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Approximate calculation of the density of electronic off-resonant states in nanoconstrictions with weak atom–lead coupling
ARTICLE IN PRESS Physica B 404 (2009) 1621–1623
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Approximate calculation of the density of electronic off-resonant states in nanoconstrictions with weak atom–lead coupling M.A. Grado-Caffaro , M. Grado-Caffaro SAPIENZA-Scientific Consultants, C/Julio Palacios 11, 9-B, 28029 Madrid, Spain
a r t i c l e in fo
abstract
Article history: Received 5 October 2008 Accepted 23 January 2009
We determine theoretically the density of electron off-resonant states in a one-atom constriction with weak coupling between the atom at the nanoconstriction (nanowire) and the corresponding leads by considering the conductance (which is found to be low) through the wire whose insulating behavior is pointed out. From the expression for conductance, the characteristic electrochemical potential versus current is calculated. & 2009 Elsevier B.V. All rights reserved.
Keywords: Nanoconstriction Off-resonant states Electrical conductance Atom–lead coupling Electrochemical potential
1. Introduction Metallic nanowires (or metallic nanoconstrictions) are structures at atomic scale whose relevance is certainly notable. As well as in other nanostructures, there are several important open questions about electrical-conduction mechanisms in metallic nanowires (metallic contacts on atomic scale). These questions are directly or indirectly related to the determination of the conductance through the aforementioned wires. Within this context, conductance through both resonant and off-resonant states plays a significant role. By assuming a Lorentzian form and a one-atom contact, the aforementioned conductance can be expressed in terms of the electronic energy, Fermi energy, and two coefficients which give a measure of the coupling between the atom at the nanoconstriction and the corresponding leads (see, for example, [1–5]). In the following, we will start from the abovementioned expression for conductance through off-resonant states under the condition that the atom–lead coupling is weak enough. Within this framework, we will show easily that the metallic character of the nanowires in question vanishes so that these structures behave as insulating. The main goal of the present paper is the determination of the density of electronic offresonant states. Under small atom–lead coupling, a different theoretical analysis has been performed on that density as well as upon the current intensity as a function of the electrochemical potential [6].
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.01.035
Certainly, speaking in general terms, theoretical techniques for calculating electronic density of states (DOS) must be improved in order to build well-grounded theories in the field of solid-state physics. Knowledge of the density of quantum states is necessary to study a number of problems; this fact becomes remarkable in nanoscience, where the quantum effects are really notorious. Consider, for example, carbon nanotubes as well as metallic and semiconducting nanowires, for which the theoretical determination of the local electronic density of states (LDOS) has a clear relevance [7–9]. In a first approximation, we may say that there exists a linear relationship between LDOS and conductance [9–12]. This fact permits to evaluate the DOS starting from a suitable mathematical expression for conductance; here we shall employ this procedure for off-resonant states and small enough coupling between the atom at the nanoconstriction and the leads. Dependence of the DOS upon the electron energy and electrochemical potential will be discussed.
2. Theory Our starting point is the consideration of the conductance (Lorentzian profile) through a, say, typical one-atom metal contact. We have (see, for instance, Refs. [1–5]): GðEÞ ¼
4G0 ab ðE EF Þ2 þ ða þ bÞ2
(1)
where G(E) denotes conductance, E is electronic energy, EF is Fermi energy, G0 designates the fundamental conductance quantum, and a and b stand for the coefficients of the atom–lead coupling. Notice that the right-hand side of Eq. (1) represents the
ARTICLE IN PRESS 1622
M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1621–1623
fundamental conductance quantum multiplied by the probability of occupation for an electronic state of energy E. Now we assume that EX0 and EaEF (off-resonance) such that a+b5EEF so we may speak of weak coupling. Under these conditions, Eq. (1) reduces to GðEÞ
4G0 ab
(2)
ðE EF Þ2
Since a and b are assumed to be negligible as compared with EEF, by formula (2) it is clear that G5G0, that is, the wire behaves as insulating. For simplicity, we express the Fermi energy level taking vacuum as reference scale so that one has that EF ¼ eV where e is the electron charge and V stands for electrochemical potential (see, for example, Ref. [13]). By inserting this expression into formula (2), it follows: GðE; VÞ
Notice that the lower limit of the integral of Eq. (6) is zero energy so that this value of electron energy must satisfy, of course, the condition a+b5E+eV. Then, inserting into this inequality the value E ¼ 0, it follows that Vb(a+b)/e. Consequently, relationship (7) is not valid for values of the electrochemical potential that do not satisfy the above condition. From expression (7) it follows that the total density of electronic off-resonant states is quasi-inversely proportional to the square of the electronic energy when EbeV. In addition, notice the dependence on the electrochemical potential of that density so that, if E5eV, the off-resonant states density is quasi-inversely proportional to the electrochemical potential. By formula (7), let us note that the density in question is practically independent of the electronic energy when E5eV.
