- Email: [email protected]
Conductance of metallic single-wall nanotubes with single magnetic impurities
Available online at www.sciencedirect.com
Physica E 22 (2004) 833 – 837 www.elsevier.com/locate/physe
Conductance of metallic single-wall nanotubes with single magnetic impurities A. Namiraniana; b;∗ , S. Jafarzadehb a Institute
b Department
for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195-159, Iran of Physics, Iran University of Science and Technology, Narmak, Teheran 16345, Iran Received 6 August 2003; accepted 22 September 2003
Abstract Using a model of conducting cylinder with a few number of impurities on its surface, we investigate the e/ects of magnetic impurity scattering on the conductance of metallic single-wall carbon nanotubes. The nonlinear part of conductance, which is due to the interaction of conduction electrons with impurities, is obtained. The signature of Kondo anomaly is found in the nonlinear conductance and it is shown that its amplitude strongly depends on the position of impurities and diameter of nanotube. ? 2003 Elsevier B.V. All rights reserved. PACS: 72.15.Qm; 73.23.−b; 73.63.Fg Keywords: Magnetic impurity; Kondo e/ect; Carbon nanotube
A single-wall nanotube (SWNT) can be simply described as a sheet of graphite coaxially rolled to create a cylindrical surface. Depending on their chirality, SWNTs vary from being metallic to semiconducting. The chirality is de
∗ Corresponding author. Tel.: +98-(0)21-745-7733; fax: +98(0)21-689-6622. E-mail address: [email protected] (A. Namiranian).
to the Landauer theory for a SWNT with no defect and ideal contacts, the bands contribute two conductance quanta (=4e2 =h) to the conductance [3,4]. However, it is argued both experimentally [5–7] and theoretically [8–10] that electrical conductance of a SWNT can be substantially modi
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.09.040
834
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
describes the interaction of electrons with electric
+ S − ck† ↑ ck↓ + S + ck† ↓ ck↑ ]: Fig. 1. The model of the SWNT in the form of a long cylindrical surface of the radius R, which smoothly connects two massive metallic reservoirs. The impurities on the surface of cylinder are shown schematically.
Theoretical study of magnetic impurity e/ects on SWNT has started recently [13,14]. In this article we investigate the problem of a metallic SWNT containing a few number of magnetic impurity, concentrating on the inJuence of impurity positions and radius of nanotube. We calculate the conductance of a metallic SWNT as a long ballistic cylindrical wire due to the presence of magnetic impurities located on its surface. The study is made of the
(1)
where † H0 =
k ck ck
(2)
k;
is the Hamiltonian of free electrons, eV † sign(vz )ck ck H1 = 2 k;
† (ck ) creates (annihilates) a conHere the operator ck duction electron with spin , wave function ’k , and energy k , k is the full set of quantum numbers of electrons, S denotes the spin of impurity; vz is the electron velocity along the axis of cylinder; and Jj; k; k is the matrix element of the exchange interaction of electron with impurity situated in the point rj Jj; k; k = dr J (r; rj )’k (r)’∗k (r): (5)
The conductance of the system is described by the Landauer formula, which applies if the wave function can spread over the whole sample. In order to investigate the inJuence of single impurities on the nonlinear quantum conductance of the wire, we use the method, which was developed by Kulik and others [15,16]. The change in the electrical current MI is related to the rate of energy dissipation by the relation: dH1 dE = : (6) MIV = dt dt Di/erential of H1 with respect to time t is obtained from Heisenberg equation. The change MI of the current due to interactions of electrons with magnetic impurities, would then be 1 MIV = [H1 (t); Hint (t)]; (7) i˝ where · · · = Tr((t) · · ·) (all operators are in the interaction representation). The statistical operator (t) satis
