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A mechanism for Coulomb blockade in scanning tunneling spectroscopy
Volume 148, number 6,7
PHYSICS LETTERS A
27 August 1990
A mechanism for Coulomb blockade in scanning tunneling spectroscopy M. Tsukada’ and J. Mahanty Department of Theoretical Physics, Research School ofPhysicalSciences, TheAu.stralian National University, Canberra, A.C. T 2501, Australia Received 21 March 1990; accepted for publication 19 June 1990 Communicated by A.A. Maradudin
The Coulomb blockade phenomenon observed in the 1—V characteristic curve in the spectroscopic mode of a STM with the simple surface—tip geometry is explained as a many-body dynamical effect occurring in the process of electron tunneling. This mechanism replaces the conventional picture based on an artificial small capacitance of the system.
Observation of the single electron tunneling process through a mesoscopic junction consisting of a tip and a surface in the scanning tunneling spectroscopy (STS) mode of a scanning tunneling microscope (STM) has attracted much attention lately [1—31.A small gap in the tunnel conductance of STS observed in the zero bias region for a metallic surface is explained by the so-called Coulomb blockade effect [1]. Another manifestation of the Coulomb blockade effect is the staircase-like I—V curve obtamed for a series junction of a metal surface—metal droplet—STM tip [3]. This phenomenon is quite significant, since it shows clearly that the charge of the carriers of the tunnel current is quantized. So far, the Coulomb blockade effect has been explained in a phenomenological way. When a charge Q is transferred between two initially neutral electrodes, the energy of the system increases by
a—, 2C 2
EE=
(1)
where C~ is the capacitance between the two electrodes. The bias voltage must exceed that corresponding to this energy gap for conduction to occur. The observed magnitude of~Eis of the order of sevOn leave from: Department of Physics, Faculty of Science, University of Tokyo, 7-3-i Hongo, Bunkyo-ku, Tokyo 113, Japan.
era! tens of meV, which corresponds to C—i 10—18 F. For the case of the series double junction including the metal droplet, such a classical picture of the Coulomb blockade would be plausible, because the stored charge in the droplet is well defined, and the capacitance C determines the energy increase AE due to the charging. However, in the case ofthe usual tip— surface geometry of STM, the mechanism of the Coulomb blockade is not well understood from the microscopic view point. The Coulomb blockade energy itself, of course, can be expressed by eq. (1), if C is regarded as a phenomenological parameter. However, the physical significance of C in this case i’s far from trivial it incorporates complicated manybody effects, as will be seen later. The transferred electron charge would rapidly spread out over a macroscopic region on the tip or on the surface, and the small value of C (—~1018 F) as observed experi—
mentally cannot be explained. Moreover, the increase in potential energy due to the charging of the capacitance is impossible, if the STM system is operated in the constant voltage mode. Hence in this case, the classical capacitance mechanism cannot explain the so-called Coulomb blockade phenomenon. The aim of the present paper is to propose a microscopic dynamic mechanism of the Coulomb blockade based on the coupftng of the tunneling electron with the plasmon modes in the tip. The tunnel current is generated as the net effect of
0375-9601/90/S 03.50 C 1990 — Elsevier Science Publishers B.V. (North-Holland)
367
Volume 148, number 6,7
PHYSICS LETTERS A
the individual electron transfer events occurring in unit time through the potential barrier region, i.e., the spatial gap between the tip and the sample surface. There are two time scales in this situation, the duration ; of each tunneling event, and the average time td between two successive events. The time scale t~of each tunneling event would be of the order of ~ where d is the width of the spatial gap, ç~is the work function and m is the electron mass [4]. For the typical case of d= 10 A, ~~=4eV, i, becomes of the order of 10—’ ~ s. On the other hand, in the usual case of STM measurements, the tunneling current is of the order of 1 nA, which implies that the average time between two successive tunneling events is about l0~s. Thus r~is about five or six orders of magnitude shorter than r~,and therefore, in the constant voltage mode each event can be considered as taking place statistically independently of the previous one. Another point to be noted is that the time scale r~ of the tunneling event is longer than the inverse plasma frequency (—~10_16 s) of the usual metals, Therefore, during the course of the tunneling event, the charge distribution in the tip has nearly enough time to reorganise to equilibrium following the temporarily created tunneling charge. We envisage the process of each tunneling event from the tip to the surface as follows. In the initial stage of tunneling there would be a pile up of charge corresponding to a wave packet state around the tunnel active region near the surface ofthe tip, as shown schematically in fig. 1. There is some evidence supporting this picture. The classical trajectory of the tunneling electron calculated by the WKB method [51reveals a focussing effect, which indicates formation of a wave packet around the apex of the tip. An important point here is that around a classical turning point on the trajectory the velocity of the electron small,the andwave this tends build up ternporarily is sustain packettoaround theand turning points of the focussed region. Some analyses of the dynamical behaviour of the tunneling electron wave also reveal the existence of the stand-by effect, before it begins to traverse into the gap region [6]. The interaction of the tunneling electron with the charge oscillation modes or plasmons of the surface and the tip produces two effects. Firstly, the shape ofthe tunneling potential barrier is governed by this 368
27 August 1990
a b
siani.z
SURFACE
Fig. 1. Wave packet formation just before tunneling.
