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Quantized conductance in a mesoscopic Tomonaga-Luttinger liquid
ELSEVIER
Physica B 227 (1996) 252-255
Quantized conductance in a mesoscopic Tomonaga-Luttinger liquid Masao Ogata
Institute of Physics, Collegeof Arts and Sciences, Universityof Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Abstract The quantized value of conductance in a quantum wire is discussed near the Mott transition. The temperature dependence of the collapse of quantized conductance in the presence of magnetic field is also discussed by taking account of both impurity scattering and mutual Coulomb interactions.
Keywords: Conductance quantization; Tomonaga Luttinger liquid; Quantum wire; Mort transition
1. Introduction
Recent progress in nanostructure technology has enabled us to study the quantum transport in very clean quantum wires with various mean-free paths and system sizes. When the system size becomes longer, the quantum wire approaches an idealistic one-dimensional interacting electron system called Tomonaga- Luttinger (TL) liquid [ 1]. This non-Fermi liquid state has not been confirmed in one-dimensional systems except for the edge state of fractional quantum Hall states. It is thus very interesting to confirm the existence of the TL liquid in quantum wires. It was predicted that the collapse of the quantized conductance has a typical temperature and size dependences due to the TL characteristics [2]. Recently, Tarucha et al. studied quantum wires with fairly long length [3, 4] to find the collapse of the quantized conductance as predicted. Analyzing the temperature dependence, they concluded that the TL liquid behavior was observed. In this paper we discuss additional ways to confirm the TL liquid behavior experimentally. The effects of the Mott transition and the magnetic field are calculated using the one-dimensional Hubbard model and realistic model for quantum wires.
2. Quantized value of conductance Our model is a quantum wire of length L and width W which is connected to perfect leads on both sides. We assume that the carrier density, n, and W are such that there exists only a single subband along the wire (one-dimensional electrons). The Hamiltonian is given as H = y ~ ~(k)a~,sak,s + 1 Z
k,s
v(q)pqp_q
q
+
u(q)p(q),
(1)
q where ~(k)--h2k2/2m *, v(q) and u(q) are onedimensional Fourier transforms of the long-range Coulomb interaction and the random impurity potential, respectively (dirty Tomonaga-Luttinger liquid) [5, 2]. First, the conductance per channel in the presence of mutual interaction and without impurity scattering becomes [6] 2e 2 g = ~-KR,
(2)
where Kp is a correlation exponent which depends on the interaction strength and electron density. This is
0921-4526/96/$15.00 @ 1996 Elsevier ScienceB.V. All rights reserved PH S092 1-4526(96)0041 3-9
253
M. Ogata/ Physica B 227 (1996) 252-255
one of the effects of TL liquid and may be observed experimentally. Kp = 1 for noninteracting electrons, giving the usual Landauer formula. For the repulsive Hubbard model, Kp decreases as the Coulomb interaction increases and K o ~ ½ as U/t -~ oc. For the actual amplitude of the interaction, however, K o may be close to 1 to be detected experimentally. However, toward half-filling (i.e., near the Mott insulator), Kp ~ 1 irrespective of the interaction strength. Thus, if we can apply a periodic potential along the quantum wire and adjust the electron density close to half-filling, we will be able to see that the conductance quantization decreases according to Eq. (2) with K~, ~ ~. 1 For example, if we
3. Magnetic field effect Next we examine the conductance due to impurity scattering in the presence of magnetic field. For the quantum wire the conductivity was evaluated based on the Mori formula which reads [2, 5]
~r--
"ctr
rt -3 2d
x 105cm -1.
This value of k F is tOO small for the lowest subband in quantum wires but may be realized, for example, in the second subband [7]. The fact that K o approaches ½ can be understood by regarding the charge degrees of freedom as spinless fermions. In the spinless fermion case, there is no spin summation in the conductance formula, leading to half of the usual Landauer result. This is a direct consequence of the spin-charge separation near the Mott insulator. The Umklapp term, which becomes important in the vicinity of the Mort transition, is usually nonlinear and it is difficult to treat in the usual bosonization scheme. However, if we use a mapping of the charge degrees of freedom into a spinless fermion Hamiltonian [8]. the Umklapp term is rewritten as
f dxg'I(x)~'z(x)e-4ik~x + h.c.,
Hu - g322rt~
(3)
where 0~ and ~92 represent the spinless fermion operators on the two branches. Generally, the forward scattering terms (g2,g4) are transformed into complex forms, but for a special parameter vv + (g41[ -~g41)/2rt = 5(2gz 5 + glll)/2rt, they take simple bilinear forms. In this case the Kubo formula for smallsize systems gives g = eZ/h, which is half the ordinary quantized value. This means that the charge degrees of freedom are regarded as the non-interacting spinless fermions for any electron density in this special case.
