- Email: [email protected]
Transverse electric fields in mesoscopic normal rings
28 December 1998
PHYSICS
LETTERS
A
Physics Letters A 250 ( 1998) 425429
Transverse electric fields in mesoscopic normal rings E. .%mhek ’ 147 Oliver
Road. Santa Barbara,
CA 93109,
USA
Received I3 January 1998; revised manuscript received 17 September 1998; accepted for publication 1 October 1998 Communicated by A. Lagendijk
Abstract A quantum mechanical estimate of the electric field within a normal metal ring is made. In the clean limit, the voltage drop across a ring threaded by Aharonov-Bohm flux contains a large flux-independent part and a small flux-periodic part.
The results agree with the Bernoulli theorem. @ 1998 Elsevier Science B.V. PACS:
73.23.-b; 0.5.4O.+j; 03.65.B~; 72.15.Gd
Euler’s equations for a steady irrotational flow show that electric fields can be present in a charged nonviscous fluid. These fields, being proportional to the gradient of the kinetic energy, are the analog of Bernoulli’s pressure of an uncharged fluid [ 11. Starting from this point of view, London [2] predicted that electric fields exist within superconductors. Experiments performed on type I superconductors [3] seem to substantiate London’s prediction. In the present Letter, we study the electric fields generated by a similar mechanism in a small normal metal ring. Since Bernoulli’s theorem holds for the Schrodinger fluid [4], a nonzero electric field in the radial direction is expected for free electrons confined to a perfect circular ring. Quantum mechanical calculations reported here confirm this expectation. In the presence of a magnetic flux through the opening, the calculated voltage difference between the inner and the outer edges of the ring consists of two parts. The first is large and independent of Hux. The other is a periodic function of the flux and is much smaller. In
the clean limit, however, it yields a voltage drop that for a submicron ring is of a measurable magnitude. If actually detected, the flux-dependent voltage drop might offer a method that is complementary to the persistent current investigations vigorously pursued since the seminal paper of Btittiker et al. [ 51. We consider a normal metal ring threaded by an Aharonov-Bohm flux @ (see Fig. 1). Assuming that the elastic mean free path is larger than the size of the ring, we are led to an idealized model of free electrons confined to a curved planar waveguide with the Dirichlet boundary conditions imposed on the wavefunction [ 6,7]. The magnitude of the flux is limited to small values such that the radius of the Landau orbit exceeds the width, a = t-2 - rl, of the ring. This allows us to treat the effective, flux-induced, potential as a small perturbation. The points in the ring are parametrized by curvilinear coordinates s = R8 and u = Y - R, where B and Y are the polar coordinates of the point, and R = f (t-1 + 1-1). Expressed in the coordinates (s, u) , the Hamiltonian of the electron of mass M, is of the form
’ E-mail: [email protected]. 0375.9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved PII SO37S-9601(98)00759-2
426
E. %uinek/Phyics
FiuC’ 1. Planar ring threaded region r 6 r,. Points in the coordinates s = RH and u The radial fill illustrates the towards the outer radius Q.
Letters A 2.50 (1998) 425-429
by a magnetic flux confined to the ring are parametrized by curviknear = r - R. where R = f (rl + t-2). displacement of the electron density
of Eq. (5) represents the potential energy associated with the centrifugal force acting on the electron in the radial direction. From a classical point of view, WCexpect this force to displace the electron orbit towards the outer edge of the ring. This picture is corroborated quantum mechanically, since the perturbation (5) admixes into the ground state the nearest excited state that is of opposite parity in the u space. Thus an excess electron density is created near the outer edge which is accompanied by a deficit near the inner edge (see Fig. ( I)). This charge density fluctuation is the source of the electric field within the metal. For a many-electron system, it is essential to invoke screening in the calculation of the admixture. Specifically. the electric field due to all other electrons must be included in the perturbation acting on the single particle states. Hence, the net perturbation is given by
H=-g[J-f~-~A.,)2+J-‘$(J$)], (1) where J = 1 +u/R, and A,y = @/2R7~. To eliminate the Jacobian, J, from the volume integrals for the matrix elements, we replace the original wave function, q?, by J-‘/‘q$ where 4 satisfies the equation [6.7] fi4 = EI$. The resealed Hamiltonian B =
(2) I? is given by
J’f”HJ-112
=-+($-;A,.)‘+;+(ZRI)-‘1. (3) The Schrodinger equation (2) is not exactly separable owing to the presence of the u-dependent Jacobian in the Hamiltonian (3). For a narrow ring, the terms which spoil the separability can be treated as a perturbation. Expanding the J-’ factors in Eq. (3) about II = 0. we have to first order in U.
l?=&+t;,,
(4)
where i?a is the unperturbed Hamiltonian defined by setting J = 1 in Eq. (3). and fir is the perturbation .,=$[(&;A,y)‘+(2R)-2].