interaction, through its contribution to the self-energy (or the “image potential”) of the tunneling electron. Secondly, it also determines the energy of the state from which the wave packet of the tunneling electron emerges at the tip (or gets into, in the reversed bias situation). We shall show that the latter effect is responsible for the Coulomb blockade. Let us denote by W(r) the tunneling wave packet formed transiently before it traverses into the barrier region. The other electrons in the tip will feel the potential due to the charge distribution of the wave packet, and react to stabilize the energy. This process is described by the interaction with the plasmons in the tip. In second order perturbation theory the stabilization energy iIE is given by [7] ~E’
~ I C~12 A
(2)
hw1
with
I C,~I
2—e2
j
I~2A(r)121 W(r) 12 dr,
(3)
where a~is the plasmon frequency of the Ath mode and 92 2(r) is the normalized potential of that mode. The above estimate is reasonable, since the time scale of the formation and duration of the wave packet is longer than the inverse plasma frequency. The physical consequence of this is illustrated in fig. 2. The electron in the tip cannot be removed from the Fermi level E~T) of the tip, but only from the (—j;)
Volume 148, number 6,7
TIP
____________
PHYSICS LETTERS A
SAMPLE 8> E’F
________
27 August 1990
Q~(k,p)exp(kz) qi~(p,z)=mw~M*~[212~(k, p)—&~ _(coscoz)+ k2w~ p[2Q~(k, p) —os]
_______
sin (pz))]
x exp(ik~p)
__________
(5)
,
respectively, with k and p representing the wave numbers parallel and perpendicular to the surface, Fig. 2. The mechanism ofCoulombblockade. The tip electron is removed from the transient wave packet state, which is situated at ~.Ebelow Er’.
wave packet state, which is situated at an energy AE below E~T). As shown below, stabilization by plasmon coupling is small for thesuch sample surface side, and we neglect it for convenience of the description, though its inclusion does not change the essential point of the results. It is then clear from fig. 2 that the electron cannot be transferred from the tip to the
and p and z are the corresponding components ofthe coordinate. The inner part of the material with the surface corresponds to the region z<0. fi is taken to be ..j~ v~,v~being the Fermi velocity. w theelectron plasma 2/m, n being0isthe frequency, with w~ = 4zne density and m being the effective mass. The normalization constant ofthe potential, Mi,,, is given by ~
=
fl4p2 [2Q~ (k, p)
—
w~]
x [4~3mnQ 8(k, p) ] —1 [Q~(k, p) 24$2k2Q~(k,p)}’. (6) TheX{[2Q~(k,p)oi~o] angular frequency Q 5 (k) and the potential (Oa (P, z) of the surface plasmon modes in the planar case are given by [81 2k2+flk~.,/2w~ +fi2k2) (7) Q~(k) +fl ~ (p, z) =~(w~ mN~ exp (ik~p) —
surface unless the negative bias applied to the tip exceedse~.E,i.e.,IeVI>~.E. The process taking place in the case ofthe electron tunneling into the tip can be described as follows. The electron coming from the sample surface to the tip will also form the wave packet around the active region of the tip surface. In the same way as before, the energy ofthe wave packet state is lowered by AE compared with the initial state at the sample surface. If the stabilized energy of the wave packet is below
X [p 5Q~(k) exp(kz) kw~exp(Psz)] Here Nk is the normalization constant, 2[l6iu2mnQs(k)k3]’ Nw~[2Q~(k)w~] x [w~ +2Q~(k)]~[Q~(k)—w2~—2 0j —
E~T),the
Fermi level of the tip, it cannot enter the tip. Therefore, unless the Fermi level E~ of the sample surface is raised by a bias voltage above E~T)+i~E,the tunnel current cannot flow. The microscopic mechanism of the Coulomb blockade is thus given above, The next question is how the Coulomb blockade energy AEthe is determined reflectingofthe the geometric tures and material properties tip. This feacan be analysed using the hydrodynamic model of plasmons [8]. The plasmon modes in the tip and the surface have been studied in this model [7,9]. In each case we have bulk and surface plasmons. For a planar surface the angular frequency ~ and the potential P*,p of the bulk plasmon mode (k, p) are given by [8] Q~(k, p) = w~+ fl2k2 + p22 (4)
.
