÞ/k ðrÞ eigenvalue problem with a complex valued k vector [3]. We usually solve this problem by working with the spatial propagator Mk ðr0 ; rÞ calculated from the Hamiltonian H ðrÞ of the wire [4]. The propagative and evanescent electronic modes which are guiding the electrons along the wire are better obtained by diagonalizing Mk ðr0 ; rÞ than by solving directly the Schr€ odinger eigenvalue problem. After calculating the Mij ðEÞ matrix elements of Mk ðr0 ; rÞ on a given molecular orbital basis set, the eigenvalues k of the MðEÞ matrix are given by the standard Kramers equation k2 trðMðEÞÞk þ 1 ¼ 0 [5]. For kðEÞ ¼ eikðEÞ and kðEÞ ¼ lðEÞ þ iqðEÞ, the Kramers equation is solved by calculating the complex valued EðkÞ roots and selecting only the real valued ones [6]. One example of a complex-band structure computed following this procedure is presented in Fig. 1 for the p and p bands structure of a polyphenyl molecular wire. In the energy range where qðEÞ ¼ 0, one finds the standard p and p bands structure of the polyphenyl wire with here a band gap v ¼ 1:5 eV (in a tight-binding approximation). In the tunneling regime where lðEÞ ¼ 0, there are two parabolo€ıd tunnel bands, one linking for example the p and p bands (at Eh and El ) via the kðEÞ imaginary axis. There are also two semi-infinite tunnel band branches b and b extending the bottom of the p and the top of p bands on the kðEÞ imaginary axis.
3. The effective mass of a tunneling electron along a molecular wire For a molecular wire, the calculation of its complex-band structure gives, for lðEÞ ¼ 0, the dispersion relationship qðEÞ of the non-resonant tunneling electrons and therefore their effective mass m ðEÞ. Let us take a single rectangular tunnel barrier of height /0 . In this case, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðEÞ ¼ 2m0 ð/0 EÞ=h with m0 the mass of the electron. For a large barrier, it comes directly cðEÞ ¼ 2qðEÞ. Returning to the case of a finite length molecular wire, but
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Fig. 1. Complex p and p band structure EðkÞ of a polyphenyl molecular wire calculated in a tight-binding approximation with k ¼ l þ iq. The resonance integrals are here a ¼ 2:04 eV in the phenyl ring and b ¼ 2:2 eV between the rings. Since the wire is 1D, the real and imaginary axes l and q are represented on the same plane with any loss of information on EðkÞ. EF is positioned in the center of the homo ðEh Þ–lumo ðEl Þ gap to respect the relative energy scale chosen. b and b are the two semi-infinite branches extending the p and p band deeply in the evanescent regime. The chemical structure of a polyphenyl molecular wire is given for reference.
Fig. 2. Comparison of the q2 ðEÞ dispersion relationship obtained from a computed complex-band structure (circles), fitted with Eq. (3) (solid line) and calculated using Eq. (4) in Eq. (3) (dashed line) in the homo–lumo energy range of the molecular wire. (a) is for a polyphenyl molecular wire computed with the full valence elastic scattering quantum chemistry (ESQC) approach and (b) is for an oligo (benzo [c] cyclo-pentadiene) molecular wire (molecule number 8 in Fig. 4) calculated also with ESQC.
