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Electronic transport in quasi-1D mesoscopic systems: the correlated electron approach
Physica E 7 (2000) 786–789
www.elsevier.nl/locate/physe
Electronic transport in quasi-1D mesoscopic systems: the correlated electron approach C.F. Destefani, G.E. Marques ∗ Departamento de Fsica, Universidade Federal de S˜ao Carlos, Via Washington Luiz, Km 235, 13565-905 S˜ao Carlos, S˜ao Paulo, Brazil
Abstract The Coulomb interaction plays a crucial role in transport properties of ultra small systems and is the most relevant energy scale for dynamical properties of mesoscopic structures. The strongly correlated model for electrons is used to calculate the tunneling current through a dot weakly coupled to 2D free electron channels. The model takes into account a bath of phonons in the island, a magnetic eld applied parallel to the current and the electron–phonon interaction in the island. ? 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the recent years, the transport properties of ultrasmall regions such as quantum dots (QD) have shown great interest [1,2] since their carrier occupation can be controlled by an external applied voltage. These islands are separated from the rest of the material by tunneling barriers which connect them to free electron leads. The intrinsic spatial con nement and the discrete energy spectrum [3,4] may in uence the transport properties of these mesoscopic systems. In metallic dots, where the pro le of the barriers can be drawn by the cross applied voltage, the electronic density may be larger than in semiconductor dots. Their energy spectra can be treated as a continuum and the average separation between single-particle levels, E; ∗ Corresponding author. Tel.: +55-16-260-8222; fax:+55-16261-4835. E-mail address: [email protected] (G.E. Marques)
is almost negligible. For semiconductor dots, the level separation becomes a crucial parameter to the study of the electronic properties. In the rst case, the usual charging model [5,6], which reduces the electron– electron interaction to a constant times the square of number of electrons in the island, can reproduce well the experimental data. For systems with small level separation it becomes necessary to consider the electron–electron interaction from a microscopic point of view in order to obtain the many-particle eigenstates. This is the main approach within the correlated electron model [7–9] that will be used in this work. The important energy scales in the transport properties of these islands are: (i) thermal uctuations, ET = kB T and (ii) charging energy, EC = e2 =C, where C is the capacitance measuring the electrostatic e ect due to the addition of an electron to the system. In metallic islands, when EC ET , the charge quantization dominates the transport properties and gives rise to Coulomb blockade (CB) and single-electron
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 0 5 8 - 8
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
tunneling oscillations (SETO) which can be seen as Coulomb steps or peaks in the I × V curves. When EC ET , the thermal uctuations tend to destroy these structures. In semiconductor dots, the ratio (EC =E) de nes a competition between charge and energy quantizations that would appear as additional ne structures over the Coulomb steps or peaks. In a linear regime, there will be transitions only between fundamental many-particle states and in a non-linear regime the transitions to excited states can also contribute to the transport properties of the island. In this work we will be using a model [10] which consider the island as a quasi-1D quantum dot of a typical size, L, which can be occupied by n electrons. The model includes a magnetic eld applied parallel to the current, a phonon thermal bath and treats the inelastic processes due to transitions generated by the electron–phonon interaction. We will also assume that the electronic motion in the island will be coherent and the transport is under ballistic regime. That is, we are assuming that the phase coherence time, ’ , is much larger than the time an electron takes to travel the typical distance L and, in this sense, the phase coherence of eigenstates can be preserved between the electron–phonon scattering processes. Certainly, the eigenstates would loose the phase coherence after ’ . The Hamiltonian for the mesoscopic double barrier structure is separated as H = (H0 + HP ), where H0 = (HD + HL + HR + HPh ) will describe the isolated systems formed by electrons in the dot (HD ), the left and right free electron channels P + (HL=R = k; kL=R cL=R;k; cL=R;k; ) and the phononic P 1 bath (HPh = q ˜!q (a+ q aq + 2 )), respectively. The term HD , measured in units of the e ective Hartree energy EH∗ , has the form ∗ 2 aB P + (m − e + gB B0 )cm; HD = cm; L m; 1 a∗B P + hm1 m2 |V |m3 m4 i 2 L mi ; j × cm+1 ;1 cm+2 ;2 cm3 ;2 cm4 ;1 :
(1)
In these expressions, kL=R is the free electron energy in the left=right channel with momentum k; is the spin, ˜!q is the phonon energy, m is the non-interacting energy of electrons in the mth state, |mi, of the island, −e is the electrostatic potential seen by carriers, g
787
Fig. 1. The e ect of the electron–phonon interaction on the mesoscopic current of a system at low temperature, for Hubbard-like (left) and correlated electron (right) models for the electron– electron interaction. The dashed (solid) lines are results without (with) the electron–phonon interaction. The voltage is measured in units of (e=C) in the left and in units of (EH∗ =e) in the right.