,
(8)
(9)
and Ps is given by 2k2). (10) flk+ ~J2co~+ fl A reasonable model of the tip is a hemispherical bump on a planar surface parallel to the sample surface. In this case, the exact forms ofthe bulk and surface plasmon potentials will be complicated, but their spatial extension particularly in the z-direction, can be expected to be qualitatively similar to that given in eqs. (5) and (8). To proceed further, let us assume a concrete form for the charge density of the wave packet, _________
Ps
=
(_
369
Volume 148, number 6,7
PHYSICS LETTERS A
W(P~z)I2=..,,J~2exP(_z_zoI) xex
~\
(_(j,_~,~)2’\
b~
~ (11)
)~
a and b are parameters defining the extent of the charge distribution in the normal and lateral directions respectively, and (Po, z0) specify the centre ofthe wave packet is taken nearthe tip. We have tried some other forms for I !P(p, z) 12, but the results do not depend much on the form, as long as the two parameters defining the spatial extent are fixed. Using eqs. (4)—(11), it is easy to show that I C,,~I 2 hQ 5(k, 2 p) / ~ ~ e = I(b/a) +O~— (12) where
27 August 1990
l6l3/2~,
forb>>a,
.~—~—ln(a/b), fora>>b.
(15)
For more general cases, i~.Ewould be given by the sum of eqs. (12) and (14). It is remarkable that if the size of the wave packet (a and b, which would be of the order of the radius of curvature of the tip) is larger than the Thomas— Fermi screening length (-.~fl/w0), the Coulomb blockage energy AE does not depend on the material properties ofthe tip or the surface, but is determined only by their geometrical features. In fact, the Cou2/ blomb for b>> a, apart from is a numerical coefficient; b here blockade energy generally given by LiE—~ e
~
~—~---
,
s—),
°
°
where 1(x) is given by 1(x) _
=
f ~exp (
J
—
~2)
(x+~)2
d~ _i._ 2x2’
for x>> 1
—
IlnxI
‘
,
forx<< 1
.
(13)
On the other hand, the contribution to the Coulomb blockade energy from the surface plasmons is calculated to be
v IC,,12 ‘r hQ
1 ~ +~(—±~— ~ (14) 3”2 b \~aw 5(k) l6x 0‘bwoJ’ If one neglects the dispersion of the plasmons, only the first terms ofthe right hand side of eqs. (12) and (14) are obtained. This is a reasonable approximation if the sizes a and b are greater than the Thomas—Fermi screening length l/FCTF—’ fl/coo. If the lateral size b of the charge distribution is much larger than the vertical size a, the dominant contribution to the Coulomb blockade energy is from the surface —
plasmons, eq. (14); but for the opposite case of a>> b, it is mainly due to the bulk plasmons, eq. (12). We thus obtain 370
is a measure of the sizeonly of the along the surface. In lateral this case the wave surfacepacket plasmons give the dominant contribution to ~.E, since the overlap between the wave packet and the bulk plasmons is small leading to small values for CA of eq. (3). For appreciable overlap of a wave packet of the form of eq. (11) with the surface plasmon p0tential function, the wave number of the surface plasmon should be smaller than a cut-off value k~, which is of the order of 1/b. The cut-off wave number lc~is much smaller than the maximum value kmax (—~kTF)of the surface plasmons. We can then, in eq. (2), regard WA as nearly constant, and CA as proportional to k I /2~ The justification for this arises from dimensional considerations, from the essentially two dimensional nature of the surface plasmons, that leads to, for instance, the specific form of —
the normalization factor N,, in eq. (9). The sum in 2k~—~e2/b, eq. (2)from is then easily estimated as ~E—~e apart numerical coefficients. With the increase of the wave packet size, a and b, the Coulomb blockade energy would be greatly reduced, as seen by eq. (15). Therefore, the interaction of the wave packet with the plasmons in the sample surface side will be negligible. If we assume the value of b, which is of the order of the radius ofcurvature of the tip, as 10 A and suppose that b> a, then the value of 1&E is estimated as 20 meV, which is of the order of the experimentally observed value. The expression for the Coulomb blockade energy, eq. (15), is similar to that for the