+
'
=-- ni Z u2(k -- kr) k,k
apply a potential with period d = 500 A, the Fermi momentum corresponding to half-filling is
kv-
m*
~k
~k 1
× I m N(kco_- k', co) .... o'
(4)
where ni is the density of impurities and N(k, co) is the density-density correlation function which takes full account of the mutual interactions. From Eq. (4), the collapse of the quantization is naturally understood from the crossover between the two cases of the finite (quantized) conductance for short wires (2VF/L >>1/ztr), and the finite conductivity for longer wires ( 2vv /L ,~ 1/'Ctr). In the TL liquid it is well known that N(2kF, co) has a power-law behavior for small co and T [1]. This leads to 1/% (x (kB T/cov)K"- 1, where the temperature is normalized by the cut-off frequency, coF, which is of the order of Fermi energy. Taking account of the finite-size cut-off, Tarucha et al. [4] found this powerlaw behavior with Kp ~ 0.7. For the repulsive Hubbard model, ½<~Kp ~ 1. On the other hand, in our model with long-range Coulomb interaction, Eq. (1), Kp is estimated as
Kp=l
1+
~:0VF
v q~
.
(5)
Because of the long-range nature of the Coulomb interaction, v(q) has a In Iql divergence as q -~ 0. However, we find that at finite temperatures K t, can be estimated from v(q ~ T/COF). Using realistic parameters for quantum wires, e0 = 10, VF ~ 2 x 107 cm/s and assuming that v(q ~ T/cot ) in Ref. [5] is about 4, we get K o ~ 0.53. The difficulty in estimating the effective Coulomb interaction v(q) gives the ambiguity
of Kp.
254
M. Ooata/ Physica B 227 (1996) 252-255
When the magnetic field is applied, the Fermi surfaces and Fermi velocities for up and down spins become different. This difference gives rise to an additional term as [9]
zxvv~ Ikl(c~sk+ s~ck),
(6)
0.55
,
,
,
,
,
,
,
,
.
,
,
,
,
,
,
,
,
,
,
0.54
0.53
k
where AVE = (VvT -Vv~)/2 and ck,sk represent the bosonized operators for the charge and spin degrees of freedom. Due to this term, the spin and charge are mixed under the magnetic field, but the total Hamiltonian is still bilinear so that it can be diagonalized. Although the transformation is complex, the resulting Hamiltonian consists of two low-lying excitations which have mixed characters of spin and charge. Using these new eigenvectors and eigenvalues, conductivity is evaluated via Mori formula (4). The charge response functions for up and down spins have singularities at NT(2kvT,O9) and N~(2kvi,og) with different exponents, K T and Kl, respectively. Correspondingly, we get two exponents for 1/Ztr. For the Hubbard model at U/t ~ cx~ [10], these two exponents are KT,.L = 1(1 ~ m) 2 q- 0 -- 1, where m = (N T - N l )/(N T + N l ) and 0 is an increasing function of m: 0--+ 1 f o r m - - - + O a n d O - - ~ 2 f o r m - - ~ 1. Using realistic parameters for quantum wires, such as Eq. (5), we calculate K i which is shown in Fig. 1. Since the smaller exponent K l gives stronger divergence 1/Ztr cx (kBT/~OF)ICi -], we assume this exponent will be observed. Although the effect is small, the exponent changes about 4%.
4. Discussion and summary
In summary we discussed a few ways to confirm the existence of the TL liquid in quantum wires. Judging from the recent systematic investigation of conductance as a function of temperature and length of the system [4], it seems possible to check these TL liquid behaviors experimentally. In order to compare in detail the experimental data with theories, the effects such as sample geometries should be taken into account. However, these effects contain subtle problems in the interacting systems, so that we have focused on the most characteristic features of the TL liquid as a first step toward the understanding. It will be of considerable interest to study the behavior of the
XO- 0 . 5 2 0
0.51
I
0.50
l
r
l
l
2
l
l
l 4
l
L
I
t
l 6
l
l
l 8
10
H(Tesla) Fig. 1. Magnetic-field dependence of the exponent.
quantized conductance in the vicinity of the Mott transition. During the preparation of this manuscript, we notice that Kimura et al. [11] also studied the effect of magnetic field to the conductivity.