3. Discussion
4G0 ab 2
ðE þ eVÞ
(3)
We will restrict ourselves to V40 because negative values of the electrochemical potential are not permitted given that the condition a+b5E+eV is not fulfilled for every E40 and Vo0 such that |V|oE/e. In effect, imagine sufficiently small values of E and |V| (E40, Vo0) such that |V|oE/e; in this case, the condition a+b5E+eV does not hold. For E ¼ 0, this condition reduces to a+b5eV (now we assume V40) so that the values of V which do not verify the above condition must not be considered. In addition, note that V ¼ 0 does not satisfy the condition in question since, for V ¼ 0, one should have that a+b50 which is absurd. Next we shall derive the characteristic VI of the wire from Eq. (3). To get this end, one has that the current intensity I(E,V) through our wire verifies that G(E,V) ¼ qI(E,V)/qV, so it follows: Z V GðE; VÞ dV (4) IðE; VÞ ¼
We have found out an approximate formula for the density of electronic off-resonant states in an insulating nanowire by starting from considerations relative to conductance. The insulating behavior arises from the fact that a+b5EEF so that offresonance is relatively very pronounced. On the left-hand side of Eq. (7), V-dependence has not been expressed explicitly because the DOS is commonly expressed as a function of the electronic energy only; therefore, mathematically speaking, V acts as a parameter so that Eq. (7) provides the values of the density of offresonant states in terms of E for a given value taken on by V. At this point, it is interesting to consider the Fermi energy of the electron gas associated with the nanoconstriction (see, for instance, Ref. [5]). As a function of temperature absolute, the Fermi energy in question is given by the well-known Sommerfeld formula namely (
0
EF ðTÞ EF ð0Þ 1
Replacing formula (3) into relation (4), one gets IðE; VÞ
4G0 abV EðE þ eVÞ
(5)
By inspecting formula (5) we see, on the one hand, that the current intensity is quasi-inversely proportional to the square of the electronic energy when EbeV and, on the other hand, that the current intensity becomes quasi-inversely proportional to the electronic energy if V-N. In Ref. [6] the current intensity has been found to be negative, contrarily to Eq. (5) by which this intensity is strictly positive (note that V40). Considering now the density of electronic off-resonant states, we can say that, in a first approximation, the LDOS is proportional to conductance (see, for instance, Refs. [9–12]). Therefore, by using Eq. (3), the total density of electronic off-resonant states (the integral of the local density extended to the spatial region enclosed by the nanowire) is given by gðEÞ
4aG0 ab ðE þ eVÞ2
(6)
where a is a positive constant corresponding to the above proportionality relationship. In order to evaluate a in terms of known quantities, let us denote by N the number of off-resonant states. Therefore, the integral of Eq. (6) between zero and infinite must be equal to N. Performing this integral with the integrand given by Eq. (6), one gets the approximate value of a which, replaced into formula (6), yields gðEÞ
NeV ðE þ eVÞ2
(7)
p2 k2 T 2 12½EF ð0Þ2
) (8)
pffiffiffi 2 (with p2 k T 2 o12½EF ð0Þ2 ; that is, To2 3EF ð0Þ=ðpkÞ), where k is the Boltzmann constant and the Fermi energy level extrapolated to zero absolute temperature is given by
EF ð0Þ ¼
_2 2m
ð3p2 nÞ2=3
(9)