is the Lande g-factor, B is the Bohr magneton, B0 is the applied magnetic eld and hm1 m2 |V |m3 m4 i is the Coulomb electron–electron interaction in the island. Furthermore, a∗B is the e ective Bohr radius which scales all distances in this work. On the other hand, HP = (HLT + HRT + Hep ) will describe the interactions between the QD and the free electron channels as P L=R + T ∗ + = (Tk; m )cL=R;k; cm; + (Tk;L=R HL=R m ) cm; cL=R;k; k; m;
and Hep =
P q; m1 ;m2 ;
p g(q; m1 ; m2 ) cm+1 ; cm2 ; (aq + a+ q)
represents the electron–phonon interaction inside the dot. Furthermore, Tk;L=R m is the transition probability between dot and free electron channels and p g(q; m1 ; m2 ) is the electron–phonon interaction within the Frohlich approximation. The transition rates from an initial ( ) to a nal ( ) state, in the QD, can be calculated as 2 |h (0) |HP | (0) i|2 (E (0) − E (0) ); (2)
; = ˜ where each state | i(0) i is a solution of full H0 with eigenvalues Ei(0) . The selection rules for transitions
788
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
Fig. 2. E ects of a magnetic eld on the mesoscopic current for B0 = 1 T (dotted lines), B0 = 5 T (dashed lines) and B0 = 10 T (solid lines). The induced magnetic uctuation is also shown for low (B0 = 1 T) and high (B0 = 10 T) magnetic elds in dotted and solid lines, respectively.
between these states, which are labeled by the number of occupation n, total spin S, and spin z-component M can be summarized as: n = (n − n ) = ±1; S = (S − S ) = ± 12 and M = (M − M ) = ± 12 . The renormalized rates of transitions between states in the island are calculated as * + P
; ; (3) ; = l ;r ;Q
th(l ;r ;Q )
where h i stands for thermal average. As a nal step, we calculate the time evolution of the occupation of a given eigenstate, | (0) i, in the dot from the solution of the occupation probability, P , in the form P d P = ( ; P − ; P ); (4) dt 6=
P under the condition P = 1. By looking for stationary conditions (dP =dt = 0) for the occupation of the eigenstates in the island one can obtain the averaged probability, P of a state , and the total DC-current through the system as P − + P ( ;L=R; − ;L=R; ); (5) I ≡ I L=R = (−=+)e ; ( 6= )
which takes into account the net balance between the number of electrons entering and leaving the island per unit of time. 2. Discussion and results We will show basically two curves for total current through the island calculated for low temperature
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
regime or ET E: (i) The current versus a voltage determined by the di erence of the chemical potentials of the leads (eV = L − R ); (ii) The current versus an external voltage, e = e(CG VG + CL VL + CR VR )=(CG + CL + CR ), for xed and equal capacitances CG=L=R and voltages VG=L=R applied to the gate and leads, respectively. In Fig. 1 we compare the results of the electron– phonon interaction on the total current according to the model used to treat the electron–electron interaction. On the left, we are showing the currents in the charging or the Hubbard-like model for zero eld (B0 = 0). On the right, we show the currents in the correlated electron model for magnetic eld B0 = 1 T. In both cases we observe that the overall e ect due to the inclusion of the electron–phonon interaction (solid lines) is small since the inelastic scattering transitions can only occur between initial and nal states with the same quantum numbers. In the charging model, the states are labeled only by the number n, whereas in the correlated model, they are labeled by n, S, M . Since the selection rules for transitions in the Hubbard model are much less restrictive, we can observe a large number of phonon suppressed peaks and steps when the occupation of carriers in the island increases. However, in the microscopic model and with a magnetic eld, the number of observed phonon suppressed steps or peaks are dramatically reduced since the probability of scattering can only occur when all n, S, M are equal. Furthermore, the region of negative di erential conductance (NDC), as induced by spin blockade (SB), can also be reduced by the electron–phonon interaction, as shown in the ne structures on the third and fourth peaks. Finally, Fig. 2 shows the e ects due to the magnetic eld, B = 1 T (dotted lines), B = 5 T (dashed lines) and B = 10 T (solid lines), on the mesoscopic current. In the top view, we also show the induced magnetic
uctuation to the current at low (dotted line) and high (solid line) magnetic elds. As can be observed, an
789
increase in the magnetic eld reduces the current or even suppress peaks or steps in NDC regions. However, the most interesting e ect can be observed in these regions since the spin blockade can occur when the system is in the state with a maximum value for the spin, S = n=2. The probability of an electron to leave this state is very small since the only possible transition for other states with n0 = n − 1 would require that S 0 = S − 12 . Therefore, this state is spin polarized and will block the current since it occurs only once for each given occupation number n. Thus, only one step of negative conductance can be observed in each Coulomb step. Other details will be discussed elsewhere.