(5)
We note that the lowest order expansion in u/R is justified as long as the ratio a/R is small. The first term
where AV = V( rl ) - V( rz) is the potential drop across the ring that needs to be calculated self-consistently. According to this equation. a given electron does not feel only the centrifugal force but also the transverse electric field produced by the other electrons in response to this force. As this field acts opposite to the bare force, it tends to screen it. The linear dependence on u of the second term in Eq. (6) is an approximation which is expected to work well when the Fermi wavelength of electron is comparable with the width of the ring. The normalized eigenstates of fin are 4:::
= (rRa)-“‘exp(iln8)
fn(u),
(7)
where f,,(zO is the wave function of the nth transverse channel. For an infinite square well, f,,(u) = sin(nru/a) or cos(nn-~/a), for II even or odd, respectively. The excess charge density produced by all occupied states is
P(U) = -e
C
C
Wr,n12 - kQil’> (8)
E. .%dnek/Phy.sics
whereb,,.,, is the wave function the first approximation
perturbed
Letters A 250 (1998)
by HI in
425-429
Substituting
pi (u)
427
from Eq. ( 1 l), we obtain
from
Eq. (13) (14)
where we limit the summation to nearest neighbor pairs. In evaluating the matrix element, the small constant term (2R)-’ in the bracket of Eq. (5) is neglected. The energy eigenvalues E, = fi%-‘n’/2m,a’ are used in Eq. (9). The quantum number me(n) is given by
1, 112
?>Z() (H) =
y(EF
- E,,)
64em,a*
(11) The potential V(r) associated with this excess charge density is given by
= s
du (R + U)PI (u)
-al2 2?r
df9[r~+(R+u)*-2r(R+u)~0~8]-“*.
X 0
(12) Performing the angular integration in Eq. ( 11), the potential drop across the ring becomes (l/2
AV=2
1 + 2u/a .I -u/2
dnpi (u) ln
1 - 2u/a’
factor
17e2m,a2m0
r=
(15)
fib-‘R
and VO is the unscreened potential obtained from Eq. ( 13) by replacing p(u) by the bare density pa(u) We note that the latter density follows from Eq. ( 1 1) by setting AV equal to zero. The sum over m, in this equation is proportional to the net kinetic energy of the Fermi sea of electrons moving along the coordinate s. As shown by Byers and Yang [ 81, fluctuations in MO tend to wash out the flux-sensitivity of the kinetic energy. Thus, it is essential that the number of electrons in the ring is constant. If m0 is odd, we have forI4/& < k,withtheuseofEqs. (11) and (13) Av
=
0
_
17a3e( $mi+ m04'/q@
( 13)
(16)
n-TR4
Taking a = IO-’ cm, R = 1O-4 cm, and using Er = 2Ei in Eq. (lo), we obtain mo = rR/a. With these values, Eq. (16) yields AV, = (0.9 x 10-j + 2.8 x 10-Q’/&
= - 27fi’ti
V(r)
screening
(10)
where E,, is the threshold energy of the nth channel and Er is the Fermi energy. The evaluation of P?,(u) using the first order perturbation in HI is possible only if the difference EF - E,, is sufficiently small. In this case, the limiting value, MO(~), of the m-sum in Eq. (8) is small ensuring that the admixture coefficients in the wave function (9) stay well below one. This condition is best realized in the single transverse mode regime for which El < EF < Ez. In what follows, we consider this case and evaluate pi (u) with the use of Eqs. (8) and (9) yielding
‘I(‘)
where r is the electron-electron
v.