where _ is the reduced Planck constant, m is the free-electron mass, and n stands for spatial electron density. Then, by changing adequately the reference scale of the electrochemical potential, the Fermi energy should be positive and therefore it should equate to formula (8) (with relation (9)) so that the temperature of the electron gas can be determined for a given value of the electrochemical potential. Nevertheless, the above equation, which involves the absolute value pffiffiffi of the electrochemical potential, is applicable only if To2 3EF ð0Þ=ðpkÞ that, by formula (9), leads to an inequality in terms of temperature and spatial electron density (note that this density refers to zero absolute temperature). We emphasize the fact that the values of V must match Eq. (8) (with formula (9)) under the restriction represented by the above quoted inequality on temperature so that the consideration of V outside the inequality in question does not make sense. Prior to calculate the density for off-resonant electronic states, we have derived an expression for the current intensity versus the electrochemical potential (Eq. (5)). In Eq. (5), the E-dependence of the current intensity can be conceived as that E acts as a parameter so that Eq. (5) defines really a family of VI curves; each curve corresponds to a given value of the parameter.
ARTICLE IN PRESS M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1621–1623
In other words, formula (5) gives the distribution of the current in terms of the electronic energy; at a given E, one has the V-variation of the current.
4. Concluding remarks Electronic transport through both resonant and off-resonant states plays a significant role in nanoscience. In particular, electrical conduction through resonant states is certainly relevant since, generally speaking, E ¼ EF (resonance) is the condition to determine conductance (consider carbon nanotubes and metallic nanowires). In practice, quasi-resonance (EEEF) takes place when we speak of, say, normal conduction in the above-mentioned nanostructures. As a particular case of resonance, we can mention tunneling resonant states whose relevance is manifest in order to tackle a wide variety of problems related to nanostructures as, for instance, quantum dots (see Ref. [14]). In Refs. [15–17], other relevant indirectly related problems on quantum transport through nanoscale systems are treated. In addition, experimental studies as Ref. [18] can be cited.
1623
References [1] A. Yazdani, D.M. Eigler, N.D. Lang, Science 272 (1996) 1921. [2] C.E.D. Chidsey, Science 251 (1991) 919. [3] C. Shu, C.Z. Li, H.X. He, A. Bogozi, J.S. Bunch, N.J. Tao, Phys. Rev. Lett. 84 (2000) 5196. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 114 (2003) 565. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 115 (2004) 476. [6] M.A. Grado-Caffaro, M. Grado-Caffaro, Phys. Lett. A 372 (2008) 4638. [7] J. Pan, T.W. Jing, S.M. Lindsay, J. Phys. Chem. 98 (1993) 4205. [8] G. Repphun, J. Halbritter, J. Vac. Sci. Tech. A 13 (1995) 1693. [9] M.A. Grado-Caffaro, M. Grado-Caffaro, Phys. A 387 (2008) 445. [10] J. Tersoff, D.R. Hamann, Phys. Rev. B 31 (1985) 805. [11] C.J. Chen, Introduction to STM, Oxford University Press, New York, 1993. [12] M.A. Grado-Caffaro, M. Grado-Caffaro, Modern Phys. Lett. B 20 (2006) 1853. [13] H. Heiss, A. Heller, J. Phys. Chem. 89 (1985) 4207. [14] V. Ermakov, S. Kruchinin, H. Hori, A. Fujiwara, Int. J. Modern Phys. B 21 (2007) 1827. [15] E.R. Bittner, J. Chem. Phys. 112 (2000) 9703. [16] D.S. Kosov, J. Chem. Phys. 116 (2002) 6368. [17] P. Pomorski, C. Roland, H. Guo, Phys. Rev. B 70 (2004) 115408. [18] S. Sahoo, C.-S. Yang, B. Doudin, Phys. Lett. A 352 (2006) 331.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Approximate calculation of the density of electronic off-resonant states in nanoconstrictions with weak atom–lead coupling M.A. Grado-Caffaro , M. Grado-Caffaro SAPIENZA-Scientific Consultants, C/Julio Palacios 11, 9-B, 28029 Madrid, Spain
a r t i c l e in fo
abstract
Article history: Received 5 October 2008 Accepted 23 January 2009
We determine theoretically the density of electron off-resonant states in a one-atom constriction with weak coupling between the atom at the nanoconstriction (nanowire) and the corresponding leads by considering the conductance (which is found to be low) through the wire whose insulating behavior is pointed out. From the expression for conductance, the characteristic electrochemical potential versus current is calculated. & 2009 Elsevier B.V. All rights reserved.