Acknowledgements
We thank H. Fukuyama, S. Tarucha, T. Tokura, Y. Hirayama and H. Aoki for useful discussions and comments. The present work is financially supported by Grant-in-Aid for Scientific Research on Priority Area "Anomalous metallic states near Mott transition".
References [1] J. S61yom, Adv. Phys. 28 (1979) 201. [2] M. Ogata and H. Fukuyama, Phys. Rev. Lett. 73 (1994) 468. [3] S. Tarucha, T. Saku, Y. Tokura and Y. Hirayama, Phys. Rev. B 47 (1993) 4064. [4] S. Tarucha, T. Honda and T. Saku, Solid State Commun. 94 (1995) 413. [5] H. Fukuyama, H. Kohno, and R. Shirasaki, J. Phys. Soc. Japan 62 (1993) 1109. [6] W. Apel and T.M. Rice, Phys. Rev. B 26 (1982) 7063; C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220. [7] M. Ogata and H. Fukuyama, Jpn. J. Appl. Phys. 34 (1995) 4284. [8] A. Luther and VJ. Emery, Phys. Rev. Lett. 33 (1974) 589. [9] H. Shiba and M. Ogata, Prog. Theor. Phys. (Suppl). 108 (1992) 265 and references therein.
M. 09ata/Physica B 227 (1996) 252-255 [10] M. Ogata, T. Sugiyama and H. Shiba, Phys. Rev. B 43 (1991) 8401. [11] T. Kimura, K. Kuroki and H. Aoki, preprint. [12] M. Ogata and H. Shiba, Phys. Rev. 13 41 (1990) 2326, Prog. Theor. Phys. (Suppl). 108 (1992) 265 and references therein. [13] H. Mori, Prog. Theor. Phys. (Suppl). 33 (1965) 423.
255
[14] A. Luther and 1. Peschel, Phys. Rev. Lett. 32 (1974) 992. [15] H. Fukuyama, H. Kohno, and R. Shirasaki, a report at Second International Symposium on "New Phenomena in Mesoscopic Structures" (December 7-1 l, 1992, Maui) and J. Phys. Soc. Japan 62 (1993) 1109.
Physica B 227 (1996) 252-255
Quantized conductance in a mesoscopic Tomonaga-Luttinger liquid Masao Ogata
Institute of Physics, Collegeof Arts and Sciences, Universityof Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Abstract The quantized value of conductance in a quantum wire is discussed near the Mott transition. The temperature dependence of the collapse of quantized conductance in the presence of magnetic field is also discussed by taking account of both impurity scattering and mutual Coulomb interactions.
Keywords: Conductance quantization; Tomonaga Luttinger liquid; Quantum wire; Mort transition
1. Introduction
Recent progress in nanostructure technology has enabled us to study the quantum transport in very clean quantum wires with various mean-free paths and system sizes. When the system size becomes longer, the quantum wire approaches an idealistic one-dimensional interacting electron system called Tomonaga- Luttinger (TL) liquid [ 1]. This non-Fermi liquid state has not been confirmed in one-dimensional systems except for the edge state of fractional quantum Hall states. It is thus very interesting to confirm the existence of the TL liquid in quantum wires. It was predicted that the collapse of the quantized conductance has a typical temperature and size dependences due to the TL characteristics [2]. Recently, Tarucha et al. studied quantum wires with fairly long length [3, 4] to find the collapse of the quantized conductance as predicted. Analyzing the temperature dependence, they concluded that the TL liquid behavior was observed. In this paper we discuss additional ways to confirm the TL liquid behavior experimentally. The effects of the Mott transition and the magnetic field are calculated using the one-dimensional Hubbard model and realistic model for quantum wires.