(17)
From Eq. (15) we obtain r x 33. The amplitude of the flux-periodic voltage, corresponding to I~/c,&I = $ is, according to Eqs. (16) and (17). AV z 2.1 x iO-9 V. The fl ux -’m d ependent part of the screened voltage is a much larger value of 2.7 x 1O-6 V. The voltage drop of such a magnitude is due to the fact that all electrons in the II = 1 channel are acted upon by a centrifugal force of the same radial direction. This is in contrast to the case of the persistent currents where contributions of opposite signs of nz cancel. To verify the consistency of our approximation, we calculate the magnitude of the admixture coefficient, (mo, l/Hllmo,2)/(Eg - El). and obtain, using the above parameters, a value of 2 x 1O-‘. This result provides a justification of the first-order approximation used in Eqs. (8)-(16). In the many-channel case, the evaluation of AV is more complicated. For E, near the Fermi level, the quantum number mo( n) can be small and the density
428
E. .%ndnek/Plz_vsics
Letters A 250 (1998) 425-329
p,,(u) can be calculated using the first order perturbation theory. However, for & well below Er, the perturbation theory fails owing to the large value of me (~1). If the transverse confinement potential is quadratic in U. the wave function (9) can be used even in this strong coupling case if a proper normalization constant is included. Using this wave function in Eq. (8), the resulting p,,(u) is very different from Eq. (18). In particular, its dependence upon AV is nonlinear which prevents us from expressing AV in the form of Eq. ( 14). By the same token the abovementioned cancellation of nearby channels is not expected for these low lying states. A rough estimate of the lower limit of AV can be made by including in the n-sum of Eq. (8) only those states near the Fermi level for which the perturbation theory holds. From Eq. (9), it follows that the admixtures of the Im, 12) and the lin. II * 1) tend to cancel in the cross products involved in the evaluation of Eq. (8). Hence, only the highest occupied transverse channel fro contributes to the rz-sum of Eq. (8). The corresponding formulas for r and AVa are similar to Eqs. ( 15) and ( 16) except for a factor of order 1/rzo coming from the energy difference in the admixture coefficient. Since 110= kFa/r, we have, for kF = IO8 cm-’ and a = lop5 cm, QJ = 3 x 10’. It should be noted that the actual value of the voltage drop may not exhibit such a large reduction since the value of mo (~0) can be made larger due to the increased transverse energy difference near the Fermi level. However, as seen from Eq. ( 16), the ratio of the flux-induced voltage to the flux-independent background goes as -2, suggesting an operation with the smallest possi1710 ble values of me. From this point of view, the singlemode regime is more suitable. In the presence of disorder, the magnitude of AV is expected to get considerably reduced compared to the present theory. This is because the free electron velocities, in Eq. ( 1 I), must be replaced by much smaller drift velocities. Consequently, an extremely pure and defect free ring is necessary for the detection of the flux-sensitive voltage drop. The R-dependence of A VO,shown in Eq. ( 16)) suggests using a ring of the smallest possible dimensions. The flux-independent part of AV, however, should be observable in practical realizations of samples with low drift velocities. We note that radial potential gradients due to centrifugal forces in spinning normal metal rotors were ob-
served by Beams [ 9 1. Capacitive probes were used to couple the voltmeter to the rotor. As pointed out in Ref. [3], capacitive coupling is essential to deal with the problem of the potential drop being cancelled by the contact potential. Washburn et al. [ 101 studied the effects of the applied transverse electric field on the magnetoresistance in an Sb loop. A similar experimental arrangement could be used for the detection of the voltage drop proposed in the present work. However, attention should be paid to the geometry of the ring. Rings that are actually squares were used in Ref. [ IO] because they are more easily fabricated with the electron-beam lithography [ II]. This geometry appears not suitable for the proposed search for the following reasons: First, there are no centrifugal forces at the sides of the square where the capacitive probes are placed. Furthermore, if the mean free path exceeds the sample dimension, the electron starts to feel the curvature-induced potential at the corners [ 61. This may result in the formation of deleterious bound states. Hence, metal rings of circular shape are necessary for the detection of the transverse electric fields. Since an ideally circular ring is difficult to produce, a question then arises to what extent is the predicted effect stable with respect to small imperfections. The connection to the Bernoulli equation points towards the problem of stability of laminar flow in