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AD-A278 338 



Annual Progress Report 
ONR/ARPA Grant # N00014-93-1-0880 


Semiconductor Single-electron Digital Devices and Circuits 


Principal Investigator: Prof. D.V. Averin 

Department of Physics 
State University of New York 
Stony Brook, NY 11794-3800 


Project Period: June 20, 1993 — June 20, 1996 

Report Period: June 20, 1993 — June 20, 1994 


DTIC 



ELECTE 
APR 2 01994 




Research Objectives: 

Theoretical investigation of single-electron tunneling phenomena in semi¬ 
conductor heterostructures, and their application possibilities in the devel¬ 
opment of ultradense logic and memory circuits. 


Report Period Objectives: 

1. To design and model simple digital circuits based on capacitively 
coupled single-electron transistors. 

2. To extend the “orthodox” theory of correlated single-electron tun¬ 
neling to ultrasmall quantvun dots. 


Accomplishments: 

1. Analysis of digital circuits based on capacitively-coupled 
single-electron transistors — Prof. K. Likharev (Co-P.I.), R. Chen, A. 
Korotkov. 

We have analyzed the possibility of using capacitively-coupled single¬ 
electron (SET) transistors to build logic and memory circuits. The aneilysis 



1 





01 5 


94 








shows that while the potential density of such circuits is very high, up to 
10'® cells/cm*, their operating temperatxure is limited to about 1/lOOth of 
the transistor charging energy. For 10-nm technology this confines possible 
circuits to liquid helium temperatures. However, the physics of SET tran¬ 
sistors perm the cells to be scaled down further, with a corresponding 
increase in operation temperature and circuit density. The approach de¬ 
veloped in our work enables us to calculate all relevant circuit parameters, 
including switching delays, power dissipation, and noise margins. 

2. Numerical analysis of quantum tunneling of charge in com¬ 
plex single-electron circuits. — Prof. K. Likharev (Co-P.I.), L. 
Fonseca, A. Korotkov. 

One of the problems of single-electron circuits based on the controlled 
transfer of electrons one-by-one is macroscopic quantum tunneling (mqt) 
of charge. It leads to imwanted electron transition through the imposed 
Coulomb energy barriers, and thus causes errors in devic' operation. Cal¬ 
culation of the mqt rates in realistic circuits presents a complicated problem 
that requires extensive numerical simulations. We have developed the soft¬ 
ware package that performs such a simulation for arbitrary circuits with the 
mjocimum mqt order 8-9. In future we plan to use this software extensively 
for the design of single-electron circuits. 

3. Extension of the “orthodox” theory of single-electron tun- 
neling to ultrasmall quantum dots. 

3.1 Tunneling in an arbitrary electromagnetic environment. — Prof. D. 
Averin, H. Imam, A. Korotkov, V. Ponomarenko (visiting scientist). 

We extended the standard theory of single-electron tunneling in two 
directions making it applicable to ultreismall quantum dots. Firstly, the 
main featiue of such dots that determines their electron transport prop¬ 
erties is the formation of discrete zero-dimensional (OD) electron states. 


2 









We developed a theory of electron tunneling via one non-degenerate OD 
state localized in such a dot in the presence of an arbitrary electromagnetic 
environment. One of the conclusions of this theory is that in sufficiently 
resistive environment (the characteristic resistance larger than the so-called 
resistance quantum equal to 13 KOhm) electron transfer between the OD 
states has an irreveTsible character, and in principle can be used to design 
single-electron circuits in complete analogy to circuits with larger dimen¬ 
sions. This result removes the main potential limitation on caling down of 
single-electron devices. 

Another limitation of the orthodox theory is related to an assumption of 
instantaneous electron tvmneling. We have developed a theory of electron 
tvmneling with finite transversal time and arbitrary electromagnetic envi¬ 
ronment. It shows that although there are quantitative corrections to the 
orthodox theory the qualitative pictiure of single-electron tuimeling remains 
valid. 


3.2 Numerical modeling of electron-electron interaction in a few-electron 
quantum dots. — Prof. J.K. Jain (Co-P.I.), L. Belkhir. 