(3)
(4)
1 (i˝)2
t
−∞
dt
t
−∞
dt
× [Hint (t ); [Hint (t ); 0 ]] + · · · :
(9)
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
We would then have MI = I1 + I2 + · · · t 1 =− 2 dt Tr(0 [[H1 ; Hint (t)]; Hint (t )]) ˝ V −∞ i + 3 ˝ V
t
−∞
dt
t
−∞
dt Tr(0 [[[H1 ; Hint (t)];
Hint (t )]; Hint (t )]) + · · · :
× (fm − fn ) ( n − m )Jj; n; m Ji; m; n ; I2 =
e s(s + 1) (sign vzk − sign vzn ) ˝ n; m; k i; j;l
× [2fn (fk − fm ) + (fm − fk )] 1 × ( n − k ) Pr
m − k 1 + ( m − k )Pr
n − k × [Jj; n; k Ji; m; n Jl; k; m + Jj; k; n Ji; n; m Jl; m; k ];
of the problem, hereafter we use following approximations:
(10)
After the simple, but cumbersome calculations we
(11)
where fn = fFD ( n + (eV=2)sign vz ) depends on the voltage. The
835
I2 =
J 3 e2 s(s + 1) (sign vzk − sign vzn ) ˝ n; m; k i; j;l
× Re[’∗k (rj )’∗n (ri )’∗m (rl )’k (rl )’m (ri )’n (rj )] 1 × ( n − k )Pr
k − m 1 + ( m − k )Pr
n − k eV × 2 sign vzn F − n − sign vzn 2 eV ×
F − k − sign vzk 2 eV −
F − m − sign vzm 2 eV +
F − n − sign vzm 2 eV −
m + sign − F vzm 2 eV × sign vzk F − k − sign vzk 2 eV − sign vzm F − m − : (12) sign vzm 2 So far, our results within the framework of perturbation theory are exact, general and applicable for a real metallic SWNT. The values for energy and wave function of electrons include all details of shape and band structure of SWNT. Carrying out numerical calculation, in this step we consider a very simple model for such values, neglecting any structure for nanotube. In this model, we use the free electron model moving in
836
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
the surface of a cylinder
- G2 / G 0
p z
1 k ’k (r) = √ exp i + k " ˝ R 2 2 pzk ˝2 k + ; 2me 2me R2
(14)
where two quantum numbers, an integer quantum number k and a continuous quantum number (momentum along wire), pzk , characterizes the electrons. Within this model and in the case of a single impurity, Eq. (12) can be analytically integrated over continuous momenta and the nonlinear part of conductance, G2 takes the following form:
(Arbitrary units)
k =
2.5
(13)
2
3 1.5
2 4 1
1 0
0.0001
0.0002
0.0003
0.0004
eV (Fermi energy units)
Fig. 2. The nonlinear conductance as a function of voltage for a wire with a single magnetic impurity. Di/erent curves correspond to wires with di/erent radius, R: curve one (R = 3$F ), curve two (R = 4$F ), curve three (R = 4:5$F ) and curve four (R = 5$F ).
e 2 m3 G2 = − 4 2 e 3 J 3 s(s + 1) ˝ R (–) (–) (–) [qk qm qn ]−1 × k ; m ; n –=±
q(–) − q(−–) q(–) n n k × ln (–) (–) q + q(−–) q n n n q(–) q(−–) − q(−–) q(–) k m m k +(1 − k m )ln (–) (−–) q q + q(−–) q(–)
k
(q(–) )2 − (q(−–) )2 k k + k m ln ; (q(−–) )2
0.25
m
(15)
k
where
2 eV ˝2 k (±) qk = 2me F ± − 2 2me R2
(Arbitrary units)
m
k
- G2 / G 0
0.2
(16) 1.725
1.75
1.775
1.8
R (Fermi wavelength units)
Eq. (15) diverges logarithmically at zero bias, V = 0 which is the character of electron-magnetic impurity scattering. This equation also diverges at qk± =0. Physically it means that in the Born approximation, the slowly moving electron is repeatedly scattered on the impurity. In this case the perturbation theory is not valid anymore. In our numerical presentation we avoid these points. In Fig. 2 the dependence of the nonlinear conductance on the applied bias is shown for cylinders with the di/erent diameters, each one including one
Fig. 3. The nonlinear conductance as a function of radius of wire. Di/erent curves correspond to wires lying in di/erent voltages V : eV = 0:15 F (solid curve) and eV = 0:2 F (dashed curve).