the presence of such deviations from perfect geometry. Recent investigations of coherent ballistic transport through a cavity [ 121 or circular bend [ 131 may serve as a guidance towards the solution of this challenging problem. In these works, a transition from the laminar to the vertical flow is seen in numerical solutions of the two-dimensional Schrodinger equation. As the electron energy increases, the vortex pattern generally grows in complexity until a highly irregular turbulent motion takes place [ 141. Furthermore, in the single-mode regime the laminar flow through the circular bend appears stable as long as the electron energy stays well below the threshold of the second subband [ 131. Hence, observations of the proposed effects should be done preferentially on rings operating in the single-mode regime. Systems of this kind may be achieved in modulation-doped semiconductor heterojunctions [ 151. In such devices, the modal occupancy may be changed by varying the gate and/or the substrate voltages. Measurements of the voltage drop as function of the Fermi energy should be of in-
E. %ndnek /Physics
Letters A 250 (1998)
terest in view of the interference blockade caused by scattering of the electrons on the imperfections [ 131. Finally, we turn to the question if the present theory, summarized in Eqs. (14)-( 16), is consistent with the Bernoulli equation [ I-31 AV = (m,/2e)[r+,)
-&Q)],
(18)
where ~1~(1.) is the velocity squared of the electron fluid moving along the circular path of radius r. This quantity can be obtained quantum mechanically from the kinetic energy density divided by one half of the mass density. Confining ourselves to the single-mode regime, we have
425-429
For a submicron ring, operating in the single-mode regime, the amplitude of the periodic part is in the nanovolt range, whereas the constant part is almost three orders of magnitude larger. After submitting this work for publication, several papers on related subjects were brought to my attention [ 16-181. In these works it has been pointed out on general grounds that electric potentials in a ring depend on the enclosed magnetic flux. Distinct from the present calculation, Btittiker [ 161 considers the electric potential in a one-dimensional ring induced by the electron density which varies only in the Sdirection (due to the lack of rotational invariance of the loop). Refs. [ 16,171 contain calculations of the flux-dependent capacitance in a Coulomb blockade model. None of these works considers the transverse electric fields generated by the centrifugal forces in electron waveguides of finite width. I thank R. Gaupsas for an instructive
Introducing a/R << I
429
discussion.
this result into Eq. ( 18), we obtain for References
eAV
= 2nrfi;RJ c
CM 171
-
d&Jo)*.
Using Eqs. ( 15 ) and ( 16)) the same result is obtained from Eq. ( 14) for r >> 1. We see that large Coulomb screening is essential for establishing the validity of the Bernoulli theorem for the normal metal ring. Screening acts as a negative feedback causing AV to be insensitive to changes of system parameters. In conclusion, the transverse electric field in a disorder-free normal metal ring threaded by an Aharonov-Bohm magnetic flux is calculated quantum mechanically. The centrifugal force acting on the electrons revolving round the ring tends to displace the electron charge towards the outer edge of the ring. The resulting potential difference between the outer and the inner edges of the ring agrees with the prediction based on the Bernoulli theorem for a charged nonviscous electron fluid. Electron-electron screening effects play an important role in the calculation. The calculated potential difference has two parts: One which is constant (independent of flux). and the other one which is a periodic function of 4.
L.D. Landau, E.M. Lifshitz, Fluid Dynamics (Pergamon Press. New York, 1975). 121 F. London. Superfluids. Vol. I (New York, 1961 ). 131 J. Bok. J. Klein, Phys. Rev. Lett. 20 (1968) 660. 141 R.P. Feynman. Statistical Mechanics (Benjamin, New York, 1972). 151 M. Biittiker. Y. Imry. R. Landauer, Phys. Lett. A 96 (1983) 365. 161 P Exner, P. Seba, J. Math. Phys. 30 ( 1989) 2574. 171 J. Goldstone, R.L. Jaffe. Phys. Rev. B 45 (1992) 14100. [Sl N. Byers, C.N. Yang, Phys. Rev. Len. 7 ( 1961) 46. 191 J.W. Beams, Phys. Rev. Lett. 21 (1968) 1093. 1101 S. Washburn. H. Schmid. D. Kern, R.A. Webb. Phys. Rev. Lett. 59 (1987) 1791. G.J. Dolan, J. Dunsmuir, Physica B 152 (1988) 7. 1111 1121 C.S. Lent, Appl. Phys. Lett 57 (1990) 1678. 1131 K.F. Berggren, Z.-L. Ji, Phys. Rev. B 47 (1993) 6390. 1141 J.O. Hirschfelder, K.T. Tang, J. Chem. Phys. 64 ( 1976) 760. 1151 C.W.J. Beenakker, H. van Houten, in: Solid State Physics, Vol. 44, eds H. Ehrenreich. D. Turnbull (Academic, New York, 1991). M. Biittiker, Phys. Ser. T 54 ( 1994) 104. M. Biittiker. C.A. Stafford. Phys. Rev. Lett. 76 (1996) 495. P. Cedraschi, M. Biittiker, J. Phys. Condens. Matter IO (1998) 3985.