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long enough for a tunnel transport regime to be stabilized [1], the proportionality between cðEÞ and qðEÞ still hold. But the tunnel dispersion relationship qðEÞ is very different from the one of a rectangular barrier case, as shown for example on the lðEÞ ¼ 0 imaginary axis presented in Fig. 1. To obtain qðEÞ, we start with the known analytical expression of cðEÞ [1] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! DðEÞ þ 4 þ DðEÞ 2 cðEÞ ¼ ln ð1Þ L0 4 with DðEÞ ¼ trðMðEÞÞ2 4 the Kramers discriminant. Using for the MðEÞ eigenvalues k the expression k ¼ eik in (1), it comes in the homo–lumo energy range of the wire pffiffiffiffiffiffiffiffiffiffi ! DðEÞ : ð2Þ qðEÞ ¼ argsh 2 DðEÞ is a polynomial which can be decomposed in a product of two parts: a second order ðE Eh ÞðEl EÞ part to describe the almost parabolo€ıd qðEÞ2 variations within the homo ðEh Þ– lumo ðEl Þ band gap and a PðEÞ polynomial [1] whose roots are outside the homo–lumo gap energy range. In the gap and for DðEF Þ < 4, qðEÞ in (2) can be expended in series of DðEÞ and cðEÞ rewritten up to the first order sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m ðEÞ ð E Eh ÞðEl EÞ cðEÞ ¼ ð3Þ v h2 with m ðEÞ ¼ ðh2 PðEÞvÞ=ð8L20 Þ. Eq. (3) is similar to the so-called non-parabolic dispersion relationship, standard for tunneling through the gap of an insulator material [7] but with an energy dependent effective mass. It is valid for a polyene molecular wire where DðEF Þ ¼ 0:86 (calculated in a tight-binding approximation) but not for a polyphenyl wire where DðEF Þ ¼ 19:07 [8]. Using the 2 hypothesis that qðEÞ is always a parabolo€ıd of the form (3) (as presented in Fig. 1 for the polyphenyl wire) even if DðEF Þ is large, the generalization of (3) for molecular wires with a large DðEF Þ is obtained by adjusting at E ¼ EF the curvature of (3) to the one of (2). By calculating the second derivative of (3) and of (2) in the middle of the gap (a procedure formally equivalent to calculate an
effective mass in a solid), the m ðEÞ analytical expression in (3) can be written for any DðEF Þ value in the gap m ðEÞ ¼
h2 vPðEÞ 1 : 8L20 1 þ v2 P2 ðEÞ 64
ð4Þ
The m ðEÞ expression (3) is simply the zero’s 2 order development of (4) in power of ðvPðEÞÞ =4. Eq. (3) with (4) generalizes the Franz two-bands dispersion relationship approximation used to describe the tunneling phenomenon in a energy range located in the gap of a two bands material [9]. The advantage of (4) is that m ðEÞ is no more a parameter as in the original Franz derivation [7,9] but can be calculated using the PðEÞ function. This function reflects the detail spectral organization of the occupied and unoccupied electronic states of the molecular wire which support the tunnel transport regime [1]. For a polyphenyl wire, qðEÞ obtained by a complex-band structure calculation (Fig. 1) is well
Fig. 3. Variation in the homo–lumo gap energy range of the molecular wire of the effective mass m ðEÞ from (4) for the molecular wire oligo (benzo [c] cyclo-pentadiene) whose chemical structure is presented in Fig. 4 (molecule number 8).
C. Joachim, M. Magoga / Chemical Physics 281 (2002) 347–352
reproduced by (3) as shown in Fig. 2(a). Here, m is independent of the electron incident energy on the wire over the gap energy range. We obtained m ¼ 0:163m0 from the best fit between (3) and the calculated qðEÞ, and m ¼ 0:159m0 using (4) in (3) (see Fig. 2(a)). When the difference between the spectral organization of the unoccupied and occupied states of the wire is too large, d2 PðEÞ 6¼ 0 dE2
for E ¼ EF :
In this case m ¼ m ðEÞ and the Franz-like approximation (3) even corrected by (4) do not fit the exact calculated q2 ðEÞ as shown in Fig. 2(b) for an oligo (benzo [c] cyclo-pentadiene) wire. In this case, (3) must be corrected by a second term depending on d2 PðEÞ : dE2
351