Acknowledgements The authors acknowledge Fundaca˜ o de Amparo a Pesquisa do Estado de S˜ao Paulo (FAPESP) for nancial support.
References [1] M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. [2] M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [3] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Harmans, C.T. Foxon, Phys. Rev. Lett. 69 (1992) 1592. [4] K. Kang, B.I. Min, Phys. Rev. B 55 (1997) 15 412. [5] H. Grabert, Z. Phys. B 85 (1991) 319. [6] L.P. Kouwenhoven, N.C. van der Vaart, A.T. Johnson, W. Kool, C.J.P.M. Harmans, J.G. Williamson, A.A.M. Staring, C.T. Foxon, Z. Phys. B 85 (1991) 367. [7] K. Jauregui, W. Hausler, D. Weinmann, B. Kramer, Phys. Rev. B 53 (1996) 1713. [8] W. Hausler, Z. Phys. B 99 (1996) 551. [9] W. Hausler, B. Kramer, Phys. Rev. B 47 (1993) 16 353. [10] D. Weinmann, W. Hausler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984.
www.elsevier.nl/locate/physe
Electronic transport in quasi-1D mesoscopic systems: the correlated electron approach C.F. Destefani, G.E. Marques ∗ Departamento de Fsica, Universidade Federal de S˜ao Carlos, Via Washington Luiz, Km 235, 13565-905 S˜ao Carlos, S˜ao Paulo, Brazil
Abstract The Coulomb interaction plays a crucial role in transport properties of ultra small systems and is the most relevant energy scale for dynamical properties of mesoscopic structures. The strongly correlated model for electrons is used to calculate the tunneling current through a dot weakly coupled to 2D free electron channels. The model takes into account a bath of phonons in the island, a magnetic eld applied parallel to the current and the electron–phonon interaction in the island. ? 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the recent years, the transport properties of ultrasmall regions such as quantum dots (QD) have shown great interest [1,2] since their carrier occupation can be controlled by an external applied voltage. These islands are separated from the rest of the material by tunneling barriers which connect them to free electron leads. The intrinsic spatial con nement and the discrete energy spectrum [3,4] may in uence the transport properties of these mesoscopic systems. In metallic dots, where the pro le of the barriers can be drawn by the cross applied voltage, the electronic density may be larger than in semiconductor dots. Their energy spectra can be treated as a continuum and the average separation between single-particle levels, E; ∗ Corresponding author. Tel.: +55-16-260-8222; fax:+55-16261-4835. E-mail address: [email protected] (G.E. Marques)
is almost negligible. For semiconductor dots, the level separation becomes a crucial parameter to the study of the electronic properties. In the rst case, the usual charging model [5,6], which reduces the electron– electron interaction to a constant times the square of number of electrons in the island, can reproduce well the experimental data. For systems with small level separation it becomes necessary to consider the electron–electron interaction from a microscopic point of view in order to obtain the many-particle eigenstates. This is the main approach within the correlated electron model [7–9] that will be used in this work. The important energy scales in the transport properties of these islands are: (i) thermal uctuations, ET = kB T and (ii) charging energy, EC = e2 =C, where C is the capacitance measuring the electrostatic e ect due to the addition of an electron to the system. In metallic islands, when EC ET , the charge quantization dominates the transport properties and gives rise to Coulomb blockade (CB) and single-electron
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 0 5 8 - 8
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