Keywords: Nanoconstriction Off-resonant states Electrical conductance Atom–lead coupling Electrochemical potential
1. Introduction Metallic nanowires (or metallic nanoconstrictions) are structures at atomic scale whose relevance is certainly notable. As well as in other nanostructures, there are several important open questions about electrical-conduction mechanisms in metallic nanowires (metallic contacts on atomic scale). These questions are directly or indirectly related to the determination of the conductance through the aforementioned wires. Within this context, conductance through both resonant and off-resonant states plays a significant role. By assuming a Lorentzian form and a one-atom contact, the aforementioned conductance can be expressed in terms of the electronic energy, Fermi energy, and two coefficients which give a measure of the coupling between the atom at the nanoconstriction and the corresponding leads (see, for example, [1–5]). In the following, we will start from the abovementioned expression for conductance through off-resonant states under the condition that the atom–lead coupling is weak enough. Within this framework, we will show easily that the metallic character of the nanowires in question vanishes so that these structures behave as insulating. The main goal of the present paper is the determination of the density of electronic offresonant states. Under small atom–lead coupling, a different theoretical analysis has been performed on that density as well as upon the current intensity as a function of the electrochemical potential [6].
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.01.035
Certainly, speaking in general terms, theoretical techniques for calculating electronic density of states (DOS) must be improved in order to build well-grounded theories in the field of solid-state physics. Knowledge of the density of quantum states is necessary to study a number of problems; this fact becomes remarkable in nanoscience, where the quantum effects are really notorious. Consider, for example, carbon nanotubes as well as metallic and semiconducting nanowires, for which the theoretical determination of the local electronic density of states (LDOS) has a clear relevance [7–9]. In a first approximation, we may say that there exists a linear relationship between LDOS and conductance [9–12]. This fact permits to evaluate the DOS starting from a suitable mathematical expression for conductance; here we shall employ this procedure for off-resonant states and small enough coupling between the atom at the nanoconstriction and the leads. Dependence of the DOS upon the electron energy and electrochemical potential will be discussed.
2. Theory Our starting point is the consideration of the conductance (Lorentzian profile) through a, say, typical one-atom metal contact. We have (see, for instance, Refs. [1–5]): GðEÞ ¼
4G0 ab ðE EF Þ2 þ ða þ bÞ2
(1)
where G(E) denotes conductance, E is electronic energy, EF is Fermi energy, G0 designates the fundamental conductance quantum, and a and b stand for the coefficients of the atom–lead coupling. Notice that the right-hand side of Eq. (1) represents the
ARTICLE IN PRESS 1622
M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1621–1623
fundamental conductance quantum multiplied by the probability of occupation for an electronic state of energy E. Now we assume that EX0 and EaEF (off-resonance) such that a+b5EEF so we may speak of weak coupling. Under these conditions, Eq. (1) reduces to GðEÞ
4G0 ab
(2)
ðE EF Þ2
Since a and b are assumed to be negligible as compared with EEF, by formula (2) it is clear that G5G0, that is, the wire behaves as insulating. For simplicity, we express the Fermi energy level taking vacuum as reference scale so that one has that EF ¼ eV where e is the electron charge and V stands for electrochemical potential (see, for example, Ref. [13]). By inserting this expression into formula (2), it follows: GðE; VÞ
Notice that the lower limit of the integral of Eq. (6) is zero energy so that this value of electron energy must satisfy, of course, the condition a+b5E+eV. Then, inserting into this inequality the value E ¼ 0, it follows that Vb(a+b)/e. Consequently, relationship (7) is not valid for values of the electrochemical potential that do not satisfy the above condition. From expression (7) it follows that the total density of electronic off-resonant states is quasi-inversely proportional to the square of the electronic energy when EbeV. In addition, notice the dependence on the electrochemical potential of that density so that, if E5eV, the off-resonant states density is quasi-inversely proportional to the electrochemical potential. By formula (7), let us note that the density in question is practically independent of the electronic energy when E5eV.
