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Tunneling times for wave packets narrow in wave-number space
Solid-State Electronics Vol. 31, No. 3/4, pp. 747 748, 1988 Printed in Great Britain
0038-1101/88 $3.00+0.00 Pergamon Journals Ltd
TUNNELING TIMES FOR WAVE PACKETS NARROW IN WAVE-NUMBER SPACE
E. H. Hauge ~, J. P. Falck ~ and T. A. Fjeldly ~ *Inst. for theor, physics, NTN, N-7034 Trondheim, Norway **Inst. for phys. electronics, NTH, N-7034 Trondheim, Norway
ABSTRACT The tunneling times for wave packets narrow in k-space are considered for an arbitrary, localized potential barrier in one dimension. To zeroth order the classic phase times are rederived. Corrections are briefly discussed. An identity between the dwell time and the phase times is presented. Finally, we give an exact expression for the transmission time at resonance for a symmetric double barrier.
INTRODUCTION We have reconsidered the simplest possible class of tunneling problems, the idealized one-body case of a wave packet narrow in wave-number space (k-space) impinging on a localized, static, but otherwise arbitrary potential barrier in one dimension. The aim has been to provide a firm basis for the definition and calculation of tunneling times for such processes. I Concepts directly related to moving wave packets, as observed in computer experiments, are used. Barker (1985, 1986) has recently studied the same class of problems. Although our methods and results are somewhat different, we agree on the essential point: To lowest order the classic phase times (Bohm, 1951; Wigner, 1955) apply. Contrary to statements in the literature, the Larmor clock (invented by Baz, 1967), when properly set, also shows the phase time to lowest order. When the effects of the finite kwidth are taken into account, however, subtle differences appear, depending on which physical quantity is used to measure time. As an application of the phase time, we have calculated exactly the transmission time at resonance for symmetric double barrier (and for a special class of asymmetric ones). Space does not allow a full exposition here, details will be provided elsewhere (Hauge, Falek and Fjeldly (1987); Falek and Hauge, to be published).
BASICS We assume that the stationary scattering problem for a localized potential, exactly for all ener ;ies. l.e., the wave functions e ikx + /Re iB-ikx
Ck(X)= X ( x ; k ) ~e
; xgb
(1)
; b
i~+ikx
V(x) , has been solved
; x~a
are taken as known. including the reflection and transmission probabilities, R(k) and T(k) , and the phase shifts B(k) and ~(k) . Furthermore~ the initial wave packet, localized to the left of b, is characterized by its Fourier transform, ¢(k) , and is assumed narrow in the sense that d2= << 2~ k 2 . Here <...> denotes an average with respect to the initial packet. In addition, T(k), a(k) c etc. are (usually) assumed slowly varying on the scale of d . The discussion of the transmitted packet is based on the (unnormalized) wave function (x>a) ¢T(X,T) = I ~
@(k)A(k)exp i[kx-flk2t/2m]
(2)
During the collision, the total wave function is very complicated. After the smoke has cleared, however, @m(x,t) describes a free packet with constant normalization moving to the right. Similarly, ~ @R(x,t) describes the reflected packet. In textbook manner, we define the positions of the packets-as their centers of gravity, This allows us to calculate space delays, from which tunneling times are inferred.
Needless to say~ other processes may require different approaches to the definition and calculation of tunneling times. See, for example, B~ttiker and Landauer (1986).