The quantitative design of single-electron devices is based on the de¬ 
scription of the charging energy in terms of the geometrical capacitances 
C. Such a description is very accurate for relatively large metallic struc¬ 
tures, but can not be, strictly speaking, justified for small semiconductor 
heterostructrures. We performed exact numericed simulations of electron- 
electron interaction in few-electron quantum dots and showed that the clas¬ 
sical expression for the charging energy should be semi-quantitatively (with 
about 30% accuracy) valid even for quantum dots which contain as few as 
five electrons. 1 Accesion 


NTIS CR 
DTIC TAB 
Ufiannci; 
JiJSlificat;': • 


for_\ 

V)’ 


By. 

Dtvt.ThA 


Dtsl 


3 









POSSIBLE EXTREMELY-HIGH-DENSITY MEMORY BASED ON 
SINGLE-ELECTRON TRANSISTORS 


R. Chen, A.N. Korotkov, and K.K. Likharev 

Department of Physics, State University of New York 
Stony Brook. NY 11794-3800, 516-632-8159 


Recently discovered effects of correlated single-electron tunneling (for reviews see, e.g.. 
Refs. 1, 2) form the physical basis for a new generation of electronic devices, in 
particular, very dense memories. Digital bits in these memories may be represented by 
either single electrons or bundles of a few electrons. In this work we are suggesting a 
static random-access memory (SRAM) of the latter type, where the number N of 
electrons in the bundle (N-10) oscillates in time by 5N=±l-2. (Larger fluctuations of N, 
implying digital errors, occur with vanishing probability.) In contrast to memory cells 
based on the trapping of single electrons [2], the new approach implies somewhat higher 
power consumption, but promises much higher speed and wider parameter margins. 

The memory cell consists of two single-electron transistors (SETs) [1.2] connected in a 
positive-feedback loop. In each of two stable states of this symmetrical flip-flop, one 
SET is open, while another is in the closed (Coulomb-blockade) state. Two additional 
SETs enable write-0, write-1 and read-out operations, in a mode very similar to that of 
the usual MOSFET SRAMs. The only substantial difference is the high output 
impedance of the SETs, which makes it necessary to use at least a two-level hierarchy of 
sense amplifiers (the first level should use SETs, while the next levels may be based on 
usual FETs). 

We have used the "orthodox" theory of single-electron tunneling, and its extension to co¬ 
tunneling processes [2], to optimize the SET SRAM cell and calculate its parameter 
tolerances, switching speed, and power consumption. These results have enabled us to 
estimate the possible performance of the SET SRAMs for several levels of their 
fabrication technology. For example, a 10-nm silicon-based technology should allow 
implementation of this type of memory with density close to 10 Gb/cm^, cycle time of 
-300 ps, and power consumption -3 W/cm^, operating at liquid helium temperatures (-4 
K). It is important that the physics of SET transistors permits the memory cells to be 
scaled down further, with a corresponding increase in cell density and operation 
temperature. 

The work was supported in part by AFOSR and ONR/ARPA. 

1. D.V. Averin and K.K. Likharev, in: "Mesoscopic Phenomena in Solids", ed. by 
B. Altshuler et al., Elsevier, 1991, pp. 173-271. 

2. "Single Charge Tunneling", ed. by H. Grabert and M.H. Devoret, Plenum, 1992. 










Figure 1. Possible circuit diagram of the SET SRAM cell. Open rectangles 
denote conducting electrodes, while gray rectangles show tunnel junction 
barriers. 



Figure 2. Parameter window of the correct operation of the cell with Cc=2C, 
CI»C, and RIslOR, for several operation temperatures T (in units e^/kgC). Qq is 
the background charge. One can see that the window closes rapidly at 
temperatures above -O.OIe^/kgC. 







Coulomb blockade of resonant tunneling 

H.T. V.V. Ponomarenko/*and D.V. 

