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Effects of barrier traversal time on escape rate through a small-capacitance tunnel junction
Super/attices and Microstructures, Vol. 16, No.3, 1994
279
EFFECTS OF BARRIER TRAVERSAL TIME ON ESCAPE RATE THROUGH A SMALL-CAPACITANCE TUNNEL JUNCTION Masahito Ueda1,2 and Tsuneya Ando? 1 NTT Basic Research Laboratories, Kanagawa 342-01, Japan 2Institute for Solid State Physics, University of Tokyo, Tokyo 106, Japan
(Received 21 August 1994) This paper describes self-consistent determination of the transfer-energy-dependent escape rate and barrier traversal time of electrons under the influence of the electromagnetic environment. This rate lets us examine dissipative and nondissipative tunneling rate separately and its integral over the transfer energy gives the total escape rate that is obtained by the instanton technique. It is found that the escape rate is exponentially enhanced as the capacitance of the junction is reduced.
1. Introduction
Physics and applications of small-capacitance tunnel junctions have enjoyed a growing interest over the last decade [1]. The elementary charging energy E e of roomtemperature single-electron devices needs to be about 1 eV, a value which is comparable to the energy barrier Uo of the junction. Barrier traversal time is thus expected
to be one of the key parameters that characterize the dynamics of these devices. This paper describes self-consistent determination of the escape rate and barrier traversal time of electrons under the influence of the electromagnetic (EM) environment [2]. Possible experimental situations for observing the predicted effects are also described. 2. Formulation of the problem
Suppose that an electron, while tunneling through a potential barrier V (x), interacts linearly with harmonicoscillator charge modes Qo on the capacitor. Starting from the Lagrangian in which Qo plays the role of the dynamical variable, we obtain the canonically quantized Hamiltonian of the whole system as
o,
and the con~llgat.e operator 4>0 satisfy the where commutation relations lQo,
where P(E,) is the initial energy distribution of the EM environment, and (E J , dle-xHt'IE" 0) denotes the joint probability amplitude for an electron to tunnel from x = o to x = d (d is the barrier width) and with the state of the environment changing from IE.) to IEJ ) . The delta function ensures that the environment gains (or if it is negative, loses) energy E, which essentially equals the energy loss (or gain) of the tunneling electron. After expressing the joint probability amplitude in terms of path integrals in coordinate space, we evaluate them in the bounce-trajectory approximation. Then, the transfer-energy-dependent escape rate can be obtained from
qE) = N
JOO -00
!!!.-eX(E+Eclt e-iSB(t),
(3)
21rh
where the bounce exponent SB(t) is given for the singlemode case as
0749-6036/94/070279 + 03 $08.00/0
© 1994 Academic Press Limited
Superlattices and Microstructures. Vol. 76. No.3. 7994
280 where w = (LC)-1/2, g(w, T) = l-w Jg'dre",(r-T)x( T)/d, and nB the Bose-Einstein distribution function. The first term in the curly brackets is associated with the emission of energy quanta liw, and the second term is associated with its absorption. The equation of motion for the bounce trajectory X(T) is determined by minimizing Eq. (4). The traversal time T is determined by the condition x(T) = d. The barrier traversal time T thus determined includes t as a parameter:. T = T(t). By substituting T(t) and X(T) into Eq. (4), we find the bounce exponent as a function of t.
SWKB (dashed curves) as a function of ratio Ec/Uo for four values of u/T«. The SWKB is calculated by defining a local effective potential (see Ref. 2) and substituting it into the usual WKB formula. Each curve decreases monotonically, which implies that the total escape rate increases exponentially as the capacitance of the junction is reduced or as wTo is increased. We observe that the deviation of the correction (So - SWKB)/So from (So - SB)/So is larger for smaller wTo. The error in the WKB estimation originates from the neglected fact that the final state energy of a tunneling electron is distributed, as shown below.