Fig. 3 provides an example of the m ðEÞ variations in the case of the oligo (benzo [c] cyclo-pentadiene molecular wire in its homo–lumo gap energy range. m ðEÞ was calculated directly using (4). As presented in Fig. 4, m can be evaluated for a large variety of regular molecular wires whose effective mass is independent of the energy in the gap range. The procedure is to fit the qðEÞ function obtained by a complex-band structure calculation with (3). The best approximation for each molecular wire is reported in Fig. 4 by plotting cðEF Þ as a function of ðvm Þ=m0 since from (3) c2 ðEF Þ is proportional to vm . For comparison, we have also reported the c2 ðEF Þ values of an alkane chain and of some v 6¼ 0 solid state materials. For a molecular wire, we have also calculate m with the analytical expression (4) as presented in Fig. 4. For solid state tunnel junctions, m is still a parameter always evaluated by comparing the experimental q2 ðEÞ with the Franz two-bands approximation
Fig. 4. Variations of the cðEF Þ inverse damping length as a function of the vm =m0 ratio for a selected number of molecular wires. The indexed open circle indicate the best m ðEF Þ fit obtained using Eq. (3) on the computed qðEÞ relation in the homo–lumo gap. The black circles are values of cðEF Þ for solid state materials (InAs ðv ¼ 0:42 eVÞ [10], GaAs ðv ¼ 1:52 eVÞ [11], AlN ðv ¼ 4:2 eVÞ [12], SiO2 ðv ¼ 8 eVÞ [7]) in a tunnel regime. The alkane chain value is indicated by a star. The solid line gives the calculated vðEF Þ value using Eq. (4) in Eq. (3). For molecules, cðEF Þ, v and m have been calculated using a full valence extended H€ uckel approximation that leads to v1 ¼ 0:49, v2 ¼ 0:88, v3 ¼ 1:11, v4 ¼ 0:875, v5 ¼ 1:25, v6 ¼ 1:64, v7 ¼ 1:01, v8 ¼ 1:52 eV.
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[7,10–12]. This explains why it fits so well on the cðEF Þ curve given in Fig. 4. As a consequence, solid state tunnel junctions and molecular wires of similar vm have equivalent cðEF Þ. But molecular wires are better in guiding very locally in space the tunnel electrons. This is an advantage over solid state junctions to pursue the study of the tunnel process.
a given v, the m analytical expression facilitates the search for a better electronic structure of the unoccupied and occupied states of the wire supporting the tunnel current. As a consequence, reducing m is as important as reducing v to transport a tunnel current on along range through a molecular wire.
Acknowledgements 4. Conclusion In the literature, the super-exchange performances of a molecular wire connected to metallic pads are often analyzed by building an effective barrier height /0 with m ¼ m0 . This usually results in a very small barrier height compared to the homo–lumo gap of the wire, giving the wrong impression that v is reduced by the wire–pads electronic interactions. For example, measurements of the effective barrier height on an ultra clean connected ‘‘lander-like molecules’’ [13] gives 1 and / ¼ 0:164 meV which has to cðEF Þ ¼ 0:4 A 0 be compared to v 3:8 eV and m ¼ 0:16m0 for this molecule. A better description of the tunnel process through a molecular wire, is to take into account the non-parabolicity of the dispersion relationship of a tunneling electron through a molecular wire and its energy dependent effective mass. In this description, the homo–lumo gap value is kept constant. The effective mass of the tunneling electron must be adapted to the tunnel process in play, that is to the electronic structure and distribution of the molecular levels supporting this process. At
We like to thank A. Gourdon for a critical reading and the IST–FET ‘‘Bottom-Up Nanomachine’’ European project for financial support during this work.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
M. Magoga, C. Joachim, Phys. Rev. B 57 (1998) 1820. A.A. Cottey, Am. J. Phys. 39 (1971) 1235. W. Kohn, Phys. Rev. 115 (1959) 809. J. Stein, C. Joachim, J. Phys. A 20 (1987) 2849. A.H. Wilson, The Theory of Metals, Cambridge University Press, Cambridge, 1965. T.B. Boykin, Phys. Rev. B 54 (1996) 8107. B. Brar, G.D. Wilk, A.C. Seabaugh, App. Phys. Lett. 69 (1996) 2728. M. Magoga, C. Joachim, Phys. Rev. B 56 (1997) 4722. W. Franz, in: S. Fl€ ugge (Ed.), Handbook of Physics, vol. 17, Springer, New York, 1956, p. 155. G.H. Parker, C.A. Mead, Phys. Rev. Lett. 21 (1968) 605. F.A. Padovani, R. Stratton, Phys. Rev. Lett. 16 (1966) 1202. G. Lewickit, C.A. Mead, Phys. Rev. Lett. 16 (1966) 939. V.J. Langlais, R.R. Schlitter, H. Tang, A. Gourdon, C. Joachim, J.K. Gimzewski, Phys. Rev. Lett. 83 (1999) 2809.