Averin^*’®) 

(*) Department of Physics, State University of New York, Stony Brook, NY 11794; 

A.F. Ioffe Physical Technical Institute, 194021 St. Petersburg, Russia; 

(^) Department of Physics, Moscow State University, Moscow 119899 GSP, Russia 


We have considered the influence of electromagnetic fluctuations on electron tun¬ 
neling via one non-degenerate resonant level, the problem that is relevant for electron 
transport through quantum dots in the Coulomb blockade regime. We show that the 
overall effect of such an influence depends on whether the electron bands in external 
electrodes are empty or filled. In the empty band case, depending on the relation be¬ 
tween the tunneling rate F and characteristic frequency fl of the fluctuations, the field 
either simply shifts the conductance peak (for rapid tunneling, F > ft) or broadens 
it (for F •< ft). In the latter case, the system can be in three different regimes for dif¬ 
ferent values of the coupling g between electrons and the field. Increasing interaction 
strength in the region </ < 1 leads to gradual suppression of the conductance peak at 
the bare energy of the resonant level £o> while at ]:> 1 it leads to the formation of 
a new peak of width Edg^^^ at the energy to + Ec, where Ec is a charging energy. 
For intermediate values of g the conductance is non-vanishing in the entire energy 
range from Sq to eo-\- Ec- These results provide a possible explanation for experimen¬ 
tally observed extra width of the conductance resonances at low temperatures. For 
filled bands the problem is essentially multi-electron in character. One consequence 
of this is that, in contrast to the situation with the empty band, the fluctuations of 


1 







the resonant level do not suppress conductance at resonance for 9 < 1. At 9 = 1 a 
phase transition occurs leading to the appearance of a Coulomb gap in the position 
of the resonant level as a function of its bare energy, and results in the suppression 
of conductance. 

PACS numbers: 73.20.Dx, 73.40.Gk, 73.40.Kp 


Typeset Using REVTEX 


2 






1. INTRODUCTION 


Several recent experiments [l]-[5] demonstrated resonant tunneling under 
Coulomb blockade conditions, when the resonant level is localized in a mesoscopic 
quantum dot and is affected by electron-electron interaction in the dot. The in¬ 
teraction shifts the energy of th resonant level by an amount which is roughly 
proportional to n, the number of electrons in the dot. This makes the effective 
energy spectrum of the dot a nearly periodic function of n, with each period corre¬ 
sponding to the addition of one electron to the dot. This periodicity leads to several 
new phenomena which are attracting considerable interest. 

The effect of the interaction has such a simple form of an energy sliift only if 
the response of electrons in the dot and/or external electrodes to the transfer of 
an electron to or from the resonant level has a time sccJe which is incompatible 
with the time scale of tunneling. Our aim in this work is to consider the situation 
when such a condition is not satisfied and the characteristic response time can be 
comparable to the tunneling time. Following the theory of electron tunneling in 
smeJl metallic tunnel junctions (see, e.g., [6]), one can model the electron-electron 
interaction in this regime as the interaction of electrons with fluctuations of the 
electromagnetic field in a given electromagnetic environment. Thus, we can reduce 
the problem imder consideration to one of inelastic resonant timneling of electrons 
coupled to bosonic degrees of freedom. The relation between the timneling time 
and characteristic frequency of the boson modes is important in this model because, 
depending on whether the boson modes have enough time to adjust to the chzmging 
charge of the resonant level or not, the effective energy of this level either coincides 
with the bare energy £o or is shifted by the charging energy E^. 

This fact, together with the renormalization of the effective tunneling rate by the 


3 








interaction, leads to several different regimes in the dynamics of tmmeling through 
the resonant level. Identification of various regimes of electron tunnehug and cal¬ 
culation of the current-voltage characteristics of the resonant level in these regimes 
are the aims of this work. 