PHYSICS
LETTERS
A
Physics Letters A 250 ( 1998) 425429
Transverse electric fields in mesoscopic normal rings E. .%mhek ’ 147 Oliver
Road. Santa Barbara,
CA 93109,
USA
Received I3 January 1998; revised manuscript received 17 September 1998; accepted for publication 1 October 1998 Communicated by A. Lagendijk
Abstract A quantum mechanical estimate of the electric field within a normal metal ring is made. In the clean limit, the voltage drop across a ring threaded by Aharonov-Bohm flux contains a large flux-independent part and a small flux-periodic part.
The results agree with the Bernoulli theorem. @ 1998 Elsevier Science B.V. PACS:
73.23.-b; 0.5.4O.+j; 03.65.B~; 72.15.Gd
Euler’s equations for a steady irrotational flow show that electric fields can be present in a charged nonviscous fluid. These fields, being proportional to the gradient of the kinetic energy, are the analog of Bernoulli’s pressure of an uncharged fluid [ 11. Starting from this point of view, London [2] predicted that electric fields exist within superconductors. Experiments performed on type I superconductors [3] seem to substantiate London’s prediction. In the present Letter, we study the electric fields generated by a similar mechanism in a small normal metal ring. Since Bernoulli’s theorem holds for the Schrodinger fluid [4], a nonzero electric field in the radial direction is expected for free electrons confined to a perfect circular ring. Quantum mechanical calculations reported here confirm this expectation. In the presence of a magnetic flux through the opening, the calculated voltage difference between the inner and the outer edges of the ring consists of two parts. The first is large and independent of Hux. The other is a periodic function of the flux and is much smaller. In
the clean limit, however, it yields a voltage drop that for a submicron ring is of a measurable magnitude. If actually detected, the flux-dependent voltage drop might offer a method that is complementary to the persistent current investigations vigorously pursued since the seminal paper of Btittiker et al. [ 51. We consider a normal metal ring threaded by an Aharonov-Bohm flux @ (see Fig. 1). Assuming that the elastic mean free path is larger than the size of the ring, we are led to an idealized model of free electrons confined to a curved planar waveguide with the Dirichlet boundary conditions imposed on the wavefunction [ 6,7]. The magnitude of the flux is limited to small values such that the radius of the Landau orbit exceeds the width, a = t-2 - rl, of the ring. This allows us to treat the effective, flux-induced, potential as a small perturbation. The points in the ring are parametrized by curvilinear coordinates s = R8 and u = Y - R, where B and Y are the polar coordinates of the point, and R = f (t-1 + 1-1). Expressed in the coordinates (s, u) , the Hamiltonian of the electron of mass M, is of the form
’ E-mail: [email protected]. 0375.9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved PII SO37S-9601(98)00759-2
426
E. %uinek/Phyics
FiuC’ 1. Planar ring threaded region r 6 r,. Points in the coordinates s = RH and u The radial fill illustrates the towards the outer radius Q.
Letters A 2.50 (1998) 425-429
by a magnetic flux confined to the ring are parametrized by curviknear = r - R. where R = f (rl + t-2). displacement of the electron density
of Eq. (5) represents the potential energy associated with the centrifugal force acting on the electron in the radial direction. From a classical point of view, WCexpect this force to displace the electron orbit towards the outer edge of the ring. This picture is corroborated quantum mechanically, since the perturbation (5) admixes into the ground state the nearest excited state that is of opposite parity in the u space. Thus an excess electron density is created near the outer edge which is accompanied by a deficit near the inner edge (see Fig. ( I)). This charge density fluctuation is the source of the electric field within the metal. For a many-electron system, it is essential to invoke screening in the calculation of the admixture. Specifically. the electric field due to all other electrons must be included in the perturbation acting on the single particle states. Hence, the net perturbation is given by
H=-g[J-f~-~A.,)2+J-‘$(J$)], (1) where J = 1 +u/R, and A,y = @/2R7~. To eliminate the Jacobian, J, from the volume integrals for the matrix elements, we replace the original wave function, q?, by J-‘/‘q$ where 4 satisfies the equation [6.7] fi4 = EI$. The resealed Hamiltonian B =
(2) I? is given by
J’f”HJ-112
=-+($-;A,.)‘+;+(ZRI)-‘1. (3) The Schrodinger equation (2) is not exactly separable owing to the presence of the u-dependent Jacobian in the Hamiltonian (3). For a narrow ring, the terms which spoil the separability can be treated as a perturbation. Expanding the J-’ factors in Eq. (3) about II = 0. we have to first order in U.
l?=&+t;,,
(4)
where i?a is the unperturbed Hamiltonian defined by setting J = 1 in Eq. (3). and fir is the perturbation .,=$[(&;A,y)‘+(2R)-2].