3. Discussion
4G0 ab 2
ðE þ eVÞ
(3)
We will restrict ourselves to V40 because negative values of the electrochemical potential are not permitted given that the condition a+b5E+eV is not fulfilled for every E40 and Vo0 such that |V|oE/e. In effect, imagine sufficiently small values of E and |V| (E40, Vo0) such that |V|oE/e; in this case, the condition a+b5E+eV does not hold. For E ¼ 0, this condition reduces to a+b5eV (now we assume V40) so that the values of V which do not verify the above condition must not be considered. In addition, note that V ¼ 0 does not satisfy the condition in question since, for V ¼ 0, one should have that a+b50 which is absurd. Next we shall derive the characteristic VI of the wire from Eq. (3). To get this end, one has that the current intensity I(E,V) through our wire verifies that G(E,V) ¼ qI(E,V)/qV, so it follows: Z V GðE; VÞ dV (4) IðE; VÞ ¼
We have found out an approximate formula for the density of electronic off-resonant states in an insulating nanowire by starting from considerations relative to conductance. The insulating behavior arises from the fact that a+b5EEF so that offresonance is relatively very pronounced. On the left-hand side of Eq. (7), V-dependence has not been expressed explicitly because the DOS is commonly expressed as a function of the electronic energy only; therefore, mathematically speaking, V acts as a parameter so that Eq. (7) provides the values of the density of offresonant states in terms of E for a given value taken on by V. At this point, it is interesting to consider the Fermi energy of the electron gas associated with the nanoconstriction (see, for instance, Ref. [5]). As a function of temperature absolute, the Fermi energy in question is given by the well-known Sommerfeld formula namely (
0
EF ðTÞ EF ð0Þ 1
Replacing formula (3) into relation (4), one gets IðE; VÞ
4G0 abV EðE þ eVÞ
(5)
By inspecting formula (5) we see, on the one hand, that the current intensity is quasi-inversely proportional to the square of the electronic energy when EbeV and, on the other hand, that the current intensity becomes quasi-inversely proportional to the electronic energy if V-N. In Ref. [6] the current intensity has been found to be negative, contrarily to Eq. (5) by which this intensity is strictly positive (note that V40). Considering now the density of electronic off-resonant states, we can say that, in a first approximation, the LDOS is proportional to conductance (see, for instance, Refs. [9–12]). Therefore, by using Eq. (3), the total density of electronic off-resonant states (the integral of the local density extended to the spatial region enclosed by the nanowire) is given by gðEÞ
4aG0 ab ðE þ eVÞ2
(6)
where a is a positive constant corresponding to the above proportionality relationship. In order to evaluate a in terms of known quantities, let us denote by N the number of off-resonant states. Therefore, the integral of Eq. (6) between zero and infinite must be equal to N. Performing this integral with the integrand given by Eq. (6), one gets the approximate value of a which, replaced into formula (6), yields gðEÞ
NeV ðE þ eVÞ2
(7)
p2 k2 T 2 12½EF ð0Þ2
) (8)
pffiffiffi 2 (with p2 k T 2 o12½EF ð0Þ2 ; that is, To2 3EF ð0Þ=ðpkÞ), where k is the Boltzmann constant and the Fermi energy level extrapolated to zero absolute temperature is given by
EF ð0Þ ¼
_2 2m
ð3p2 nÞ2=3
(9)