The effective mass of an electron when tunneling through a molecular wire C. Joachim a,*, M. Magoga b a
CEMES–CNRS, 29 rue J. Marvig, BP 4347, Toulouse Cedex 31055, France b Nanotimes, 29 rue J. Marvig, Toulouse Cedex 31400, France Received 5 October 2001
Abstract Observed from the electrodes, the electronic tunnel transport regime though a molecular wire is described by a generalized non-parabolic dispersion relationship. The calculation of the complex-band structure of a molecular wire leads to an analytical expression for the effective mass m of a tunneling electron through this wire. m is calculated for different molecular wires and compared to the ones of standard solid state tunnel junction. In the tunneling regime, the conductance optimization of a molecular wire depends equally on this m and on its homo–lumo gap. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction When connected between two metallic pads, a conjugated molecular wire forms a well-located conduit in space for the electrons to be exchanged between these pads through the wire. In term of the pad–molecular wire–pad junction conductance, the efficiency of such a molecular pathway is orders of magnitude better than the vacuum for an equivalent distance between the pads. At low bias voltage, the pad–molecular wire–pad conductance G is given by the well-known expression GðEF Þ ¼ G0 ðEF Þ ecðEF ÞL with G0 ðEF Þ the contact conductance, cðEF Þ the tunnel inverse damping
*
Corresponding author. Fax: +33-5-6225-7999. E-mail address: [email protected] (C. Joachim).
length of the tunnel process at the pads Fermi level EF and L the length of the molecular wire [1]. In the standard ballistic and diffusive transport regimes, cðEF Þ ¼ 0. The electrons are exchanged between the pads in a resonant regime via the electronic band structure of the material embedded between the pads. The electron effective mass m (and the average time between two collisions, if any) governs the transport in this regime. In a tunnel transport regime through a molecular wire i.e., for a bias voltage much lower that the homo– lumo gap of the molecule, cðEF Þ 6¼ 0. There is no resonance at EF between the pads and the molecular energy levels of the wire. But the super-exchange mechanism of the electrons between the pads via such a wire still depends on the distribution of the molecular levels opening and supporting the tunnel pathway. For example, cðEÞ depends on the highest occupied–lowest unoccu-
0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 3 7 2 - 5
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pied (homo–lumo) gap v of the wire and on a spectral function PðEÞ which takes into account the distribution of the molecular levels of the wire each side of the gap [1]. We demonstrate here that the dependence of cðEÞ on v and PðEÞ defines an effective mass m ðEÞ for the tunneling electrons. First, the calculation technique of the complex electronic band structure EðkÞ (with k the complex valued wave vector of the tunneling electron) for a molecular wire is presented and applied to a simple polyphenyl molecular wire. In a second part, the relationship between cðEÞ and the electron dispersion function kðEÞ is given. Then, kðEÞ is expended in power of the Kramers determinant to demonstrate how m ðEÞ depends on its parabolo€ıdal shape in the homo–lumo gap of the wire. The effective mass m for different wires are compared with those of solid state materials included in the junction and studied in the tunnel regime. In conclusion, we show how a tunnel process through a molecular wire must be described by a two band model with an energy dependent m ðEÞ rather than by a simple effective tunnel barrier whose height is adjusted to the observed cðEF Þ. In the following, it is supposed that the tunnel transport regime is elastic i.e., the electron tunneling time is much smaller than intramolecular relaxation times.