The paper is organized as follows. In Sec. 2 we derive the general expression 
for the current through the resonant level, and prove that in the large-bias hmit 
the cvurent reaches an asymptotic value which is independent of interaction. In 
Sec. 3 we simplify the general expression by adopting a one-electron approximation 
which was originally developed in [7]-[9], and establish limits on the validity of this 
approximation. It is shown that the one-electron approximation is valid when the 
electron band in at least one of external electrodes is empty and the rate of timneling 
into this electrode is much larger than the rate of tunneling into the other electrode. 
Both of these conditions are typically satisfied in vertical resonant tunneling struc¬ 
tures. In Sec. 4 we calculate the current-voltage characteristic of such a structure 
in the one-electron approximation. In Sec. 5 we consider the situation (typical for 
lateral quantum dots) when the bands in both externeil electrodes are filled zuid 
the one-electron approximation is incorrect, and C2ilculate the linear conductance 
zind the current within two approximation scliemes, adiabatic approximation and 
perturbation theory in tumieling. It is shown that if interaction with the bosonic 
mode is strong enough and the bare tunnehng rate is small, then, due to a strong 
renormalization of the tunneling amplitude a Coulomb gap appears in the effective 
position of the resonant level as a function of its bare energy . In Conclusion we 
discuss the relation of oiu results to existing experiments on electron tuimeling in 
quantum dots, and also discuss them from the point of view of general approaches 
to tremsport of interacting electrons. 


4 








II. MODEL AND BASIC EXPRESSION FOR CURRENT 


We consider tunneling of electrons between two bulk electrodes via a quantum 
dot with a resonant non-degenerate state using a standard tunneling Hamiltonian: 

Hei = ^^kjcijCkj + eoc^c -I- + h.c.). (1) 

j.fcj },kj 

Here index j = 1,2 denotes, respectively, left and right electrode, and eo is the 
energy of the resonant level. The assumption of a non-degenerate level can be 
justified in several situations. In particular, strong magnetic field applied to the dot 
may result in the Zeeman splitting of the spin-degenerate levels. Another possibility 
is a strong intrasite electron repulsion which may lead to significant splitting of the 
initially degenerate levels when the Kondo resonance is destroyed by non-vanishing 
temperature or bias voltage. 

To describe the relevant low-energy properties of tunneling through the resonant 
level, we need to tahe into account the interaction of the tunneling electron with 
low-energy excitations in the quantum dot structure. In the mesoscopic structures, 
the relevant excitations are low-frequency modes of the longitudinal electromagnetic 
field associated with fluctuations of the total electric charge of electrodes (see, e.g.. 
Ref. [6]). Using standard gauge transformation we can express the interaction with 
these modes in the form of field-induced phases of the electron tunneling amplitudes 
ty. 

tj , Mi) = eVi(<). (2) 

where Vj is the fluctuating voltage between the resonant level and jth external 
electrode. 

Below we consider a standard structure with two bias external electrodes and 
one gate electrode (Fig. 1). In such a structure there are two types of low-frequency 


5 











modes, one associated with fluctuations of the bias voltage, and another one asso¬ 
ciated with the fluctuations of the gate voltage. Adding the energy of these two 
types of modes to the electron Hamiltonian (1) with tunneling amplitudes (2) we 
get the total Hamiltonian of om model: 

H = + + ( 3 ) 

An important characteristic of the low-frequency photon modes is that the spa¬ 
tial distribution of the electric field in them is determined solely by the geometry 
of the system and is frequency-independent. Hence, we can introduce frequency- 
independent factors A, Aj 2 which determine the distribution of the voltages in the 
electrodes of the system and write the phases <f>j as sums of the two terms corre¬ 
sponding to fluctuations of the bias voltage and of the gate voltage: 

<i>l = X<f)v + ^l4>g , <t >2 = —0- - X)<f>v + X 2 <l>g . (4) 

Here A and Aj give, respectively, the fraction of the bias voltage and of the gate 
voltage that drop between the dot aiid left external electrode; (1 — A) and A 2 have 
the same meanings for the dot and right external electrode. 

The phases <j>v aaid can be directly expressed in terms of the photon modes 
a and b: 

+ fcw). (5) 

u; w 

where the coupling constants and i/„ together with the density of modes axe 
related to the effective impedances of the structure [6]: 



Here spectral densities Fv,g{u)) = (e^/27r^)ReZv,j(a>) zire given by the real part of 
impedances of the bias circuit and of the gate electrode — see Fig. 1. 