Volume 148, number 6,7
PHYSICS LETTERS A
charging energy of a small capacitor, but it is obtamed here by considering microscopic dynamic processes and not through use of a phenomenological expression like eq. (1). In conclusion, we have proposed here a microscopic mechanism to explain the phenomenon of Coulomb blockade observed in STM geometry, based on the charge response of the tip to the tunneling electron. Formation of a transient wave packet state is assumed to be an indispensable precursor of the tunneling process, and the spatial extension of this packet determines the magnitude ofthe blockade energy. The picture described in this paper is very qua!itative, and needs further work to provide a firm theoretical basis for the transient wave packet states. One of the authors (M.T.) is grateful to the Australian Academy of Science and the Japan Society of Promotion of Science for supporting his visit to
27 August 1990
A.N.U., and to the members of the Department of Theoretical Physics, Research School of Physical Sciences at A.N.U. for their hospitality.
References [1] P.J.M. van Bentum, H. van Kempen, L.E.C. van de Leemput and P.A.A. Teunissen, Phys. Rev. Lett. 60 (1988) 369. [2] D.V. Averin and K.K. Likharev, J. Low Temp. Phys. 62 (1986) 345. [3] K. Mullen, E. Ben-Jacob, R.C. Jakievic and Z. Schuss, Phys. B 37 (1988) 98.Landauer, Phys. Rev. Lett. 49 (1982) [4] Rev. M. Bflttiker and R. 1739. [5] B. Das and J. Mahanty, Phys. Rev. B 36 (1987) 898. [6] W. Elberfeld and M. Kieber, Am. J. Phys. 56 (1988) 154. [7] J. Mahanty and M.T. Michalewicz, J. Phys. C 19 (1986) 5005. [8] G. Barton, Rep. Prog. Phys. 42 (1979) 963. [9] J. Mahanty and M.T. Michalewicz, Aust. J. Phys. 40 (1987) 413.
371
PHYSICS LETTERS A
27 August 1990
A mechanism for Coulomb blockade in scanning tunneling spectroscopy M. Tsukada’ and J. Mahanty Department of Theoretical Physics, Research School ofPhysicalSciences, TheAu.stralian National University, Canberra, A.C. T 2501, Australia Received 21 March 1990; accepted for publication 19 June 1990 Communicated by A.A. Maradudin
The Coulomb blockade phenomenon observed in the 1—V characteristic curve in the spectroscopic mode of a STM with the simple surface—tip geometry is explained as a many-body dynamical effect occurring in the process of electron tunneling. This mechanism replaces the conventional picture based on an artificial small capacitance of the system.
Observation of the single electron tunneling process through a mesoscopic junction consisting of a tip and a surface in the scanning tunneling spectroscopy (STS) mode of a scanning tunneling microscope (STM) has attracted much attention lately [1—31.A small gap in the tunnel conductance of STS observed in the zero bias region for a metallic surface is explained by the so-called Coulomb blockade effect [1]. Another manifestation of the Coulomb blockade effect is the staircase-like I—V curve obtamed for a series junction of a metal surface—metal droplet—STM tip [3]. This phenomenon is quite significant, since it shows clearly that the charge of the carriers of the tunnel current is quantized. So far, the Coulomb blockade effect has been explained in a phenomenological way. When a charge Q is transferred between two initially neutral electrodes, the energy of the system increases by
a—, 2C 2
EE=
(1)
where C~ is the capacitance between the two electrodes. The bias voltage must exceed that corresponding to this energy gap for conduction to occur. The observed magnitude of~Eis of the order of sevOn leave from: Department of Physics, Faculty of Science, University of Tokyo, 7-3-i Hongo, Bunkyo-ku, Tokyo 113, Japan.