2. The molecular wire complex valued band structure Through a molecular wire with v 6¼ 0, the tunnel electrons are guided by the electronic evanescent waves coming from each pad and over-lapping on the molecular wire. Those waves are not special solutions of the electronic eigenvalue Schr€ odinger equation [2] but are parts (together with the ordinary propagative electronic states) of the electronic states associated with the complex-band structure of the pad–molecule–pad junction. In this formalism, EðkÞ is an analytic function of the complex valued k vector of the tunneling electrons. A molecular wire is usually made of a regular but finite repetition of the same monomer of unit length L0 . Therefore, in a mono-electronic approximation, the electronic complex-band struc-
ture EðkÞ of the periodic extension of this finite chain is obtained by solving the electronic H ðrÞ/k ðrÞ ¼ EðkÞ/k ðrÞ eigenvalue problem with a complex valued k vector [3]. We usually solve this problem by working with the spatial propagator Mk ðr0 ; rÞ calculated from the Hamiltonian H ðrÞ of the wire [4]. The propagative and evanescent electronic modes which are guiding the electrons along the wire are better obtained by diagonalizing Mk ðr0 ; rÞ than by solving directly the Schr€ odinger eigenvalue problem. After calculating the Mij ðEÞ matrix elements of Mk ðr0 ; rÞ on a given molecular orbital basis set, the eigenvalues k of the MðEÞ matrix are given by the standard Kramers equation k2 trðMðEÞÞk þ 1 ¼ 0 [5]. For kðEÞ ¼ eikðEÞ and kðEÞ ¼ lðEÞ þ iqðEÞ, the Kramers equation is solved by calculating the complex valued EðkÞ roots and selecting only the real valued ones [6]. One example of a complex-band structure computed following this procedure is presented in Fig. 1 for the p and p bands structure of a polyphenyl molecular wire. In the energy range where qðEÞ ¼ 0, one finds the standard p and p bands structure of the polyphenyl wire with here a band gap v ¼ 1:5 eV (in a tight-binding approximation). In the tunneling regime where lðEÞ ¼ 0, there are two parabolo€ıd tunnel bands, one linking for example the p and p bands (at Eh and El ) via the kðEÞ imaginary axis. There are also two semi-infinite tunnel band branches b and b extending the bottom of the p and the top of p bands on the kðEÞ imaginary axis.
3. The effective mass of a tunneling electron along a molecular wire For a molecular wire, the calculation of its complex-band structure gives, for lðEÞ ¼ 0, the dispersion relationship qðEÞ of the non-resonant tunneling electrons and therefore their effective mass m ðEÞ. Let us take a single rectangular tunnel barrier of height /0 . In this case, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðEÞ ¼ 2m0 ð/0 EÞ=h with m0 the mass of the electron. For a large barrier, it comes directly cðEÞ ¼ 2qðEÞ. Returning to the case of a finite length molecular wire, but
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Fig. 1. Complex p and p band structure EðkÞ of a polyphenyl molecular wire calculated in a tight-binding approximation with k ¼ l þ iq. The resonance integrals are here a ¼ 2:04 eV in the phenyl ring and b ¼ 2:2 eV between the rings. Since the wire is 1D, the real and imaginary axes l and q are represented on the same plane with any loss of information on EðkÞ. EF is positioned in the center of the homo ðEh Þ–lumo ðEl Þ gap to respect the relative energy scale chosen. b and b are the two semi-infinite branches extending the p and p band deeply in the evanescent regime. The chemical structure of a polyphenyl molecular wire is given for reference.
Fig. 2. Comparison of the q2 ðEÞ dispersion relationship obtained from a computed complex-band structure (circles), fitted with Eq. (3) (solid line) and calculated using Eq. (4) in Eq. (3) (dashed line) in the homo–lumo energy range of the molecular wire. (a) is for a polyphenyl molecular wire computed with the full valence elastic scattering quantum chemistry (ESQC) approach and (b) is for an oligo (benzo [c] cyclo-pentadiene) molecular wire (molecule number 8 in Fig. 4) calculated also with ESQC.