6 









Below we will study in detail two specific examples of the spectral densities 
In the Ohmic case, the impedance is reduced to the frequency-independent 
resistor R. Then 

f{u,) = 9/(l + (a,/!l)^), (6) 

where g = e^R/n, and the cut-off frequency fi is determined by the finite capacitance 
C between the electrodes, fl = X/RC. Another example corresponds to an inductor 
L in the external circuit which leads to a one-mode spectr 2 d density: 

/’(w) = 9fl£<(u.±!>), (7) 

± 

where Q. = (£0)”*^^ is the mode frequency, and g = (e^/7r)(L/C)*^^ is the coupling 
constant. Spectral densities (6), (7) satisfy the sum rule: 

J duF(uj) ~ = 2Ec. (8) 

which should be satisfied by arbitrary F{u}) [6]. 

Although we have used, so far, the language appropriate specifically for the pho¬ 
ton modes, the model is obviously valid for other bosonic modes. Moreover, to the 
extent that the “bosonization” procedure is applicable to excitations of arbitrary 
system, the model describes resonant tunneling of electrons interacting with arbi¬ 
trary dissipative environment. As an example, one can note that the microscopic 
model of Coulomb screening of the resonant level by a Fermi sea of electrons [10], 
[11] czin be reduced to the bosonic form considered in this work. In the microscopic 
model an electron on the resonant level creates a self-consistent potential for elec¬ 
trons in the Fermi sea, which gives rise to shift of scattering phases of these 
electrons. Bosonization of electron-hole excitations of the Fermi sea reduces this 
model to the one discussed above with 


7 









( 9 ) 


} 

where the exponential cut-ofF is commonly used for simplicity. 

FVom a general point of view, the spectral density (6) corresponds to an environ¬ 
ment with a relaxation-type response to charging of the resonant level characterized 
by a single relaxation-time constant while the spectral density (7) corresponds 
to the environment with an oscillatory response and a single frequency fl. 

Our aim is to calculate the dc current I through the resonant level which can 
be expressed in terms of either left or right currents, 

/ = /l=/2, 

where 

/j = —2elm^ti{cj[,c‘^‘c), /2 = 2elm]^t2(cl2«‘'^*c). (10) 

kl k2 

Writing down the expansion of the averages in eq. (10) using perturbation theory 
with respect to the tunneling terms in the Hamiltonian, and constructing the cor¬ 
responding Dyson equation, we express the ctirrents in terms of non-equilibrium 
Green’s functions of the resonant level. (A similar transformation was used in [12].) 
For instance, the left current is: 

/. = -2e ^(G< (£)GS(£) - G',(£)Gf (£)), (11) 

where Gi(e) is the exact (i.e., calculated in presence of tunneling) Green’s fimction 
of the resonant level dressed by the phase e’^*, e.g., 

G,<(£) = J d<e‘'‘G<(t), G<(t) = i(c^e-‘^(°)e‘^<‘>c(i)), 

while Gfci(e) is a free Green’s function of the left electrode. (Notations for G’s in eq. 
(11) are the same as in the book by Mahan [13], Sec. 2.9.) Using the fact that G^{e) 


8 






and G'*^(£) are purely imaginary and ImG‘(£) = IG^(£) + G^(£)]/2i, and plugging 
expressions for Gn, Gf^ie) = 27ri/k,6(e - e^,), ImGii(e) = rr(2/i, - 1)<5(£ - e*i) in 
eq. (11) we get finally: 

I< = -|^ Jds[M^)aHe) + (1 - Me))G<(e)]. (12) 

Here Fj = 27r | 53fci ~ £/ti) is the rate of tunneling into the left electrode, 
and flic) is Fermi distribution function in this electrode. 

The current between the resonant level and the right electrode can be expressed 
similarly: 


^2 = ^ jdilMs)GU‘) + (1 - /2(£))G<(0) , (13) 

where Gaie) is the Green’s function of the resonant level dressed by the phase 

Equations (12) and (13) will be the starting point of our calculations in the next 
sections. They show, in particular, that in the large-bias limit the current reaches 
a constant value that is independent of interaction. Indeed, in this limit fi{e) = 1 
and / 2 (e) = 0 for all relevant energies, and we have from (12) and (13): 

I, = ~JdeG>M = eT,(l-n), h = ^JdeG<(e) = eT,n, (14) 

where n = (c^c) is the occupation probability of the resonant level. Since the left 
and right currents should be equal, we get from eq. (14) that in the large-bias hmit 
both n and I are the same as in the non-interacting ceise: 


n 


Fi 

Fi+Fj’ 


/ = 


riF2 

^Ti + T^' 


(15) 


Using eqs. (12) and (13) we can in principle determine the current for arbitrary 
bias voltage adopting various approximations for Green’s functions of the resonant 
level. 