(5)
We note that the lowest order expansion in u/R is justified as long as the ratio a/R is small. The first term
where AV = V( rl ) - V( rz) is the potential drop across the ring that needs to be calculated self-consistently. According to this equation. a given electron does not feel only the centrifugal force but also the transverse electric field produced by the other electrons in response to this force. As this field acts opposite to the bare force, it tends to screen it. The linear dependence on u of the second term in Eq. (6) is an approximation which is expected to work well when the Fermi wavelength of electron is comparable with the width of the ring. The normalized eigenstates of fin are 4:::
= (rRa)-“‘exp(iln8)
fn(u),
(7)
where f,,(zO is the wave function of the nth transverse channel. For an infinite square well, f,,(u) = sin(nru/a) or cos(nn-~/a), for II even or odd, respectively. The excess charge density produced by all occupied states is
P(U) = -e
C
C
Wr,n12 - kQil’> (8)
E. .%dnek/Phy.sics
whereb,,.,, is the wave function the first approximation
perturbed
Letters A 250 (1998)
by HI in
425-429
Substituting
pi (u)
427
from Eq. ( 1 l), we obtain
from
Eq. (13) (14)
where we limit the summation to nearest neighbor pairs. In evaluating the matrix element, the small constant term (2R)-’ in the bracket of Eq. (5) is neglected. The energy eigenvalues E, = fi%-‘n’/2m,a’ are used in Eq. (9). The quantum number me(n) is given by
1, 112
?>Z() (H) =
y(EF
- E,,)
64em,a*
(11) The potential V(r) associated with this excess charge density is given by
= s
du (R + U)PI (u)
-al2 2?r
df9[r~+(R+u)*-2r(R+u)~0~8]-“*.
X 0
(12) Performing the angular integration in Eq. ( 11), the potential drop across the ring becomes (l/2
AV=2
1 + 2u/a .I -u/2
dnpi (u) ln
1 - 2u/a’
factor
17e2m,a2m0
r=
(15)
fib-‘R
and VO is the unscreened potential obtained from Eq. ( 13) by replacing p(u) by the bare density pa(u) We note that the latter density follows from Eq. ( 1 1) by setting AV equal to zero. The sum over m, in this equation is proportional to the net kinetic energy of the Fermi sea of electrons moving along the coordinate s. As shown by Byers and Yang [ 81, fluctuations in MO tend to wash out the flux-sensitivity of the kinetic energy. Thus, it is essential that the number of electrons in the ring is constant. If m0 is odd, we have forI4/& < k,withtheuseofEqs. (11) and (13) Av
=
0
_
17a3e( $mi+ m04'/q@
( 13)
(16)
n-TR4
Taking a = IO-’ cm, R = 1O-4 cm, and using Er = 2Ei in Eq. (lo), we obtain mo = rR/a. With these values, Eq. (16) yields AV, = (0.9 x 10-j + 2.8 x 10-Q’/&
= - 27fi’ti
V(r)
screening
(10)
where E,, is the threshold energy of the nth channel and Er is the Fermi energy. The evaluation of P?,(u) using the first order perturbation in HI is possible only if the difference EF - E,, is sufficiently small. In this case, the limiting value, MO(~), of the m-sum in Eq. (8) is small ensuring that the admixture coefficients in the wave function (9) stay well below one. This condition is best realized in the single transverse mode regime for which El < EF < Ez. In what follows, we consider this case and evaluate pi (u) with the use of Eqs. (8) and (9) yielding
‘I(‘)
where r is the electron-electron
v.