747
748
RESULTS i. To zeroth order in
~2
one rederives the classic phase times
T T = (a-b+~')/v
where a'= d~/dk and evaluated at k=k c 2. To
~(~2)
;
~R = ( - 2 b + B ' ) / v
(3)
v=~k/m . It is understood that in (311 and (4) all quantities are to be
tile average velocity of (e.g.) the transmitted packe~ ~s shifted
v ~ v[I+T'~2/Tk ] c
(4)
Qualitatively, this feature was pointed out by B[ttiker (1983). The shift necessitates a careful discussion of what is meant by spatial and temporal delays. New, explicit expressions for the leading corrections to the phase times, as given by (3), resist (Hauge et al., 1987). These corrections have two sources, the tunneling process itself, and the shifts in the average velocities of the free motion. In the interpretation of computer experiments, it is important to distinguish between these two types of corrections. 3. The Larmor clock has been used for the present class of problems by Ryhachemko (1967) and by B~ttiker (1983). Their results on the stationary problem seem in conflict with those quoted here. However, a reanalysis reveals that the Larmor clock also shows the phase time to zeroth order. Subtle differences to d~(~ 2) can be traced to the fact that the Larmor clock measures whereas space delays give -I 4. The now standard definition of the d~ell time [see, for example, B~ttiker (1983)], reads, in our notation rD(k) = v(k)-l i ~xl×(x;k) 12
(5)
b By similar methods as above we have derived, to zeroth order, the following new identity relating TD(k) to ~T(k) and rR(k) • D = T rT + R T R + (~/kv)sin(B-2kb)
(6)
Barker (1986) has found a similar connection, but without the interference term. Our belief that (6) is indeed the correct general relation is strengthened by the fact that it checks for a square barrier, for which all quantities are known explicitly. T~len the interference term ~ is negligible (R~N0 , sin( )~0 , or kv large), the dwell time has a clear physical meaning. However, for small energies the interference term is generally large, and the dwell time becomes useless. 5. For a symmetric double barrier with widths b , heights V n , and well width w , the exact coherent transmission time at the n'th resonance is (the dwell time coincides with the phase time here, since R=0) _ v-l{~2sinh2Kb[~w + - i TTn- n where
V0= ~2k~/2m
K 2= k2-k 2 , and '
coth
~ = (K/k) + (k/K) . ~ e n
(7)
n
~b>>l
(7) agrees well, hut not
0
perfectly, with the approximate result found earlier by Ricco and Azbel (1984).
REFERENCES Barker, J. R. (1985). Ph~sica 134B, 22-31. Barker, J. R. (1986). In M. J. Kelly and C. Weisbuch, Eds., ~ s i c s structures and Microdevices. Springer, Berlin, 210-230. Baz, A. I. (1967). Soy. J. Nuel. Phys. 4, 182-188. BolLm, D. (1951). Quantum Theory. Prentice-Hall, New York. B~ttiker, M. (1983), Phys. Rev. B 27 , 6178-6188. B~ttiker, M. and R. Landauer (1986). ~BM J. Res. Develo~ 30, 451-45~. Hauge, E. H., J. P. Falck and T. A. Fjeldly (1987). P_hys. Rev. B. Ricco, B. and M. Ya. Azbel (198h). Phys. Rev. B 29, 1970-1981. Rybachenko, V. F. (1967). So___v.J. Nucl. Phys. 5, 635-639. Wigner, E. P. (1955). Phys. Rev. 98, IL5-147.
and Fabrication of Micro-
0038-1101/88 $3.00+0.00 Pergamon Journals Ltd
TUNNELING TIMES FOR WAVE PACKETS NARROW IN WAVE-NUMBER SPACE
E. H. Hauge ~, J. P. Falck ~ and T. A. Fjeldly ~ *Inst. for theor, physics, NTN, N-7034 Trondheim, Norway **Inst. for phys. electronics, NTH, N-7034 Trondheim, Norway
ABSTRACT The tunneling times for wave packets narrow in k-space are considered for an arbitrary, localized potential barrier in one dimension. To zeroth order the classic phase times are rederived. Corrections are briefly discussed. An identity between the dwell time and the phase times is presented. Finally, we give an exact expression for the transmission time at resonance for a symmetric double barrier.
INTRODUCTION We have reconsidered the simplest possible class of tunneling problems, the idealized one-body case of a wave packet narrow in wave-number space (k-space) impinging on a localized, static, but otherwise arbitrary potential barrier in one dimension. The aim has been to provide a firm basis for the definition and calculation of tunneling times for such processes. I Concepts directly related to moving wave packets, as observed in computer experiments, are used. Barker (1985, 1986) has recently studied the same class of problems. Although our methods and results are somewhat different, we agree on the essential point: To lowest order the classic phase times (Bohm, 1951; Wigner, 1955) apply. Contrary to statements in the literature, the Larmor clock (invented by Baz, 1967), when properly set, also shows the phase time to lowest order. When the effects of the finite kwidth are taken into account, however, subtle differences appear, depending on which physical quantity is used to measure time. As an application of the phase time, we have calculated exactly the transmission time at resonance for symmetric double barrier (and for a special class of asymmetric ones). Space does not allow a full exposition here, details will be provided elsewhere (Hauge, Falek and Fjeldly (1987); Falek and Hauge, to be published).