9 







III. ONE-ELECTRON APPROXIMATION 


We begin by considering the one-electron approximation, which was originally 
developed in [7] - [9]. The starting point of this approach is an assumption of free 
evolution of the bosonic modes. With tins assumption the dynamics of electron 
operators can be determined straightforwardly. In order to obtain the limits on the 
validity of this approximation we start with a complete set of equations of motion, 
both for electron operators c and the bose fields <f)j that follow from Hamiltonian 
(1) with tunneling amplitudes (2): 


icicj — £kj^kj d" , ic — £oC -L ^ ^ tjC ^Ckj 1 


j.fcj 


• JO CJ J—oo 

Here denotes free bose field, and are operators of the left and right 

currents (10). Appropriate sum over the two spectral densities Fg{ij) and Fy{u}) 
is assmned in the last equation (16). Equations of motion for electron operators 
can be solved explicitly provided that the density of states in external electrodes is 
constant on the energy scale associated with tunneling; 

c{t) = -i • (17) 

Here F = (Fi -I- F2)/2 is the tunneling width of the resonant level, zmd are free 
electron operators which determine the occupation of electron states in external 
electrodes. Equation (17) gives for the Green’s functions of the resonant level: 

Gf(£) = r 

ij •' 

{4i + r)), (18) 

Gr(£) = -* E J f 

ij 


10 







In the one-electron approximation, the evolution of the hose field in eq. (18) is taken 
to be free. Under this assumption eq. (18) is reduced to: 


C?<(e) = i f dte^* f 5^ 1 1, p /,(£*,) 

J J-oc 

Giie) = -i /d<c‘*‘ r | tj p (1 - 

J J-co -I^- 

Combining eq. (19) with eq. (12) we find the current: 
eFiFa 


(19) 


1 = 


2ir 


Jded£'[f^{e){l - f2{e'))Pu{e,£^) - /2(£')(1 - /x(£))P2,(£',e)], 


( 20 ) 


where Puie,^') and P 2 i(£i£') are transition probabilities from the state with the 
energy e in one electrode to the state e' in another electrode. They can be written 
as follows: 

Pu(€,c’) = , />2i(£',e) = dte«-<"»{A<A{t )), 

( 21 ) 


where A has the meaning of transition amplitude: 

A = r dTe^‘(''>-'0+n’-e«>i(o)g-.>2(T) ^22) 

J—oo 

Representation (21) shows straitforwardly that the left—aright and the right—»left 
transitions are related by the detailed balance condition: 

P^2ie,£') = e^^‘-^'^P2^{£\£). (23) 

Due to this condition the terms that describe transition within one electrode drop 
out from expression for the current (20). It should be noted that eq. (23) does not 


11 










imply left<-+right symmetry of transition probabilities, and the current can be an 
asymmetric function of the voltage if the resonant level is placed asymmetriczdly 
with respect to the electrodes (i.e. A ^ 1/2). 

To understand why the assumption of free bosonic evolution constitutes one- 
electron approximation one can note that deviations from free evolution are medi¬ 
ating electron-electron interaction, since disturbance of the bosonic mode by one 
electron affects other electrons. This interaction is disregarded, when the bosonic 
evolution is assumed to be free. 