2. Quantized value of conductance Our model is a quantum wire of length L and width W which is connected to perfect leads on both sides. We assume that the carrier density, n, and W are such that there exists only a single subband along the wire (one-dimensional electrons). The Hamiltonian is given as H = y ~ ~(k)a~,sak,s + 1 Z
k,s
v(q)pqp_q
q
+
u(q)p(q),
(1)
q where ~(k)--h2k2/2m *, v(q) and u(q) are onedimensional Fourier transforms of the long-range Coulomb interaction and the random impurity potential, respectively (dirty Tomonaga-Luttinger liquid) [5, 2]. First, the conductance per channel in the presence of mutual interaction and without impurity scattering becomes [6] 2e 2 g = ~-KR,
(2)
where Kp is a correlation exponent which depends on the interaction strength and electron density. This is
0921-4526/96/$15.00 @ 1996 Elsevier ScienceB.V. All rights reserved PH S092 1-4526(96)0041 3-9
253
M. Ogata/ Physica B 227 (1996) 252-255
one of the effects of TL liquid and may be observed experimentally. Kp = 1 for noninteracting electrons, giving the usual Landauer formula. For the repulsive Hubbard model, Kp decreases as the Coulomb interaction increases and K o ~ ½ as U/t -~ oc. For the actual amplitude of the interaction, however, K o may be close to 1 to be detected experimentally. However, toward half-filling (i.e., near the Mott insulator), Kp ~ 1 irrespective of the interaction strength. Thus, if we can apply a periodic potential along the quantum wire and adjust the electron density close to half-filling, we will be able to see that the conductance quantization decreases according to Eq. (2) with K~, ~ ~. 1 For example, if we
3. Magnetic field effect Next we examine the conductance due to impurity scattering in the presence of magnetic field. For the quantum wire the conductivity was evaluated based on the Mori formula which reads [2, 5]
~r--
"ctr
rt -3 2d
x 105cm -1.
This value of k F is tOO small for the lowest subband in quantum wires but may be realized, for example, in the second subband [7]. The fact that K o approaches ½ can be understood by regarding the charge degrees of freedom as spinless fermions. In the spinless fermion case, there is no spin summation in the conductance formula, leading to half of the usual Landauer result. This is a direct consequence of the spin-charge separation near the Mott insulator. The Umklapp term, which becomes important in the vicinity of the Mort transition, is usually nonlinear and it is difficult to treat in the usual bosonization scheme. However, if we use a mapping of the charge degrees of freedom into a spinless fermion Hamiltonian [8]. the Umklapp term is rewritten as
f dxg'I(x)~'z(x)e-4ik~x + h.c.,
Hu - g322rt~
(3)
where 0~ and ~92 represent the spinless fermion operators on the two branches. Generally, the forward scattering terms (g2,g4) are transformed into complex forms, but for a special parameter vv + (g41[ -~g41)/2rt = 5(2gz 5 + glll)/2rt, they take simple bilinear forms. In this case the Kubo formula for smallsize systems gives g = eZ/h, which is half the ordinary quantized value. This means that the charge degrees of freedom are regarded as the non-interacting spinless fermions for any electron density in this special case.
+
'
=-- ni Z u2(k -- kr) k,k
apply a potential with period d = 500 A, the Fermi momentum corresponding to half-filling is
kv-
m*
~k
~k 1
× I m N(kco_- k', co) .... o'
(4)
where ni is the density of impurities and N(k, co) is the density-density correlation function which takes full account of the mutual interactions. From Eq. (4), the collapse of the quantization is naturally understood from the crossover between the two cases of the finite (quantized) conductance for short wires (2VF/L >>1/ztr), and the finite conductivity for longer wires ( 2vv /L ,~ 1/'Ctr). In the TL liquid it is well known that N(2kF, co) has a power-law behavior for small co and T [1]. This leads to 1/% (x (kB T/cov)K"- 1, where the temperature is normalized by the cut-off frequency, coF, which is of the order of Fermi energy. Taking account of the finite-size cut-off, Tarucha et al. [4] found this powerlaw behavior with Kp ~ 0.7. For the repulsive Hubbard model, ½<~Kp ~ 1. On the other hand, in our model with long-range Coulomb interaction, Eq. (1), Kp is estimated as
Kp=l
1+
~:0VF
v q~
.
(5)
Because of the long-range nature of the Coulomb interaction, v(q) has a In Iql divergence as q -~ 0. However, we find that at finite temperatures K t, can be estimated from v(q ~ T/COF). Using realistic parameters for quantum wires, e0 = 10, VF ~ 2 x 107 cm/s and assuming that v(q ~ T/cot ) in Ref. [5] is about 4, we get K o ~ 0.53. The difficulty in estimating the effective Coulomb interaction v(q) gives the ambiguity
of Kp.
254
M. Ooata/ Physica B 227 (1996) 252-255
When the magnetic field is applied, the Fermi surfaces and Fermi velocities for up and down spins become different. This difference gives rise to an additional term as [9]
zxvv~ Ikl(c~sk+ s~ck),
(6)
0.55
,
,
,
,
,
,
,
,
.