where _ is the reduced Planck constant, m is the free-electron mass, and n stands for spatial electron density. Then, by changing adequately the reference scale of the electrochemical potential, the Fermi energy should be positive and therefore it should equate to formula (8) (with relation (9)) so that the temperature of the electron gas can be determined for a given value of the electrochemical potential. Nevertheless, the above equation, which involves the absolute value pffiffiffi of the electrochemical potential, is applicable only if To2 3EF ð0Þ=ðpkÞ that, by formula (9), leads to an inequality in terms of temperature and spatial electron density (note that this density refers to zero absolute temperature). We emphasize the fact that the values of V must match Eq. (8) (with formula (9)) under the restriction represented by the above quoted inequality on temperature so that the consideration of V outside the inequality in question does not make sense. Prior to calculate the density for off-resonant electronic states, we have derived an expression for the current intensity versus the electrochemical potential (Eq. (5)). In Eq. (5), the E-dependence of the current intensity can be conceived as that E acts as a parameter so that Eq. (5) defines really a family of VI curves; each curve corresponds to a given value of the parameter.
ARTICLE IN PRESS M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1621–1623
In other words, formula (5) gives the distribution of the current in terms of the electronic energy; at a given E, one has the V-variation of the current.
4. Concluding remarks Electronic transport through both resonant and off-resonant states plays a significant role in nanoscience. In particular, electrical conduction through resonant states is certainly relevant since, generally speaking, E ¼ EF (resonance) is the condition to determine conductance (consider carbon nanotubes and metallic nanowires). In practice, quasi-resonance (EEEF) takes place when we speak of, say, normal conduction in the above-mentioned nanostructures. As a particular case of resonance, we can mention tunneling resonant states whose relevance is manifest in order to tackle a wide variety of problems related to nanostructures as, for instance, quantum dots (see Ref. [14]). In Refs. [15–17], other relevant indirectly related problems on quantum transport through nanoscale systems are treated. In addition, experimental studies as Ref. [18] can be cited.
1623
References [1] A. Yazdani, D.M. Eigler, N.D. Lang, Science 272 (1996) 1921. [2] C.E.D. Chidsey, Science 251 (1991) 919. [3] C. Shu, C.Z. Li, H.X. He, A. Bogozi, J.S. Bunch, N.J. Tao, Phys. Rev. Lett. 84 (2000) 5196. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 114 (2003) 565. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 115 (2004) 476. [6] M.A. Grado-Caffaro, M. Grado-Caffaro, Phys. Lett. A 372 (2008) 4638. [7] J. Pan, T.W. Jing, S.M. Lindsay, J. Phys. Chem. 98 (1993) 4205. [8] G. Repphun, J. Halbritter, J. Vac. Sci. Tech. A 13 (1995) 1693. [9] M.A. Grado-Caffaro, M. Grado-Caffaro, Phys. A 387 (2008) 445. [10] J. Tersoff, D.R. Hamann, Phys. Rev. B 31 (1985) 805. [11] C.J. Chen, Introduction to STM, Oxford University Press, New York, 1993. [12] M.A. Grado-Caffaro, M. Grado-Caffaro, Modern Phys. Lett. B 20 (2006) 1853. [13] H. Heiss, A. Heller, J. Phys. Chem. 89 (1985) 4207. [14] V. Ermakov, S. Kruchinin, H. Hori, A. Fujiwara, Int. J. Modern Phys. B 21 (2007) 1827. [15] E.R. Bittner, J. Chem. Phys. 112 (2000) 9703. [16] D.S. Kosov, J. Chem. Phys. 116 (2002) 6368. [17] P. Pomorski, C. Roland, H. Guo, Phys. Rev. B 70 (2004) 115408. [18] S. Sahoo, C.-S. Yang, B. Doudin, Phys. Lett. A 352 (2006) 331.