magnetic impurity. Obviously the curves have similar shapes, but the values of G2 are di/erent, con
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
In summary, magnetic impurity e/ects on metallic SWNT have been investigated theoretically. Signatures of the Kondo e/ect are predicted in conductance of such systems. We have shown that in metallic SWNT the spatial distribution of magnetic impurities inJuences the nonlinear dependence of the conductance on the applied voltage. The nonlinear part of conductance also depends on the diameter of SWNT. One of the authors (A.N.) would like to thank the Institute for Advanced Studies in Basic Sciences, Zanjan, for support. References [1] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerines and Carbon Nanotubes, Academic Press, San Diego, CA, 1996. [2] C. Dekker, Phys. Today 52 (5) (1999) 22. [3] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 632.
837
[4] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [5] L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 76 (1996) 971. [6] M. Bockrath, W. Liang, D. Bozovic, J.H. Hafner, C.H. Lieber, M. Tinkham, H. Park, Science 291 (2001) 283. [7] Z. Yao, H.W.Ch. Postma, L. Balents, C. Dekker, Nature 402, (99) 273. [8] M.B. Nardelli, Phys. Rev. B 60 (1999) 7828. [9] H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 84 (2000) 2917. [10] L. Chico, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. B 54 (1996) 2600. [11] L. Grigorian, G.U. Sumanasekera, A.L. Loper, S.L. Fang, J.L. Allen, P.C. Eklund, Phys. Rev. B 60 (1999) R11309. [12] T.A. Odom, J.-l. Huang, C.L. Cheung, C.M. Lieber, Science 290 (2000) 1549. [13] G.A. Fiete, G. Zarand, B.I. Halperin, Y. Oreg, Phys. Rev. B 66 (2002) 024431. [14] K. Harigaya, New J. Phys. 2 (2000) 9.1. [15] I.O. Kulik, JETP Lett. 5 (1967) 345. [16] I.O. Kulik, A.N. Omelyanchouk, I.G. Tuluzov, Sov. J. Low Temp. Phys. 14 (1988) 149.
Physica E 22 (2004) 833 – 837 www.elsevier.com/locate/physe
Conductance of metallic single-wall nanotubes with single magnetic impurities A. Namiraniana; b;∗ , S. Jafarzadehb a Institute
b Department
for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195-159, Iran of Physics, Iran University of Science and Technology, Narmak, Teheran 16345, Iran Received 6 August 2003; accepted 22 September 2003
Abstract Using a model of conducting cylinder with a few number of impurities on its surface, we investigate the e/ects of magnetic impurity scattering on the conductance of metallic single-wall carbon nanotubes. The nonlinear part of conductance, which is due to the interaction of conduction electrons with impurities, is obtained. The signature of Kondo anomaly is found in the nonlinear conductance and it is shown that its amplitude strongly depends on the position of impurities and diameter of nanotube. ? 2003 Elsevier B.V. All rights reserved. PACS: 72.15.Qm; 73.23.−b; 73.63.Fg Keywords: Magnetic impurity; Kondo e/ect; Carbon nanotube
A single-wall nanotube (SWNT) can be simply described as a sheet of graphite coaxially rolled to create a cylindrical surface. Depending on their chirality, SWNTs vary from being metallic to semiconducting. The chirality is de
∗ Corresponding author. Tel.: +98-(0)21-745-7733; fax: +98(0)21-689-6622. E-mail address: [email protected] (A. Namiranian).
to the Landauer theory for a SWNT with no defect and ideal contacts, the bands contribute two conductance quanta (=4e2 =h) to the conductance [3,4]. However, it is argued both experimentally [5–7] and theoretically [8–10] that electrical conductance of a SWNT can be substantially modi
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.09.040
834
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
describes the interaction of electrons with electric
+ S − ck† ↑ ck↓ + S + ck† ↓ ck↑ ]: Fig. 1. The model of the SWNT in the form of a long cylindrical surface of the radius R, which smoothly connects two massive metallic reservoirs. The impurities on the surface of cylinder are shown schematically.