3. Total escape rate 4. Transfer-energy-dependent escape rate
The total escape rate is obtained by integrating f( E) over transfer energy E. Figure 1 compares the bounce exponent SB (solid curves) and the WKB exponent
In the limit wTo ~ I, the transfer-energy-dependent escape rate is calculated to be s
1-£
IV
f(E) = Ne-*s"(O) e -i5~
[k I-Ee/fa L ,,'" (",~a) 00
n=O
'\ '....
--------------l------ _ (h)
a
wT o
(cl 0.05 Ib10.1 (e11 (d110
(f)
0.5
-,
-SB/S O ........ SWKB/S O
o
1
2
3
8 (E+Ec-nliw), (5)
n.
which shows that most tunneling events occur elastically (i.e., via the n = 0 channel) under normal conditions where (E c-Uo)/[liw(wTo)2] 0), where
(a)
~ co
r
4
EC/U O I. Bounce exponent SB (solid curves) and WKB exponent SWKB (dashed curves) as a function of the ratio of the elementary charging energy E c to the barrier height V o for four values of wTo. These curves show SB and SWKB values normalized by the bounce exponent So= 4UoTo in the absence of coupling between the tunneling electron and the electromagnetic environment.
f(E) = Ne- t
it =
eoo f(E)EdE = -E [ _ eoo f(E)dE c 1
1 - Ec/Uo]
(wToJ2
.
(6)
In the opposite limit wTo
s8
E
(0)
e-*""(l-",To)
00
[k (1 _ wTo)] n
"=0
n.
L ""'
I
8 (E+Ec-nliw), (7)
281
Superlattices and Microstructures, Vol. 16, No.3, 1994
manifestation of the orthogonality catastrophe. Equation (7) shows that the nonzero traversal time decreases this effect by a factor of 1 - wTo in comparison with the prediction of Ref. 4.
E
5. Possible experimental situations
dynamic limit
roT=O Q
static limit
x
roT = 00
2. Schematic diagram of the transfer energy E as a function of the position x of a tunneling electron and the polarization charge Q on the capacitor. For clarity, the origin of E is displaced by E c . Each parabola shows the E - Q dispersion relation parametrized by x. The Gaussian-like wave packets illustrate the wave functions of Q at positions where they are shown (scale arbitrary).
which shows that when the average number of excited quanta n = E c (1 - wTo)/hw is much greater than unity, the elastic tunneling is prohibited as a consequence of the orthogonality catastrophe, i.e., the two ground-state wave functions become orthogonal . The Coulomb blockade of tunneling may be regarded as a
An experimental situation in which the predicted effects of traversal time may be observed is a GaAs/AlGaAs semiconductor tunnel junction, which can cover a wide range of barrier heights and thicknesses. For example, with Vo ~ 50 meV, d ~ 100 A, c ~ 10- 16 F, and L :::: 10- 10 H, we have wTo :::: 0.5. Another candidate situation is a single-electron box consisting of a tunnel junction with one electrode terminated by a capacitor. When the middle electrode is a metal with good conductivity, its electromagnetic response is dominated by an inductance given by ml[ner S, where n is the electron density,S is the cross-sectional area, and I the length. By assuming that 5 = (0.1 pm)2, I = 0.1 um, and n = 1023 cm- J , we get L ~ 10- 14 H. If the total capacitance of the middle electrode is of the order of C = 10- 16 F, we have wTo ~ 1 for typical normal tunnel junctions with Vo ~ 10 eV and d:::: 10 A. Since the inductance is proportional to the length of the electrode, wTo can be varied to provide a sufficient change in the escape rate.
REFERENCES [1] For a collection of recent review articles, see Single Charge Tunneling, edited by H. Grabert and M. Devoret (Plenum, New York, 1992). [2] Part of the present work was reported in: M. Ueda and T. Ando, Phys. Rev. Lett. 72, 1726 (1994). [3] M. Veda and T. Ando, Phys. Rev. B, Sep. 15 (1994). [4] M. H. Devoret, et al., Phys. Rev. Lett 64, 1824 (1990).