BASICS We assume that the stationary scattering problem for a localized potential, exactly for all ener ;ies. l.e., the wave functions e ikx + /Re iB-ikx
Ck(X)= X ( x ; k ) ~e
; xgb
(1)
; b
i~+ikx
V(x) , has been solved
; x~a
are taken as known. including the reflection and transmission probabilities, R(k) and T(k) , and the phase shifts B(k) and ~(k) . Furthermore~ the initial wave packet, localized to the left of b, is characterized by its Fourier transform, ¢(k) , and is assumed narrow in the sense that d2=
@(k)A(k)exp i[kx-flk2t/2m]
(2)
During the collision, the total wave function is very complicated. After the smoke has cleared, however, @m(x,t) describes a free packet with constant normalization moving to the right. Similarly, ~ @R(x,t) describes the reflected packet. In textbook manner, we define the positions of the packets-as their centers of gravity, This allows us to calculate space delays, from which tunneling times are inferred.
Needless to say~ other processes may require different approaches to the definition and calculation of tunneling times. See, for example, B~ttiker and Landauer (1986).
747
748
RESULTS i. To zeroth order in
~2
one rederives the classic phase times
T T = (a-b+~')/v
where a'= d~/dk and evaluated at k=k c 2. To
~(~2)
;
~R = ( - 2 b + B ' ) / v
(3)
v=~k/m . It is understood that in (311 and (4) all quantities are to be
tile average velocity of (e.g.) the transmitted packe~ ~s shifted
v ~ v[I+T'~2/Tk ] c
(4)
Qualitatively, this feature was pointed out by B[ttiker (1983). The shift necessitates a careful discussion of what is meant by spatial and temporal delays. New, explicit expressions for the leading corrections to the phase times, as given by (3), resist (Hauge et al., 1987). These corrections have two sources, the tunneling process itself, and the shifts in the average velocities of the free motion. In the interpretation of computer experiments, it is important to distinguish between these two types of corrections. 3. The Larmor clock has been used for the present class of problems by Ryhachemko (1967) and by B~ttiker (1983). Their results on the stationary problem seem in conflict with those quoted here. However, a reanalysis reveals that the Larmor clock also shows the phase time to zeroth order. Subtle differences to d~(~ 2) can be traced to the fact that the Larmor clock measures
(5)
b By similar methods as above we have derived, to zeroth order, the following new identity relating TD(k) to ~T(k) and rR(k) • D = T rT + R T R + (~/kv)sin(B-2kb)
(6)
Barker (1986) has found a similar connection, but without the interference term. Our belief that (6) is indeed the correct general relation is strengthened by the fact that it checks for a square barrier, for which all quantities are known explicitly. T~len the interference term ~ is negligible (R~N0 , sin( )~0 , or kv large), the dwell time has a clear physical meaning. However, for small energies the interference term is generally large, and the dwell time becomes useless. 5. For a symmetric double barrier with widths b , heights V n , and well width w , the exact coherent transmission time at the n'th resonance is (the dwell time coincides with the phase time here, since R=0) _ v-l{~2sinh2Kb[~w + - i TTn- n where
V0= ~2k~/2m
K 2= k2-k 2 , and '
coth
~ = (K/k) + (k/K) . ~ e n
(7)
n
~b>>l
(7) agrees well, hut not
0
perfectly, with the approximate result found earlier by Ricco and Azbel (1984).
REFERENCES Barker, J. R. (1985). Ph~sica 134B, 22-31. Barker, J. R. (1986). In M. J. Kelly and C. Weisbuch, Eds., ~ s i c s structures and Microdevices. Springer, Berlin, 210-230. Baz, A. I. (1967). Soy. J. Nuel. Phys. 4, 182-188. BolLm, D. (1951). Quantum Theory. Prentice-Hall, New York. B~ttiker, M. (1983), Phys. Rev. B 27 , 6178-6188. B~ttiker, M. and R. Landauer (1986). ~BM J. Res. Develo~ 30, 451-45~. Hauge, E. H., J. P. Falck and T. A. Fjeldly (1987). P_hys. Rev. B. Ricco, B. and M. Ya. Azbel (198h). Phys. Rev. B 29, 1970-1981. Rybachenko, V. F. (1967). So___v.J. Nucl. Phys. 5, 635-639. Wigner, E. P. (1955). Phys. Rev. 98, IL5-147.
and Fabrication of Micro-