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long enough for a tunnel transport regime to be stabilized [1], the proportionality between cðEÞ and qðEÞ still hold. But the tunnel dispersion relationship qðEÞ is very different from the one of a rectangular barrier case, as shown for example on the lðEÞ ¼ 0 imaginary axis presented in Fig. 1. To obtain qðEÞ, we start with the known analytical expression of cðEÞ [1] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! DðEÞ þ 4 þ DðEÞ 2 cðEÞ ¼ ln ð1Þ L0 4 with DðEÞ ¼ trðMðEÞÞ2 4 the Kramers discriminant. Using for the MðEÞ eigenvalues k the expression k ¼ eik in (1), it comes in the homo–lumo energy range of the wire pffiffiffiffiffiffiffiffiffiffi ! DðEÞ : ð2Þ qðEÞ ¼ argsh 2 DðEÞ is a polynomial which can be decomposed in a product of two parts: a second order ðE Eh ÞðEl EÞ part to describe the almost parabolo€ıd qðEÞ2 variations within the homo ðEh Þ– lumo ðEl Þ band gap and a PðEÞ polynomial [1] whose roots are outside the homo–lumo gap energy range. In the gap and for DðEF Þ < 4, qðEÞ in (2) can be expended in series of DðEÞ and cðEÞ rewritten up to the first order sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m ðEÞ ð E Eh ÞðEl EÞ cðEÞ ¼ ð3Þ v h2 with m ðEÞ ¼ ðh2 PðEÞvÞ=ð8L20 Þ. Eq. (3) is similar to the so-called non-parabolic dispersion relationship, standard for tunneling through the gap of an insulator material [7] but with an energy dependent effective mass. It is valid for a polyene molecular wire where DðEF Þ ¼ 0:86 (calculated in a tight-binding approximation) but not for a polyphenyl wire where DðEF Þ ¼ 19:07 [8]. Using the 2 hypothesis that qðEÞ is always a parabolo€ıd of the form (3) (as presented in Fig. 1 for the polyphenyl wire) even if DðEF Þ is large, the generalization of (3) for molecular wires with a large DðEF Þ is obtained by adjusting at E ¼ EF the curvature of (3) to the one of (2). By calculating the second derivative of (3) and of (2) in the middle of the gap (a procedure formally equivalent to calculate an
effective mass in a solid), the m ðEÞ analytical expression in (3) can be written for any DðEF Þ value in the gap m ðEÞ ¼
h2 vPðEÞ 1 : 8L20 1 þ v2 P2 ðEÞ 64
ð4Þ
The m ðEÞ expression (3) is simply the zero’s 2 order development of (4) in power of ðvPðEÞÞ =4. Eq. (3) with (4) generalizes the Franz two-bands dispersion relationship approximation used to describe the tunneling phenomenon in a energy range located in the gap of a two bands material [9]. The advantage of (4) is that m ðEÞ is no more a parameter as in the original Franz derivation [7,9] but can be calculated using the PðEÞ function. This function reflects the detail spectral organization of the occupied and unoccupied electronic states of the molecular wire which support the tunnel transport regime [1]. For a polyphenyl wire, qðEÞ obtained by a complex-band structure calculation (Fig. 1) is well
Fig. 3. Variation in the homo–lumo gap energy range of the molecular wire of the effective mass m ðEÞ from (4) for the molecular wire oligo (benzo [c] cyclo-pentadiene) whose chemical structure is presented in Fig. 4 (molecule number 8).