279
EFFECTS OF BARRIER TRAVERSAL TIME ON ESCAPE RATE THROUGH A SMALL-CAPACITANCE TUNNEL JUNCTION Masahito Ueda1,2 and Tsuneya Ando? 1 NTT Basic Research Laboratories, Kanagawa 342-01, Japan 2Institute for Solid State Physics, University of Tokyo, Tokyo 106, Japan
(Received 21 August 1994) This paper describes self-consistent determination of the transfer-energy-dependent escape rate and barrier traversal time of electrons under the influence of the electromagnetic environment. This rate lets us examine dissipative and nondissipative tunneling rate separately and its integral over the transfer energy gives the total escape rate that is obtained by the instanton technique. It is found that the escape rate is exponentially enhanced as the capacitance of the junction is reduced.
1. Introduction
Physics and applications of small-capacitance tunnel junctions have enjoyed a growing interest over the last decade [1]. The elementary charging energy E e of roomtemperature single-electron devices needs to be about 1 eV, a value which is comparable to the energy barrier Uo of the junction. Barrier traversal time is thus expected
to be one of the key parameters that characterize the dynamics of these devices. This paper describes self-consistent determination of the escape rate and barrier traversal time of electrons under the influence of the electromagnetic (EM) environment [2]. Possible experimental situations for observing the predicted effects are also described. 2. Formulation of the problem
Suppose that an electron, while tunneling through a potential barrier V (x), interacts linearly with harmonicoscillator charge modes Qo on the capacitor. Starting from the Lagrangian in which Qo plays the role of the dynamical variable, we obtain the canonically quantized Hamiltonian of the whole system as
o,
and the con~llgat.e operator 4>0 satisfy the where commutation relations lQo,
where P(E,) is the initial energy distribution of the EM environment, and (E J , dle-xHt'IE" 0) denotes the joint probability amplitude for an electron to tunnel from x = o to x = d (d is the barrier width) and with the state of the environment changing from IE.) to IEJ ) . The delta function ensures that the environment gains (or if it is negative, loses) energy E, which essentially equals the energy loss (or gain) of the tunneling electron. After expressing the joint probability amplitude in terms of path integrals in coordinate space, we evaluate them in the bounce-trajectory approximation. Then, the transfer-energy-dependent escape rate can be obtained from
qE) = N
JOO -00
!!!.-eX(E+Eclt e-iSB(t),
(3)
21rh
where the bounce exponent SB(t) is given for the singlemode case as
0749-6036/94/070279 + 03 $08.00/0
© 1994 Academic Press Limited
Superlattices and Microstructures. Vol. 76. No.3. 7994
280 where w = (LC)-1/2, g(w, T) = l-w Jg'dre",(r-T)x( T)/d, and nB the Bose-Einstein distribution function. The first term in the curly brackets is associated with the emission of energy quanta liw, and the second term is associated with its absorption. The equation of motion for the bounce trajectory X(T) is determined by minimizing Eq. (4). The traversal time T is determined by the condition x(T) = d. The barrier traversal time T thus determined includes t as a parameter:. T = T(t). By substituting T(t) and X(T) into Eq. (4), we find the bounce exponent as a function of t.
SWKB (dashed curves) as a function of ratio Ec/Uo for four values of u/T«. The SWKB is calculated by defining a local effective potential (see Ref. 2) and substituting it into the usual WKB formula. Each curve decreases monotonically, which implies that the total escape rate increases exponentially as the capacitance of the junction is reduced or as wTo is increased. We observe that the deviation of the correction (So - SWKB)/So from (So - SB)/So is larger for smaller wTo. The error in the WKB estimation originates from the neglected fact that the final state energy of a tunneling electron is distributed, as shown below.