,
,
,
,
,
,
,
,
,
,
0.54
0.53
k
where AVE = (VvT -Vv~)/2 and ck,sk represent the bosonized operators for the charge and spin degrees of freedom. Due to this term, the spin and charge are mixed under the magnetic field, but the total Hamiltonian is still bilinear so that it can be diagonalized. Although the transformation is complex, the resulting Hamiltonian consists of two low-lying excitations which have mixed characters of spin and charge. Using these new eigenvectors and eigenvalues, conductivity is evaluated via Mori formula (4). The charge response functions for up and down spins have singularities at NT(2kvT,O9) and N~(2kvi,og) with different exponents, K T and Kl, respectively. Correspondingly, we get two exponents for 1/Ztr. For the Hubbard model at U/t ~ cx~ [10], these two exponents are KT,.L = 1(1 ~ m) 2 q- 0 -- 1, where m = (N T - N l )/(N T + N l ) and 0 is an increasing function of m: 0--+ 1 f o r m - - - + O a n d O - - ~ 2 f o r m - - ~ 1. Using realistic parameters for quantum wires, such as Eq. (5), we calculate K i which is shown in Fig. 1. Since the smaller exponent K l gives stronger divergence 1/Ztr cx (kBT/~OF)ICi -], we assume this exponent will be observed. Although the effect is small, the exponent changes about 4%.
4. Discussion and summary
In summary we discussed a few ways to confirm the existence of the TL liquid in quantum wires. Judging from the recent systematic investigation of conductance as a function of temperature and length of the system [4], it seems possible to check these TL liquid behaviors experimentally. In order to compare in detail the experimental data with theories, the effects such as sample geometries should be taken into account. However, these effects contain subtle problems in the interacting systems, so that we have focused on the most characteristic features of the TL liquid as a first step toward the understanding. It will be of considerable interest to study the behavior of the
XO- 0 . 5 2 0
0.51
I
0.50
l
r
l
l
2
l
l
l 4
l
L
I
t
l 6
l
l
l 8
10
H(Tesla) Fig. 1. Magnetic-field dependence of the exponent.
quantized conductance in the vicinity of the Mott transition. During the preparation of this manuscript, we notice that Kimura et al. [11] also studied the effect of magnetic field to the conductivity.
Acknowledgements
We thank H. Fukuyama, S. Tarucha, T. Tokura, Y. Hirayama and H. Aoki for useful discussions and comments. The present work is financially supported by Grant-in-Aid for Scientific Research on Priority Area "Anomalous metallic states near Mott transition".
References [1] J. S61yom, Adv. Phys. 28 (1979) 201. [2] M. Ogata and H. Fukuyama, Phys. Rev. Lett. 73 (1994) 468. [3] S. Tarucha, T. Saku, Y. Tokura and Y. Hirayama, Phys. Rev. B 47 (1993) 4064. [4] S. Tarucha, T. Honda and T. Saku, Solid State Commun. 94 (1995) 413. [5] H. Fukuyama, H. Kohno, and R. Shirasaki, J. Phys. Soc. Japan 62 (1993) 1109. [6] W. Apel and T.M. Rice, Phys. Rev. B 26 (1982) 7063; C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220. [7] M. Ogata and H. Fukuyama, Jpn. J. Appl. Phys. 34 (1995) 4284. [8] A. Luther and VJ. Emery, Phys. Rev. Lett. 33 (1974) 589. [9] H. Shiba and M. Ogata, Prog. Theor. Phys. (Suppl). 108 (1992) 265 and references therein.
M. 09ata/Physica B 227 (1996) 252-255 [10] M. Ogata, T. Sugiyama and H. Shiba, Phys. Rev. B 43 (1991) 8401. [11] T. Kimura, K. Kuroki and H. Aoki, preprint. [12] M. Ogata and H. Shiba, Phys. Rev. 13 41 (1990) 2326, Prog. Theor. Phys. (Suppl). 108 (1992) 265 and references therein. [13] H. Mori, Prog. Theor. Phys. (Suppl). 33 (1965) 423.
255
[14] A. Luther and 1. Peschel, Phys. Rev. Lett. 32 (1974) 992. [15] H. Fukuyama, H. Kohno, and R. Shirasaki, a report at Second International Symposium on "New Phenomena in Mesoscopic Structures" (December 7-1 l, 1992, Maui) and J. Phys. Soc. Japan 62 (1993) 1109.