C. Joachim, M. Magoga / Chemical Physics 281 (2002) 347–352
reproduced by (3) as shown in Fig. 2(a). Here, m is independent of the electron incident energy on the wire over the gap energy range. We obtained m ¼ 0:163m0 from the best fit between (3) and the calculated qðEÞ, and m ¼ 0:159m0 using (4) in (3) (see Fig. 2(a)). When the difference between the spectral organization of the unoccupied and occupied states of the wire is too large, d2 PðEÞ 6¼ 0 dE2
for E ¼ EF :
In this case m ¼ m ðEÞ and the Franz-like approximation (3) even corrected by (4) do not fit the exact calculated q2 ðEÞ as shown in Fig. 2(b) for an oligo (benzo [c] cyclo-pentadiene) wire. In this case, (3) must be corrected by a second term depending on d2 PðEÞ : dE2
351
Fig. 3 provides an example of the m ðEÞ variations in the case of the oligo (benzo [c] cyclo-pentadiene molecular wire in its homo–lumo gap energy range. m ðEÞ was calculated directly using (4). As presented in Fig. 4, m can be evaluated for a large variety of regular molecular wires whose effective mass is independent of the energy in the gap range. The procedure is to fit the qðEÞ function obtained by a complex-band structure calculation with (3). The best approximation for each molecular wire is reported in Fig. 4 by plotting cðEF Þ as a function of ðvm Þ=m0 since from (3) c2 ðEF Þ is proportional to vm . For comparison, we have also reported the c2 ðEF Þ values of an alkane chain and of some v 6¼ 0 solid state materials. For a molecular wire, we have also calculate m with the analytical expression (4) as presented in Fig. 4. For solid state tunnel junctions, m is still a parameter always evaluated by comparing the experimental q2 ðEÞ with the Franz two-bands approximation
Fig. 4. Variations of the cðEF Þ inverse damping length as a function of the vm =m0 ratio for a selected number of molecular wires. The indexed open circle indicate the best m ðEF Þ fit obtained using Eq. (3) on the computed qðEÞ relation in the homo–lumo gap. The black circles are values of cðEF Þ for solid state materials (InAs ðv ¼ 0:42 eVÞ [10], GaAs ðv ¼ 1:52 eVÞ [11], AlN ðv ¼ 4:2 eVÞ [12], SiO2 ðv ¼ 8 eVÞ [7]) in a tunnel regime. The alkane chain value is indicated by a star. The solid line gives the calculated vðEF Þ value using Eq. (4) in Eq. (3). For molecules, cðEF Þ, v and m have been calculated using a full valence extended H€ uckel approximation that leads to v1 ¼ 0:49, v2 ¼ 0:88, v3 ¼ 1:11, v4 ¼ 0:875, v5 ¼ 1:25, v6 ¼ 1:64, v7 ¼ 1:01, v8 ¼ 1:52 eV.
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[7,10–12]. This explains why it fits so well on the cðEF Þ curve given in Fig. 4. As a consequence, solid state tunnel junctions and molecular wires of similar vm have equivalent cðEF Þ. But molecular wires are better in guiding very locally in space the tunnel electrons. This is an advantage over solid state junctions to pursue the study of the tunnel process.
a given v, the m analytical expression facilitates the search for a better electronic structure of the unoccupied and occupied states of the wire supporting the tunnel current. As a consequence, reducing m is as important as reducing v to transport a tunnel current on along range through a molecular wire.
Acknowledgements 4. Conclusion In the literature, the super-exchange performances of a molecular wire connected to metallic pads are often analyzed by building an effective barrier height /0 with m ¼ m0 . This usually results in a very small barrier height compared to the homo–lumo gap of the wire, giving the wrong impression that v is reduced by the wire–pads electronic interactions. For example, measurements of the effective barrier height on an ultra clean connected ‘‘lander-like molecules’’ [13] gives 1 and / ¼ 0:164 meV which has to cðEF Þ ¼ 0:4 A 0 be compared to v 3:8 eV and m ¼ 0:16m0 for this molecule. A better description of the tunnel process through a molecular wire, is to take into account the non-parabolicity of the dispersion relationship of a tunneling electron through a molecular wire and its energy dependent effective mass. In this description, the homo–lumo gap value is kept constant. The effective mass of the tunneling electron must be adapted to the tunnel process in play, that is to the electronic structure and distribution of the molecular levels supporting this process. At
We like to thank A. Gourdon for a critical reading and the IST–FET ‘‘Bottom-Up Nanomachine’’ European project for financial support during this work.
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