In order to obtain conditions for the validity of the one-electron approximation, 
the full evolution of the bose field should be taken into account. Deviations of 
(/? from give rise to corrections in the expressions for G\ (19) and the current 
(20). In the first nonvanishing order in electron-boson couphng, correction to the 
current contains the terms of the following structure: 

A7 oc r.rj^ri / / - (i - /!(.))/.(e)l|...], 

where denotes a function of c, c' precise form of which is not relevant for 
our argument. This expression shows that all corrections resulting from deviations 
of the boson evolution from free evolution can be omitted when we can neglect 
products of the type r,/,(€), e.g. if the conduction band of the second electrode is 
empty and transpmency of the first baxrier is mucli smaller than that of the second: 

/,(£) = 0, r, «r,. (24) 

In simple terms conditions (24) mean that the resonant level is practically empty, so 
that electrons indeed tunnel independently. Another way to understand eq. (24) is 
to note that in the one-electron approximation the tunneling width of the resonant 
level is unaffected by interaction. However, it is obvious on simple physical grounds 
that since the interaction changes the tuimeling rates, the width F should also be 


12 








modified. Thus, the conditions (24) mean that F is determined solely by the rate 
of tunneling from the resonant level into the empty band which is really unaffected 
by interaction. 

If the conditions (24) are not satisfied, the one-electron approximation is vedid 
only for energies away from resonance, where we can neglect the width of the reso¬ 
nant level altogether, and eq. (20) reduces basically to an expression for the current 
that follows from the second order perturbation theory in tunneling (see Sec. 5). 

Now we proceed with the calculation of transition rates (21). Since the phases 
^ 1 , <l >2 are combinations of free boson operators, the averages of the four exponents 
in eq. (19) can be evaluated directly (see, e.g., [13], Sec. 4.3). For instance: 

= exp{((<i,(<) - <^,(0))<^,(0) -|- (Mt + - r')- 

^2(0))<^2(0) -f <^l(t)(^2(^ + "r) ~ — <^2(t -f r))<^j(0))} . (25) 

Together with expression for equilibrium phase correlators, 

. 

eq. (19) gives the finally: 

Pi2(e,e') = /“ 

ZtT J «/—oo 

exp{/ - 1) + - 1) + 

+ e-’""' - . (26) 

Here 

Fn(a;) = A2 F,(u;) -|- X^Fviu ), F 22 (a;) = XlF,{u;) -^(1 - A)2Fv(a;), 

F,2(u;) = A, A2F,(a;) - A(1 - A)Fv(u;). 

For Fv(w) = 0 equation (26) coincides with the results of [7]—[9]. 


13 





IV. TUNNELING INTO AN EMPTY BAND 
In this section we will deal with the case when the conditions (24) axe satisfied 
and the one-electron approximation (expressed by eqs. (20) and (26)) is valid for 
all energies including those close to eo- Under these conditions the expression for 
the current can be simphfied further. Making a change of variables in integral over 
t in eq. (26), t —* t — r + t^ and integrating Pi 2 {£,e') over the final energy e' we get 
(in this Section we will suppress indices 1,2 of P 12 and Fn): 

I= — [ defii£)P(£) , -PCs) = Re r ^ 

TT J J—ca 



F{u) - 1 
u> 1 - ■ 


(27) 


Equation (27) describes the current step as the current increases from 0, for 
voltages below the resonance, to eFi for voltages above the resonance. We limit 
ourselves mostly to the calculation of the zero-temperature differential conductance 
of the structure G = dl/dV which characterizes the shape of the current step and 
coincides with the transition probability P(£): 

G{£) = ^P{£). (28) 


Substitution £ = XeV in eq. (28) gives the conductance as a function of the bias 
voltage V. 

The shape of the conductance-versus-energy peak depends on the spectral den¬ 
sity F{u) which characterizes the dissipative environment of the resonant level. In 
the one-mode case (10) we get from eqs. (27), (28): 


^ ~ nfl)2 -f-17^ 


(29) 


where 6 = £ — Eq, and r/ = F 2 / 2 . For large coupling constant g » max{l, (F 2 /ft)*} 


eq. (29) shows that the envelope of the system of elementary peaks is 


14 






(30) 


G = 




n{2irEcny/^''"^^ 2Ecn 
so that the total conductance peak is Gaussian and centered around eo + Ec- 

However, when Fj ^ max{l,g''/*}ft (i.e., response of the photon mode is much 
slower than the tunneling rate), the peak becomes Lorentzian and has no fine struc¬ 
ture. The conductance exhibits then the same peak as in the non-interacting case, 
but with the peak position shifted to the energy £o -I- Ec'. 

e^AFiFj 1 


G = 


(31) 


27rh {S - Ecf + ry ■ 

Equation (31) can be obtained at large r 2 directly from the sum rule (8) and eq. 
(27), and, hence, is valid for arbitrary frequency dependence of F{u). Thus, we see 
that for the one-mode environment, the conductance peak evolves, as a function of 
the coupling constant g (for fixed energy Ec = from the Lorentzian at e = Co 
for ^ » 0 to a similar peak centered at £ = £o + -E-c for ^ 1 . 