era! tens of meV, which corresponds to C—i 10—18 F. For the case of the series double junction including the metal droplet, such a classical picture of the Coulomb blockade would be plausible, because the stored charge in the droplet is well defined, and the capacitance C determines the energy increase AE due to the charging. However, in the case ofthe usual tip— surface geometry of STM, the mechanism of the Coulomb blockade is not well understood from the microscopic view point. The Coulomb blockade energy itself, of course, can be expressed by eq. (1), if C is regarded as a phenomenological parameter. However, the physical significance of C in this case i’s far from trivial it incorporates complicated manybody effects, as will be seen later. The transferred electron charge would rapidly spread out over a macroscopic region on the tip or on the surface, and the small value of C (—~1018 F) as observed experi—
mentally cannot be explained. Moreover, the increase in potential energy due to the charging of the capacitance is impossible, if the STM system is operated in the constant voltage mode. Hence in this case, the classical capacitance mechanism cannot explain the so-called Coulomb blockade phenomenon. The aim of the present paper is to propose a microscopic dynamic mechanism of the Coulomb blockade based on the coupftng of the tunneling electron with the plasmon modes in the tip. The tunnel current is generated as the net effect of
0375-9601/90/S 03.50 C 1990 — Elsevier Science Publishers B.V. (North-Holland)
367
Volume 148, number 6,7
PHYSICS LETTERS A
the individual electron transfer events occurring in unit time through the potential barrier region, i.e., the spatial gap between the tip and the sample surface. There are two time scales in this situation, the duration ; of each tunneling event, and the average time td between two successive events. The time scale t~of each tunneling event would be of the order of ~ where d is the width of the spatial gap, ç~is the work function and m is the electron mass [4]. For the typical case of d= 10 A, ~~=4eV, i, becomes of the order of 10—’ ~ s. On the other hand, in the usual case of STM measurements, the tunneling current is of the order of 1 nA, which implies that the average time between two successive tunneling events is about l0~s. Thus r~is about five or six orders of magnitude shorter than r~,and therefore, in the constant voltage mode each event can be considered as taking place statistically independently of the previous one. Another point to be noted is that the time scale r~ of the tunneling event is longer than the inverse plasma frequency (—~10_16 s) of the usual metals, Therefore, during the course of the tunneling event, the charge distribution in the tip has nearly enough time to reorganise to equilibrium following the temporarily created tunneling charge. We envisage the process of each tunneling event from the tip to the surface as follows. In the initial stage of tunneling there would be a pile up of charge corresponding to a wave packet state around the tunnel active region near the surface ofthe tip, as shown schematically in fig. 1. There is some evidence supporting this picture. The classical trajectory of the tunneling electron calculated by the WKB method [51reveals a focussing effect, which indicates formation of a wave packet around the apex of the tip. An important point here is that around a classical turning point on the trajectory the velocity of the electron small,the andwave this tends build up ternporarily is sustain packettoaround theand turning points of the focussed region. Some analyses of the dynamical behaviour of the tunneling electron wave also reveal the existence of the stand-by effect, before it begins to traverse into the gap region [6]. The interaction of the tunneling electron with the charge oscillation modes or plasmons of the surface and the tip produces two effects. Firstly, the shape ofthe tunneling potential barrier is governed by this 368
27 August 1990
a b
siani.z
SURFACE
Fig. 1. Wave packet formation just before tunneling.
