JOURNAL DE PHYSIQUE Colloque C6, supplbment

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Colloque C6, supplbment au n0ll, Tome 49, novembre 1988. SPIN PRECESSION AND PHASE DELAY TUNNELING TIMES. Z. HUANG, P.H. CUTLER, T.E. ...
JOURNAL DE PHYSIQUE Colloque C6, supplbment au n0ll, Tome 49, novembre 1988

SPIN PRECESSION AND PHASE DELAY TUNNELING TIMES Z. HUANG, P.H. CUTLER, T.E. FEUCHTWANG, R.H. GOOD, Jr., E. KAZES, H.Q. NGUYEN and S.K. PARK The Pennsylvania State University, 104 Davey Laboratory, University Park, Pennsylvania, PA 16802, U.S.A. Abstract - Spin precession and phase delay are two methods for defining "clocks" which can measure tunneling times. That these two methods in some cases do not predict the same tunneling time has caused much controversy. In this paper, we re-analyze the spin precession method of Buttiker and find that if the magnetic field is applied throughout the space, the spin precession and phase delay times agree with each other. 1. INTRODUCTION The concept of time in classical mechanics is well defined. The traversal time, or the duration of time for a parcicle to travel from point "a" to point "b" in one dimension is given as follows:

where E and V(x) are the total energy and the potential energy of the particle, respectively. However, such a unique determination is not available in quantum mechanics. Formally, there is no operator for time as for the other dynamical variables. A physical picture is obtained if the particle is described by some wave function. For a particle in a given stationary energy state, 'ZE, che probability does not change with time if the potential is static. These states cannot, therefore, be used to describe any time-dependent dynamical motion of the particle. Instead, it is necessary to construct a wave packet consisting of different energy eigenfunctions to analyze the dynamic motion of the particle. Since a wave packet does not have a unique energy (or unique position) at any given time, the variables E and t only have statistical meanings. If the particle is traveling in a classically allowed region, one can determine the traversal time by following the center of the packet. When the particle can be treated as purely classical (i.e. fi-to), then, according to the correspondence principle, Eq. 1 is the classical limit for the wave packet moving in a one-dimensional potential. However, tunneling is a quantum phenomenon, which does not have a classical counterpart. In a typical tunneling process, the incident wave packet is split into a reflected and a transmitted packet. Even though a well defined transmitted wave packet can be found[l], the identification of the transmitted packet as the tunneling "particle" is somewhat ambiguous because the transmitted packet is only one part of the total wave function. It cannot represent the particle by itself. To avoid this conceptual difficulty, two interpretations suggest themselves; in the first, the transmitted wave packet can be identified as the probability density of the total wave function to the right of the barrier. In the second interpretation, the incident wave packet is assumed to describe a flux of particles. The transmitted wave packet corresponds to the number of particles that have tunneled. Stevens[2] has discussed this interpretation with some cogent observations. From these simple considerations, it is evident that the concept of tunneling time is a fundamental and still unresolved problem in quantum mechanics. Nevertheless, there are important technological and scientific implications in a knowledge of tunneling time. For example, the study of this phenomenon will be important in describing; a) tunneling through a time-dependent barrier, such as laser beams incident on junctions[3.4], b) the limit of the response time of a tunnel junction device[5], c) dynamic response of a surface when an electron tunnels out to the vacuum[6,7], d) resonant tunneling[8,10], and e) macroscopic quantum tunneling[ll]. Currently, two dynamic methods for measuring tunneling time have been proposed[ll,l2], but only the STM experiment has provided preliminary results[l,5]. An elegant but static experiment using magnetic field dependent tunneling through heterostructures has been done by Gukret et a1[13]. The subject of tunneling time has been controversial. In 1932 MacColl[l4] suggested that a tunneling process is almost instantaneous, and the tunneling time, if any, is very small. Since then, many expressions for tunneling time have been derived or proposed based on

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988604

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different approaches. One of these is the so-called phase delay method introduced by Bohm[l5] and Wigner[l6]. When applied to a one-dimensional square barrier of height Vo and width d, the phase delay tunneling time is[17]

Jm~.