tunneling oscillations (SETO) which can be seen as Coulomb steps or peaks in the I × V curves. When EC ET , the thermal uctuations tend to destroy these structures. In semiconductor dots, the ratio (EC =E) de nes a competition between charge and energy quantizations that would appear as additional ne structures over the Coulomb steps or peaks. In a linear regime, there will be transitions only between fundamental many-particle states and in a non-linear regime the transitions to excited states can also contribute to the transport properties of the island. In this work we will be using a model [10] which consider the island as a quasi-1D quantum dot of a typical size, L, which can be occupied by n electrons. The model includes a magnetic eld applied parallel to the current, a phonon thermal bath and treats the inelastic processes due to transitions generated by the electron–phonon interaction. We will also assume that the electronic motion in the island will be coherent and the transport is under ballistic regime. That is, we are assuming that the phase coherence time, ’ , is much larger than the time an electron takes to travel the typical distance L and, in this sense, the phase coherence of eigenstates can be preserved between the electron–phonon scattering processes. Certainly, the eigenstates would loose the phase coherence after ’ . The Hamiltonian for the mesoscopic double barrier structure is separated as H = (H0 + HP ), where H0 = (HD + HL + HR + HPh ) will describe the isolated systems formed by electrons in the dot (HD ), the left and right free electron channels P + (HL=R = k; kL=R cL=R;k; cL=R;k; ) and the phononic P 1 bath (HPh = q ˜!q (a+ q aq + 2 )), respectively. The term HD , measured in units of the e ective Hartree energy EH∗ , has the form ∗ 2 aB P + (m − e + gB B0 )cm; HD = cm; L m; 1 a∗B P + hm1 m2 |V |m3 m4 i 2 L mi ; j × cm+1 ;1 cm+2 ;2 cm3 ;2 cm4 ;1 :
(1)
In these expressions, kL=R is the free electron energy in the left=right channel with momentum k; is the spin, ˜!q is the phonon energy, m is the non-interacting energy of electrons in the mth state, |mi, of the island, −e is the electrostatic potential seen by carriers, g
787
Fig. 1. The e ect of the electron–phonon interaction on the mesoscopic current of a system at low temperature, for Hubbard-like (left) and correlated electron (right) models for the electron– electron interaction. The dashed (solid) lines are results without (with) the electron–phonon interaction. The voltage is measured in units of (e=C) in the left and in units of (EH∗ =e) in the right.
is the Lande g-factor, B is the Bohr magneton, B0 is the applied magnetic eld and hm1 m2 |V |m3 m4 i is the Coulomb electron–electron interaction in the island. Furthermore, a∗B is the e ective Bohr radius which scales all distances in this work. On the other hand, HP = (HLT + HRT + Hep ) will describe the interactions between the QD and the free electron channels as P L=R + T ∗ + = (Tk; m )cL=R;k; cm; + (Tk;L=R HL=R m ) cm; cL=R;k; k; m;
and Hep =
P q; m1 ;m2 ;
p g(q; m1 ; m2 ) cm+1 ; cm2 ; (aq + a+ q)
represents the electron–phonon interaction inside the dot. Furthermore, Tk;L=R m is the transition probability between dot and free electron channels and p g(q; m1 ; m2 ) is the electron–phonon interaction within the Frohlich approximation. The transition rates from an initial ( ) to a nal ( ) state, in the QD, can be calculated as 2 |h (0) |HP | (0) i|2 (E (0) − E (0) ); (2)
; = ˜ where each state | i(0) i is a solution of full H0 with eigenvalues Ei(0) . The selection rules for transitions
788
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
Fig. 2. E ects of a magnetic eld on the mesoscopic current for B0 = 1 T (dotted lines), B0 = 5 T (dashed lines) and B0 = 10 T (solid lines). The induced magnetic uctuation is also shown for low (B0 = 1 T) and high (B0 = 10 T) magnetic elds in dotted and solid lines, respectively.