interaction, through its contribution to the self-energy (or the “image potential”) of the tunneling electron. Secondly, it also determines the energy of the state from which the wave packet of the tunneling electron emerges at the tip (or gets into, in the reversed bias situation). We shall show that the latter effect is responsible for the Coulomb blockade. Let us denote by W(r) the tunneling wave packet formed transiently before it traverses into the barrier region. The other electrons in the tip will feel the potential due to the charge distribution of the wave packet, and react to stabilize the energy. This process is described by the interaction with the plasmons in the tip. In second order perturbation theory the stabilization energy iIE is given by [7] ~E’
~ I C~12 A
(2)
hw1
with
I C,~I
2—e2
j
I~2A(r)121 W(r) 12 dr,
(3)
where a~is the plasmon frequency of the Ath mode and 92 2(r) is the normalized potential of that mode. The above estimate is reasonable, since the time scale of the formation and duration of the wave packet is longer than the inverse plasma frequency. The physical consequence of this is illustrated in fig. 2. The electron in the tip cannot be removed from the Fermi level E~T) of the tip, but only from the (—j;)
Volume 148, number 6,7
TIP
____________
PHYSICS LETTERS A
SAMPLE 8> E’F
________
27 August 1990
Q~(k,p)exp(kz) qi~(p,z)=mw~M*~[212~(k, p)—&~ _(coscoz)+ k2w~ p[2Q~(k, p) —os]
_______
sin (pz))]
x exp(ik~p)
__________
(5)
,
respectively, with k and p representing the wave numbers parallel and perpendicular to the surface, Fig. 2. The mechanism ofCoulombblockade. The tip electron is removed from the transient wave packet state, which is situated at ~.Ebelow Er’.
wave packet state, which is situated at an energy AE below E~T). As shown below, stabilization by plasmon coupling is small for thesuch sample surface side, and we neglect it for convenience of the description, though its inclusion does not change the essential point of the results. It is then clear from fig. 2 that the electron cannot be transferred from the tip to the
and p and z are the corresponding components ofthe coordinate. The inner part of the material with the surface corresponds to the region z<0. fi is taken to be ..j~ v~,v~being the Fermi velocity. w theelectron plasma 2/m, n being0isthe frequency, with w~ = 4zne density and m being the effective mass. The normalization constant ofthe potential, Mi,,, is given by ~
=
fl4p2 [2Q~ (k, p)
—
w~]
x [4~3mnQ 8(k, p) ] —1 [Q~(k, p) 24$2k2Q~(k,p)}’. (6) TheX{[2Q~(k,p)oi~o] angular frequency Q 5 (k) and the potential (Oa (P, z) of the surface plasmon modes in the planar case are given by [81 2k2+flk~.,/2w~ +fi2k2) (7) Q~(k) +fl ~ (p, z) =~(w~ mN~ exp (ik~p) —
surface unless the negative bias applied to the tip exceedse~.E,i.e.,IeVI>~.E. The process taking place in the case ofthe electron tunneling into the tip can be described as follows. The electron coming from the sample surface to the tip will also form the wave packet around the active region of the tip surface. In the same way as before, the energy ofthe wave packet state is lowered by AE compared with the initial state at the sample surface. If the stabilized energy of the wave packet is below
X [p 5Q~(k) exp(kz) kw~exp(Psz)] Here Nk is the normalization constant, 2[l6iu2mnQs(k)k3]’ Nw~[2Q~(k)w~] x [w~ +2Q~(k)]~[Q~(k)—w2~—2 0j —
E~T),the
Fermi level of the tip, it cannot enter the tip. Therefore, unless the Fermi level E~ of the sample surface is raised by a bias voltage above E~T)+i~E,the tunnel current cannot flow. The microscopic mechanism of the Coulomb blockade is thus given above, The next question is how the Coulomb blockade energy AEthe is determined reflectingofthe the geometric tures and material properties tip. This feacan be analysed using the hydrodynamic model of plasmons [8]. The plasmon modes in the tip and the surface have been studied in this model [7,9]. In each case we have bulk and surface plasmons. For a planar surface the angular frequency ~ and the potential P*,p of the bulk plasmon mode (k, p) are given by [8] Q~(k, p) = w~+ fl2k2 + p22 (4)
.
,
(8)
(9)
and Ps is given by 2k2). (10) flk+ ~J2co~+ fl A reasonable model of the tip is a hemispherical bump on a planar surface parallel to the sample surface. In this case, the exact forms ofthe bulk and surface plasmon potentials will be complicated, but their spatial extension particularly in the z-direction, can be expected to be qualitatively similar to that given in eqs. (5) and (8). To proceed further, let us assume a concrete form for the charge density of the wave packet, _________
Ps
=
(_
369
Volume 148, number 6,7
PHYSICS LETTERS A
W(P~z)I2=..,,J~2exP(_z_zoI) xex
~\
(_(j,_~,~)2’\
b~
~ (11)
)~
a and b are parameters defining the extent of the charge distribution in the normal and lateral directions respectively, and (Po, z0) specify the centre ofthe wave packet is taken nearthe tip. We have tried some other forms for I !P(p, z) 12, but the results do not depend much on the form, as long as the two parameters defining the spatial extent are fixed. Using eqs. (4)—(11), it is easy to show that I C,,~I 2 hQ 5(k, 2 p) / ~ ~ e = I(b/a) +O~— (12) where
27 August 1990
l6l3/2~,
forb>>a,
.~—~—ln(a/b), fora>>b.