where k = d m , n = &m(~~-~)/fi~, and ko = m and E are respectively the mass and energy of the electron. A detailed analysis of wave packet tunneling through a square barrier has recently been presented by Hauge et a1[9]. The analysis is based on the series expansion in powers of the energy half-width of the wave packet. They found that the lowest order approximation in their analysis recovers Eq. 2. However, the phase-delay method of analyzing tunneling time has been severely criticized by Biittiker[l8] who maintains that "the strong deformation of a wave packet when it interacts with the barrier makes the procedure of following the peak of the wave packet not meaningful." The author of Ref. 18 uses another method to determine tunneling time which relies on the Larmor spin precession as a "clock". This method was first introduced by Baz1[19] for determining scattering time. Rybachenko[20] applied the same analysis to tunneling through one-dimensional barriers. Buttiker[l8] extended the Bazl-Rybachenko work to include an additional spin rotation contribution, in the direction of the magnetic field. In his model the magnetic field is assumed to be applied only in the barrier region. The incident beam has a spin polarization perpendicular to the magnetic field. By determining the spin direction of the transmitted beam, he derives the following expression for the Larmor spin precession time

It should be noted that

T , ,

is not interpreted by Buttiker as the tunneling time. Instead, he

argues that the correct tunneling time,

TT,

is given by TT

=

J-,

where

T~~

is the spin

rotation time. Eqs. 2 and 3 are plotted in Fig. 1. It can be seen that they are almost identical except in the low energy region, where, in general, the tunneling probability is vanishingly small. Eqs. 2 and 3 have been discussed in detail by Leavens and Aers[21]. The above analysis by Buttiker was criticized by Collins, Lowe, and Barker as "erroneous on several points"[22]. Very recently, a rebuttal has been presented by Biittiker and Landauer[23].

Fig. 1. Comparison of phase delay tunneling time (-) and Buttiker's spin precession time (- - -), for a one-dimensional square barrier with Vo 10eV and d = 10A.

-

In this paper, we reexamine the Larmor spin precession time. By applying the field throughout the space, in contrast to Rybachenko and the Biittiker model where the field only exists in the barrier region, we are able to show that the spin precession time is identical to the phase delay time. 2. LARMOR CLOCK Consider a square barrier model as shown in Fig. 2. The magnetic field is applied in the z-direction throughout the space, i.e.,

Fig. 2. Tunneling through a square barrier potential with a magnetic field applied in z-direction throughout the space. The Hamiltonian for an electron, charge -e, mass m, is H = + eA). (p + eA) + V ( z ) + 71 fiwlp,,

k(p

(5)

eB where wL is the Larmor frequency -, as defined by Biittiker[l8], and mc

[i -!]

is a ~ a u l i 1 differs from the usual Larmor frequency by a factor of 3 [24]. o

=

matrix. We note that wL We may choose the gauge such that

Following Schmit and Good[24], and writing the wave function Ir(r)=I+(x,y)I(z), we can separate the x-y motion from the spin dependence and the z-motion. The equation for the zdependence is

where ET is the total energy and motion,

p

and

v

are the quantum numbers characterizing the x-y

Subtracting the kinetic energy for the x-y motion, the energy is

Defining 2 k+

2m

=

1

1(E T 7 fiwL),

and

R

the Schrodinger equation becomes

2

n+ =

-

2m

1

- - ( E - V o T 7 f i wL) fi2

,

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The incident beam is spin polarized in the x-direction, with the z-dependence given by the 2x2 spinor,

The normalization of the wave function is not required in this scattering problem because we are primary interested in the spin precession angle which depends only on the ratio of the wave functions. The transmitted beam is assumed to be

where Dk are complex transmission coefficients. Solving the Schrodinger equation yields the following:

D?r.

=

T+ -1/2eiA4L-e-k+d

(12)

where

and 2 tan(A$+)

-

=

2

k+ -n+ tanh(n+d)

-

2np+

In a polar coordinate system, once a wavefunction, [:.I] is,known, the polar angles 6' and 4 are determined from the following equation,

Applying it to the transmitted wave function, we have 6 ei4 tan -

D-exp (ik-z) =

2

The polar coordinates, 8 and tan(8/2)

=

4, can be

(T-/T+)'/~,

evaluated using Eqs. 12-14.

and

For small fields, 4 can be expressed by a Taylor expansion

where Ak=k--k+, and An

and

=

An--n

. Consistent with the small field approximation,

Using Eqs. 17.and 18, the expression for 4 becomes

By identifying d-bLr4, we have

-

In Eq. 16, k and n are treated as independent variables, even though they satisfy relation, n2 -k2 + constant. Using the derivative for an implicit function and suppressing the subscripts k and n, we obtain