between these states, which are labeled by the number of occupation n, total spin S, and spin z-component M can be summarized as: n = (n − n ) = ±1; S = (S − S ) = ± 12 and M = (M − M ) = ± 12 . The renormalized rates of transitions between states in the island are calculated as * + P
; ; (3) ; = l ;r ;Q
th(l ;r ;Q )
where h i stands for thermal average. As a nal step, we calculate the time evolution of the occupation of a given eigenstate, | (0) i, in the dot from the solution of the occupation probability, P , in the form P d P = ( ; P − ; P ); (4) dt 6=
P under the condition P = 1. By looking for stationary conditions (dP =dt = 0) for the occupation of the eigenstates in the island one can obtain the averaged probability, P of a state , and the total DC-current through the system as P − + P ( ;L=R; − ;L=R; ); (5) I ≡ I L=R = (−=+)e ; ( 6= )
which takes into account the net balance between the number of electrons entering and leaving the island per unit of time. 2. Discussion and results We will show basically two curves for total current through the island calculated for low temperature
C.F. Destefani, G.E. Marques / Physica E 7 (2000) 786–789
regime or ET E: (i) The current versus a voltage determined by the di erence of the chemical potentials of the leads (eV = L − R ); (ii) The current versus an external voltage, e = e(CG VG + CL VL + CR VR )=(CG + CL + CR ), for xed and equal capacitances CG=L=R and voltages VG=L=R applied to the gate and leads, respectively. In Fig. 1 we compare the results of the electron– phonon interaction on the total current according to the model used to treat the electron–electron interaction. On the left, we are showing the currents in the charging or the Hubbard-like model for zero eld (B0 = 0). On the right, we show the currents in the correlated electron model for magnetic eld B0 = 1 T. In both cases we observe that the overall e ect due to the inclusion of the electron–phonon interaction (solid lines) is small since the inelastic scattering transitions can only occur between initial and nal states with the same quantum numbers. In the charging model, the states are labeled only by the number n, whereas in the correlated model, they are labeled by n, S, M . Since the selection rules for transitions in the Hubbard model are much less restrictive, we can observe a large number of phonon suppressed peaks and steps when the occupation of carriers in the island increases. However, in the microscopic model and with a magnetic eld, the number of observed phonon suppressed steps or peaks are dramatically reduced since the probability of scattering can only occur when all n, S, M are equal. Furthermore, the region of negative di erential conductance (NDC), as induced by spin blockade (SB), can also be reduced by the electron–phonon interaction, as shown in the ne structures on the third and fourth peaks. Finally, Fig. 2 shows the e ects due to the magnetic eld, B = 1 T (dotted lines), B = 5 T (dashed lines) and B = 10 T (solid lines), on the mesoscopic current. In the top view, we also show the induced magnetic
uctuation to the current at low (dotted line) and high (solid line) magnetic elds. As can be observed, an
789
increase in the magnetic eld reduces the current or even suppress peaks or steps in NDC regions. However, the most interesting e ect can be observed in these regions since the spin blockade can occur when the system is in the state with a maximum value for the spin, S = n=2. The probability of an electron to leave this state is very small since the only possible transition for other states with n0 = n − 1 would require that S 0 = S − 12 . Therefore, this state is spin polarized and will block the current since it occurs only once for each given occupation number n. Thus, only one step of negative conductance can be observed in each Coulomb step. Other details will be discussed elsewhere.
Acknowledgements The authors acknowledge Fundaca˜ o de Amparo a Pesquisa do Estado de S˜ao Paulo (FAPESP) for nancial support.
References [1] M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. [2] M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [3] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Harmans, C.T. Foxon, Phys. Rev. Lett. 69 (1992) 1592. [4] K. Kang, B.I. Min, Phys. Rev. B 55 (1997) 15 412. [5] H. Grabert, Z. Phys. B 85 (1991) 319. [6] L.P. Kouwenhoven, N.C. van der Vaart, A.T. Johnson, W. Kool, C.J.P.M. Harmans, J.G. Williamson, A.A.M. Staring, C.T. Foxon, Z. Phys. B 85 (1991) 367. [7] K. Jauregui, W. Hausler, D. Weinmann, B. Kramer, Phys. Rev. B 53 (1996) 1713. [8] W. Hausler, Z. Phys. B 99 (1996) 551. [9] W. Hausler, B. Kramer, Phys. Rev. B 47 (1993) 16 353. [10] D. Weinmann, W. Hausler, B. Kramer, Phys. Rev. Lett. 74 (1995) 984.