Theoretical study of magnetic impurity e/ects on SWNT has started recently [13,14]. In this article we investigate the problem of a metallic SWNT containing a few number of magnetic impurity, concentrating on the inJuence of impurity positions and radius of nanotube. We calculate the conductance of a metallic SWNT as a long ballistic cylindrical wire due to the presence of magnetic impurities located on its surface. The study is made of the
(1)
where † H0 =
k ck ck
(2)
k;
is the Hamiltonian of free electrons, eV † sign(vz )ck ck H1 = 2 k;
† (ck ) creates (annihilates) a conHere the operator ck duction electron with spin , wave function ’k , and energy k , k is the full set of quantum numbers of electrons, S denotes the spin of impurity; vz is the electron velocity along the axis of cylinder; and Jj; k; k is the matrix element of the exchange interaction of electron with impurity situated in the point rj Jj; k; k = dr J (r; rj )’k (r)’∗k (r): (5)
The conductance of the system is described by the Landauer formula, which applies if the wave function can spread over the whole sample. In order to investigate the inJuence of single impurities on the nonlinear quantum conductance of the wire, we use the method, which was developed by Kulik and others [15,16]. The change in the electrical current MI is related to the rate of energy dissipation by the relation: dH1 dE = : (6) MIV = dt dt Di/erential of H1 with respect to time t is obtained from Heisenberg equation. The change MI of the current due to interactions of electrons with magnetic impurities, would then be 1 MIV = [H1 (t); Hint (t)]; (7) i˝ where · · · = Tr((t) · · ·) (all operators are in the interaction representation). The statistical operator (t) satis
(3)
(4)
1 (i˝)2
t
−∞
dt
t
−∞
dt
× [Hint (t ); [Hint (t ); 0 ]] + · · · :
(9)
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
We would then have MI = I1 + I2 + · · · t 1 =− 2 dt Tr(0 [[H1 ; Hint (t)]; Hint (t )]) ˝ V −∞ i + 3 ˝ V
t
−∞
dt
t
−∞
dt Tr(0 [[[H1 ; Hint (t)];
Hint (t )]; Hint (t )]) + · · · :
× (fm − fn ) ( n − m )Jj; n; m Ji; m; n ; I2 =
e s(s + 1) (sign vzk − sign vzn ) ˝ n; m; k i; j;l
× [2fn (fk − fm ) + (fm − fk )] 1 × ( n − k ) Pr
m − k 1 + ( m − k )Pr
n − k × [Jj; n; k Ji; m; n Jl; k; m + Jj; k; n Ji; n; m Jl; m; k ];
of the problem, hereafter we use following approximations:
(10)
After the simple, but cumbersome calculations we
(11)
where fn = fFD ( n + (eV=2)sign vz ) depends on the voltage. The
835
I2 =
J 3 e2 s(s + 1) (sign vzk − sign vzn ) ˝ n; m; k i; j;l
× Re[’∗k (rj )’∗n (ri )’∗m (rl )’k (rl )’m (ri )’n (rj )] 1 × ( n − k )Pr
k − m 1 + ( m − k )Pr
n − k eV × 2 sign vzn F − n − sign vzn 2 eV ×
F − k − sign vzk 2 eV −
F − m − sign vzm 2 eV +
F − n − sign vzm 2 eV −
m + sign − F vzm 2 eV × sign vzk F − k − sign vzk 2 eV − sign vzm F − m − : (12) sign vzm 2 So far, our results within the framework of perturbation theory are exact, general and applicable for a real metallic SWNT. The values for energy and wave function of electrons include all details of shape and band structure of SWNT. Carrying out numerical calculation, in this step we consider a very simple model for such values, neglecting any structure for nanotube. In this model, we use the free electron model moving in
836
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
the surface of a cylinder
- G2 / G 0
p z
1 k ’k (r) = √ exp i + k " ˝ R 2 2 pzk ˝2 k + ; 2me 2me R2
(14)
where two quantum numbers, an integer quantum number k and a continuous quantum number (momentum along wire), pzk , characterizes the electrons. Within this model and in the case of a single impurity, Eq. (12) can be analytically integrated over continuous momenta and the nonlinear part of conductance, G2 takes the following form:
(Arbitrary units)
k =
2.5
(13)
2
3 1.5
2 4 1
1 0
0.0001
0.0002
0.0003
0.0004
eV (Fermi energy units)
Fig. 2. The nonlinear conductance as a function of voltage for a wire with a single magnetic impurity. Di/erent curves correspond to wires with di/erent radius, R: curve one (R = 3$F ), curve two (R = 4$F ), curve three (R = 4:5$F ) and curve four (R = 5$F ).