(15)
For more general cases, i~.Ewould be given by the sum of eqs. (12) and (14). It is remarkable that if the size of the wave packet (a and b, which would be of the order of the radius of curvature of the tip) is larger than the Thomas— Fermi screening length (-.~fl/w0), the Coulomb blockage energy AE does not depend on the material properties ofthe tip or the surface, but is determined only by their geometrical features. In fact, the Cou2/ blomb for b>> a, apart from is a numerical coefficient; b here blockade energy generally given by LiE—~ e
~
~—~---
,
s—),
°
°
where 1(x) is given by 1(x) _
=
f ~exp (
J
—
~2)
(x+~)2
d~ _i._ 2x2’
for x>> 1
—
IlnxI
‘
,
forx<< 1
.
(13)
On the other hand, the contribution to the Coulomb blockade energy from the surface plasmons is calculated to be
v IC,,12 ‘r hQ
1 ~ +~(—±~— ~ (14) 3”2 b \~aw 5(k) l6x 0‘bwoJ’ If one neglects the dispersion of the plasmons, only the first terms ofthe right hand side of eqs. (12) and (14) are obtained. This is a reasonable approximation if the sizes a and b are greater than the Thomas—Fermi screening length l/FCTF—’ fl/coo. If the lateral size b of the charge distribution is much larger than the vertical size a, the dominant contribution to the Coulomb blockade energy is from the surface —
plasmons, eq. (14); but for the opposite case of a>> b, it is mainly due to the bulk plasmons, eq. (12). We thus obtain 370
is a measure of the sizeonly of the along the surface. In lateral this case the wave surfacepacket plasmons give the dominant contribution to ~.E, since the overlap between the wave packet and the bulk plasmons is small leading to small values for CA of eq. (3). For appreciable overlap of a wave packet of the form of eq. (11) with the surface plasmon p0tential function, the wave number of the surface plasmon should be smaller than a cut-off value k~, which is of the order of 1/b. The cut-off wave number lc~is much smaller than the maximum value kmax (—~kTF)of the surface plasmons. We can then, in eq. (2), regard WA as nearly constant, and CA as proportional to k I /2~ The justification for this arises from dimensional considerations, from the essentially two dimensional nature of the surface plasmons, that leads to, for instance, the specific form of —
the normalization factor N,, in eq. (9). The sum in 2k~—~e2/b, eq. (2)from is then easily estimated as ~E—~e apart numerical coefficients. With the increase of the wave packet size, a and b, the Coulomb blockade energy would be greatly reduced, as seen by eq. (15). Therefore, the interaction of the wave packet with the plasmons in the sample surface side will be negligible. If we assume the value of b, which is of the order of the radius ofcurvature of the tip, as 10 A and suppose that b> a, then the value of 1&E is estimated as 20 meV, which is of the order of the experimentally observed value. The expression for the Coulomb blockade energy, eq. (15), is similar to that for the
Volume 148, number 6,7
PHYSICS LETTERS A
charging energy of a small capacitor, but it is obtamed here by considering microscopic dynamic processes and not through use of a phenomenological expression like eq. (1). In conclusion, we have proposed here a microscopic mechanism to explain the phenomenon of Coulomb blockade observed in STM geometry, based on the charge response of the tip to the tunneling electron. Formation of a transient wave packet state is assumed to be an indispensable precursor of the tunneling process, and the spatial extension of this packet determines the magnitude ofthe blockade energy. The picture described in this paper is very qua!itative, and needs further work to provide a firm theoretical basis for the transient wave packet states. One of the authors (M.T.) is grateful to the Australian Academy of Science and the Japan Society of Promotion of Science for supporting his visit to
27 August 1990
A.N.U., and to the members of the Department of Theoretical Physics, Research School of Physical Sciences at A.N.U. for their hospitality.
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