After some algebra and use of Eq. 20, Eq. 19 becomes

"

rnallO-m

4 2 2 2 ko sinh(2nd)-2k nd(k -n )

fikn

fikak

4k2n2cosh2(nd)

+

2 2 2 2 (k -n ) sinh (nd)

(21)

This is identical with Eq. 2. Hence the two methods, phase delay and the spin precession, agree with each other in this application. Similarly, we can define a time quantity associated with spin rotation (i.e., the change in angle 9) by writing A9 = wLrg (22) as is done in Ref. 18. A derivation similar to the above leads to the following expression, arn~l'~ '9 =mfik ak '

where T

-

T+ or T as B

-+

0. Here we have assumed that the spin rotation has the same angular

velocity, o L , as the spin precession, which, however, has not been justified[l8,25].

For an

-opaquebarrier (i.e.. nd >> 1)

'4

m l , * fikn

v

and

where Ivl = magnitude of the imaginary velocity. Although, in the above calculation, the magnetic field B is in the z-direction, the result is the same for B in the x-direction if the field is assumed vanishingly small. Then, one can This is ignore the motion due to the Lorentz force as has been assumed by Buttiker[l8]. justified because the only energy term in the Hamiltonian associated with B is o-B, which has two values according to whether the spin is parallel or anti-parallel to the field. By making the coordinate transformation,

one can derive the same expression for the tunneling time.

3. DISCUSSIONS AND CONCLUSIONS To determine a tunneling time quantum mechanically, one first calculates either the change in phase or the change in angle in the Larmor spin precession, from which the tunneling times can be extracted. Therefore, in principle, we have two "clocks" which, to be physically acceptable, must be synchronous and record identical times for the same event. The above derivation does indeed show that the two clocks give identical tunneling time when the field is applied throughout the space. The two "clocks" also agree in free space (i.e., no barrier) as can be easily verified. Note, however, that a B field has to be present in order to have a Larmor clock. Now we may ask the question, why is there a difference between the spin precession and phase delay times when the field is applied only in the barrier region? The following argument suggests an explanation which has also been given by Falck, et a1.[25].

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They, however, consider a wave packet moving in a magnetic field region extending only a finite distance beyond the barrier. We stress that we are contrasting our model with Buttiker's in which the field is confined to the barrier region. The model used in the present work is a plane wave moving in a field covering all space. If the field is applied only in the barrier region, then in order for the Larmor clock to work properly (i.e., monitor precisely the motion of the particle), a "point" particle (e.g., a spatially localized wave packet) model should be used. It then initiates the clock precisely when it enters the barrier and it turns off the clock as it exits. In the present calculation, as in Buttiker, the particle is treated as a mono-energetic wave. According to the uncertainty principle, the position of the electron is indeterminate. Therefore, when the region spanned by the field is confined to only the barrier, one encounters the difficulty as to when the particle enters the field to "initiate" the clock and when it is turned off. On the other hand, when the field is applied throughout the space, the clock is "on" all the time and registering continually the motion of the particle. The change in the spin precession angle can now be calculated and used to extract the tunneling time. The same reasoning was used by Falck and Hauge, who assume the packet and the tunneling process is entirely within a field region, albeit finite. The wave packet starts from -zo (which is very large but still within the field), approaches and tunnels through the barrier, and goes to +zo also within the field. The change in the spin precession angle is then used to calculate the tunneling time. This procedure is precisely the same as the phase delay method except that the spin precession angle is replaced by the phase. In conclusion, we have shown that the phase delay and the Larmor spin precession are identical clocks for measuring tunneling times when the measuring "clocks" are properly set. However, a unique definition of tunneling time is still not yet available. It is interesting to note that Sokolovski and Baskint261, using the path integral method, have shown that the transit time in a barrier leads to a complex time. The real part corresponds to the phase delay time for spin precession, while the modulus is essentially Buttiker's tunneling time, TT = ,/rzP + rir. However, Biittiker maintains that during the tunneling process, Tsr >> rsp and therefore T T = T S ~ . REFERENCES

* This research was

supported in part by the Office of Naval Research,.Arlington, Virginia Contract No. N00014-86K-0160.

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