e 2 m3 G2 = − 4 2 e 3 J 3 s(s + 1) ˝ R (–) (–) (–) [qk qm qn ]−1 × k ; m ; n –=±
q(–) − q(−–) q(–) n n k × ln (–) (–) q + q(−–) q n n n q(–) q(−–) − q(−–) q(–) k m m k +(1 − k m )ln (–) (−–) q q + q(−–) q(–)
k
(q(–) )2 − (q(−–) )2 k k + k m ln ; (q(−–) )2
0.25
m
(15)
k
where
2 eV ˝2 k (±) qk = 2me F ± − 2 2me R2
(Arbitrary units)
m
k
- G2 / G 0
0.2
(16) 1.725
1.75
1.775
1.8
R (Fermi wavelength units)
Eq. (15) diverges logarithmically at zero bias, V = 0 which is the character of electron-magnetic impurity scattering. This equation also diverges at qk± =0. Physically it means that in the Born approximation, the slowly moving electron is repeatedly scattered on the impurity. In this case the perturbation theory is not valid anymore. In our numerical presentation we avoid these points. In Fig. 2 the dependence of the nonlinear conductance on the applied bias is shown for cylinders with the di/erent diameters, each one including one
Fig. 3. The nonlinear conductance as a function of radius of wire. Di/erent curves correspond to wires lying in di/erent voltages V : eV = 0:15 F (solid curve) and eV = 0:2 F (dashed curve).
magnetic impurity. Obviously the curves have similar shapes, but the values of G2 are di/erent, con
A. Namiranian, S. Jafarzadeh / Physica E 22 (2004) 833 – 837
In summary, magnetic impurity e/ects on metallic SWNT have been investigated theoretically. Signatures of the Kondo e/ect are predicted in conductance of such systems. We have shown that in metallic SWNT the spatial distribution of magnetic impurities inJuences the nonlinear dependence of the conductance on the applied voltage. The nonlinear part of conductance also depends on the diameter of SWNT. One of the authors (A.N.) would like to thank the Institute for Advanced Studies in Basic Sciences, Zanjan, for support. References [1] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerines and Carbon Nanotubes, Academic Press, San Diego, CA, 1996. [2] C. Dekker, Phys. Today 52 (5) (1999) 22. [3] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 632.
837
[4] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [5] L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 76 (1996) 971. [6] M. Bockrath, W. Liang, D. Bozovic, J.H. Hafner, C.H. Lieber, M. Tinkham, H. Park, Science 291 (2001) 283. [7] Z. Yao, H.W.Ch. Postma, L. Balents, C. Dekker, Nature 402, (99) 273. [8] M.B. Nardelli, Phys. Rev. B 60 (1999) 7828. [9] H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 84 (2000) 2917. [10] L. Chico, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. B 54 (1996) 2600. [11] L. Grigorian, G.U. Sumanasekera, A.L. Loper, S.L. Fang, J.L. Allen, P.C. Eklund, Phys. Rev. B 60 (1999) R11309. [12] T.A. Odom, J.-l. Huang, C.L. Cheung, C.M. Lieber, Science 290 (2000) 1549. [13] G.A. Fiete, G. Zarand, B.I. Halperin, Y. Oreg, Phys. Rev. B 66 (2002) 024431. [14] K. Harigaya, New J. Phys. 2 (2000) 9.1. [15] I.O. Kulik, JETP Lett. 5 (1967) 345. [16] I.O. Kulik, A.N. Omelyanchouk, I.G. Tuluzov, Sov. J. Low Temp. Phys. 14 (1988) 149.