(17)
From Eq. (15) we obtain r x 33. The amplitude of the flux-periodic voltage, corresponding to I~/c,&I = $ is, according to Eqs. (16) and (17). AV z 2.1 x iO-9 V. The fl ux -’m d ependent part of the screened voltage is a much larger value of 2.7 x 1O-6 V. The voltage drop of such a magnitude is due to the fact that all electrons in the II = 1 channel are acted upon by a centrifugal force of the same radial direction. This is in contrast to the case of the persistent currents where contributions of opposite signs of nz cancel. To verify the consistency of our approximation, we calculate the magnitude of the admixture coefficient, (mo, l/Hllmo,2)/(Eg - El). and obtain, using the above parameters, a value of 2 x 1O-‘. This result provides a justification of the first-order approximation used in Eqs. (8)-(16). In the many-channel case, the evaluation of AV is more complicated. For E, near the Fermi level, the quantum number mo( n) can be small and the density
428
E. .%ndnek/Plz_vsics
Letters A 250 (1998) 425-329
p,,(u) can be calculated using the first order perturbation theory. However, for & well below Er, the perturbation theory fails owing to the large value of me (~1). If the transverse confinement potential is quadratic in U. the wave function (9) can be used even in this strong coupling case if a proper normalization constant is included. Using this wave function in Eq. (8), the resulting p,,(u) is very different from Eq. (18). In particular, its dependence upon AV is nonlinear which prevents us from expressing AV in the form of Eq. ( 14). By the same token the abovementioned cancellation of nearby channels is not expected for these low lying states. A rough estimate of the lower limit of AV can be made by including in the n-sum of Eq. (8) only those states near the Fermi level for which the perturbation theory holds. From Eq. (9), it follows that the admixtures of the Im, 12) and the lin. II * 1) tend to cancel in the cross products involved in the evaluation of Eq. (8). Hence, only the highest occupied transverse channel fro contributes to the rz-sum of Eq. (8). The corresponding formulas for r and AVa are similar to Eqs. ( 15) and ( 16) except for a factor of order 1/rzo coming from the energy difference in the admixture coefficient. Since 110= kFa/r, we have, for kF = IO8 cm-’ and a = lop5 cm, QJ = 3 x 10’. It should be noted that the actual value of the voltage drop may not exhibit such a large reduction since the value of mo (~0) can be made larger due to the increased transverse energy difference near the Fermi level. However, as seen from Eq. ( 16), the ratio of the flux-induced voltage to the flux-independent background goes as -2, suggesting an operation with the smallest possi1710 ble values of me. From this point of view, the singlemode regime is more suitable. In the presence of disorder, the magnitude of AV is expected to get considerably reduced compared to the present theory. This is because the free electron velocities, in Eq. ( 1 I), must be replaced by much smaller drift velocities. Consequently, an extremely pure and defect free ring is necessary for the detection of the flux-sensitive voltage drop. The R-dependence of A VO,shown in Eq. ( 16)) suggests using a ring of the smallest possible dimensions. The flux-independent part of AV, however, should be observable in practical realizations of samples with low drift velocities. We note that radial potential gradients due to centrifugal forces in spinning normal metal rotors were ob-
served by Beams [ 9 1. Capacitive probes were used to couple the voltmeter to the rotor. As pointed out in Ref. [3], capacitive coupling is essential to deal with the problem of the potential drop being cancelled by the contact potential. Washburn et al. [ 101 studied the effects of the applied transverse electric field on the magnetoresistance in an Sb loop. A similar experimental arrangement could be used for the detection of the voltage drop proposed in the present work. However, attention should be paid to the geometry of the ring. Rings that are actually squares were used in Ref. [ IO] because they are more easily fabricated with the electron-beam lithography [ II]. This geometry appears not suitable for the proposed search for the following reasons: First, there are no centrifugal forces at the sides of the square where the capacitive probes are placed. Furthermore, if the mean free path exceeds the sample dimension, the electron starts to feel the curvature-induced potential at the corners [ 61. This may result in the formation of deleterious bound states. Hence, metal rings of circular shape are necessary for the detection of the transverse electric fields. Since an ideally circular ring is difficult to produce, a question then arises to what extent is the predicted effect stable with respect to small imperfections. The connection to the Bernoulli equation points towards the problem of stability of laminar flow in