3. Total escape rate 4. Transfer-energy-dependent escape rate
The total escape rate is obtained by integrating f( E) over transfer energy E. Figure 1 compares the bounce exponent SB (solid curves) and the WKB exponent
In the limit wTo ~ I, the transfer-energy-dependent escape rate is calculated to be s
1-£
IV
f(E) = Ne-*s"(O) e -i5~
[k I-Ee/fa L ,,'" (",~a) 00
n=O
'\ '....
--------------l------ _ (h)
a
wT o
(cl 0.05 Ib10.1 (e11 (d110
(f)
0.5
-,
-SB/S O ........ SWKB/S O
o
1
2
3
8 (E+Ec-nliw), (5)
n.
which shows that most tunneling events occur elastically (i.e., via the n = 0 channel) under normal conditions where (E c-Uo)/[liw(wTo)2]
(a)
~ co
r
4
EC/U O I. Bounce exponent SB (solid curves) and WKB exponent SWKB (dashed curves) as a function of the ratio of the elementary charging energy E c to the barrier height V o for four values of wTo. These curves show SB and SWKB values normalized by the bounce exponent So= 4UoTo in the absence of coupling between the tunneling electron and the electromagnetic environment.
f(E) = Ne- t
it =
eoo f(E)EdE = -E [ _ eoo f(E)dE c 1
1 - Ec/Uo]
(wToJ2
.
(6)
In the opposite limit wTo
s8
E
(0)
e-*""(l-",To)
00
[k (1 _ wTo)] n
"=0
n.
L ""'
I
8 (E+Ec-nliw), (7)
281
Superlattices and Microstructures, Vol. 16, No.3, 1994
manifestation of the orthogonality catastrophe. Equation (7) shows that the nonzero traversal time decreases this effect by a factor of 1 - wTo in comparison with the prediction of Ref. 4.
E
5. Possible experimental situations
dynamic limit
roT=O Q
static limit
x
roT = 00
2. Schematic diagram of the transfer energy E as a function of the position x of a tunneling electron and the polarization charge Q on the capacitor. For clarity, the origin of E is displaced by E c . Each parabola shows the E - Q dispersion relation parametrized by x. The Gaussian-like wave packets illustrate the wave functions of Q at positions where they are shown (scale arbitrary).
which shows that when the average number of excited quanta n = E c (1 - wTo)/hw is much greater than unity, the elastic tunneling is prohibited as a consequence of the orthogonality catastrophe, i.e., the two ground-state wave functions become orthogonal . The Coulomb blockade of tunneling may be regarded as a
An experimental situation in which the predicted effects of traversal time may be observed is a GaAs/AlGaAs semiconductor tunnel junction, which can cover a wide range of barrier heights and thicknesses. For example, with Vo ~ 50 meV, d ~ 100 A, c ~ 10- 16 F, and L :::: 10- 10 H, we have wTo :::: 0.5. Another candidate situation is a single-electron box consisting of a tunnel junction with one electrode terminated by a capacitor. When the middle electrode is a metal with good conductivity, its electromagnetic response is dominated by an inductance given by ml[ner S, where n is the electron density,S is the cross-sectional area, and I the length. By assuming that 5 = (0.1 pm)2, I = 0.1 um, and n = 1023 cm- J , we get L ~ 10- 14 H. If the total capacitance of the middle electrode is of the order of C = 10- 16 F, we have wTo ~ 1 for typical normal tunnel junctions with Vo ~ 10 eV and d:::: 10 A. Since the inductance is proportional to the length of the electrode, wTo can be varied to provide a sufficient change in the escape rate.
REFERENCES [1] For a collection of recent review articles, see Single Charge Tunneling, edited by H. Grabert and M. Devoret (Plenum, New York, 1992). [2] Part of the present work was reported in: M. Ueda and T. Ando, Phys. Rev. Lett. 72, 1726 (1994). [3] M. Veda and T. Ando, Phys. Rev. B, Sep. 15 (1994). [4] M. H. Devoret, et al., Phys. Rev. Lett 64, 1824 (1990).