the presence of such deviations from perfect geometry. Recent investigations of coherent ballistic transport through a cavity [ 121 or circular bend [ 131 may serve as a guidance towards the solution of this challenging problem. In these works, a transition from the laminar to the vertical flow is seen in numerical solutions of the two-dimensional Schrodinger equation. As the electron energy increases, the vortex pattern generally grows in complexity until a highly irregular turbulent motion takes place [ 141. Furthermore, in the single-mode regime the laminar flow through the circular bend appears stable as long as the electron energy stays well below the threshold of the second subband [ 131. Hence, observations of the proposed effects should be done preferentially on rings operating in the single-mode regime. Systems of this kind may be achieved in modulation-doped semiconductor heterojunctions [ 151. In such devices, the modal occupancy may be changed by varying the gate and/or the substrate voltages. Measurements of the voltage drop as function of the Fermi energy should be of in-
E. %ndnek /Physics
Letters A 250 (1998)
terest in view of the interference blockade caused by scattering of the electrons on the imperfections [ 131. Finally, we turn to the question if the present theory, summarized in Eqs. (14)-( 16), is consistent with the Bernoulli equation [ I-31 AV = (m,/2e)[r+,)
-&Q)],
(18)
where ~1~(1.) is the velocity squared of the electron fluid moving along the circular path of radius r. This quantity can be obtained quantum mechanically from the kinetic energy density divided by one half of the mass density. Confining ourselves to the single-mode regime, we have
425-429
For a submicron ring, operating in the single-mode regime, the amplitude of the periodic part is in the nanovolt range, whereas the constant part is almost three orders of magnitude larger. After submitting this work for publication, several papers on related subjects were brought to my attention [ 16-181. In these works it has been pointed out on general grounds that electric potentials in a ring depend on the enclosed magnetic flux. Distinct from the present calculation, Btittiker [ 161 considers the electric potential in a one-dimensional ring induced by the electron density which varies only in the Sdirection (due to the lack of rotational invariance of the loop). Refs. [ 16,171 contain calculations of the flux-dependent capacitance in a Coulomb blockade model. None of these works considers the transverse electric fields generated by the centrifugal forces in electron waveguides of finite width. I thank R. Gaupsas for an instructive
Introducing a/R << I
429
discussion.
this result into Eq. ( 18), we obtain for References
eAV
= 2nrfi;RJ c
CM 171
-
d&Jo)*.
Using Eqs. ( 15 ) and ( 16)) the same result is obtained from Eq. ( 14) for r >> 1. We see that large Coulomb screening is essential for establishing the validity of the Bernoulli theorem for the normal metal ring. Screening acts as a negative feedback causing AV to be insensitive to changes of system parameters. In conclusion, the transverse electric field in a disorder-free normal metal ring threaded by an Aharonov-Bohm magnetic flux is calculated quantum mechanically. The centrifugal force acting on the electrons revolving round the ring tends to displace the electron charge towards the outer edge of the ring. The resulting potential difference between the outer and the inner edges of the ring agrees with the prediction based on the Bernoulli theorem for a charged nonviscous electron fluid. Electron-electron screening effects play an important role in the calculation. The calculated potential difference has two parts: One which is constant (independent of flux). and the other one which is a periodic function of 4.
L.D. Landau, E.M. Lifshitz, Fluid Dynamics (Pergamon Press. New York, 1975). 121 F. London. Superfluids. Vol. I (New York, 1961 ). 131 J. Bok. J. Klein, Phys. Rev. Lett. 20 (1968) 660. 141 R.P. Feynman. Statistical Mechanics (Benjamin, New York, 1972). 151 M. Biittiker. Y. Imry. R. Landauer, Phys. Lett. A 96 (1983) 365. 161 P Exner, P. Seba, J. Math. Phys. 30 ( 1989) 2574. 171 J. Goldstone, R.L. Jaffe. Phys. Rev. B 45 (1992) 14100. [Sl N. Byers, C.N. Yang, Phys. Rev. Len. 7 ( 1961) 46. 191 J.W. Beams, Phys. Rev. Lett. 21 (1968) 1093. 1101 S. Washburn. H. Schmid. D. Kern, R.A. Webb. Phys. Rev. Lett. 59 (1987) 1791. G.J. Dolan, J. Dunsmuir, Physica B 152 (1988) 7. 1111 1121 C.S. Lent, Appl. Phys. Lett 57 (1990) 1678. 1131 K.F. Berggren, Z.-L. Ji, Phys. Rev. B 47 (1993) 6390. 1141 J.O. Hirschfelder, K.T. Tang, J. Chem. Phys. 64 ( 1976) 760. 1151 C.W.J. Beenakker, H. van Houten, in: Solid State Physics, Vol. 44, eds H. Ehrenreich. D. Turnbull (Academic, New York, 1991). M. Biittiker, Phys. Ser. T 54 ( 1994) 104. M. Biittiker. C.A. Stafford. Phys. Rev. Lett. 76 (1996) 495. P. Cedraschi, M. Biittiker, J. Phys. Condens. Matter IO (1998) 3985.