A similar transition takes place for the Ohmic environment (3). To follow this 
transition we first consider the shape of G(£) at small energies, | £ — £o (<C 
r 2 -C fl, and g < 1. In this case the conductance is determined by the long-time 
asymptote of J(r) in eq. (27). Using the well-known expression for this asymptote 
(see, e.g., [ 6 ]), we get: 


G = 


e^AFi 


r dre^- 

^ —OO 


Re 

TT/l ^/-oo 


sin[(l - g)Q]{ 




(Qt)s 


nhil 


e-^®r(l -g)x 


ft* 




(32) 


where 7 is Euler’s constant, and r(l — g) is the gamma function. Equation (32) 
shows that for 5 ^ r/ the conductance peak at £ = £0 is suppressed and turns into 
a plato at y = 1 : 


G = 


e^AFie-^ 

irhU 



(33) 


15 








For arbitrary g, the conductance can be calculated either numerically or ana¬ 
lytically if the Lorentzian cut-off 1/(1 - 1 - (ui/U)^) in the Ohmic spectrum is replaced 
with the exponential cut-off The integrals in (27) can then be calculated 

explicitly and give: 


G = 


®^^^'lm{2«-*r(l - 5,2)} , 2 S expi-iOl^*^" 


(34) 


Trhfl ^ ^ ft 

where r(l — g, z) is the incomplete gamma fimction. At low energies, | £ — Co |<C ft, 
Fa ft eq. (34) gives: 


G= ^!^{r(l -<,)sin({l - j) 9 )(-^)(■-»)/> + ’ 


Equation (35) generalizes eq. (32) to 5 > 1 . It shows that as 5 is increased 
further in the region 5 > 1 the conductance around £0 is suppressed stronger and, 
moreover, starts to grow as one goes away from the original resonance at £ = £ 0 . 
This indicates that another conductance peak is formed at larger energies. The next 
(after 5 = 1 ) change in the asymptotic behavior of conductance at low energies occur 
at 5 = 2, when the first term in brackets in eq. (35) should be neglected and G is 
dominated by the second term. This means that the point 5 = 2 can be viewed as 
the starting point for the formation of the conductance pealc at larger energies. To 
describe this peak we can expand J(r) at small r in eq. (27) and get: 

^" r 

We see that the new conductance peak is formed at the energy of the resonant 
level shifted by the cheirging energy, £ = £0 -I- Ec and has a width 5 *^^ft. Hence, it 
becomes well-defined at 5 ^ max{l,(r 2 /ft)*}. 

Figvire 2 shows numerically calculated conductance of the resonant level coupled 
to the Ohmic environment (without the approximation of exponential cutoff), which 
exhibits transition from the peak at £ = £0 to £ = £0 -f Ec with increasing 5 . The 


16 








formation of the new peak is described quantitatively by eq. (36), but is slightly 
slower because of the divergence of X (36) for the Ohmic environment. 

Another interesting aspect of the above results is that, for intermediate values 
of fli, the conductance is non-vanishing in the entire energy interval between e — Sq 
and e = Co + Ec- This means that the width of the current step in this regime is 
determined by the charging energy Ec and can be much larger than the timneling 
width of the resonant level r 2 or temperature T. Carrying out the integration in 
the first of eqs. (27) we get for the Ohmic environment: 


I(e)=cT,[\-T f_Jr 


sin[(£o - £)t + (irg/2)(e^'^ - 1 )] 
sinhfTrTT] 


exp{r2r/2-f Re[J(T)]}]. 


Figure 3 show the current (37) for several values of the temperature. At T •C 
the width of the current step saturates, and the current step becomes asymmetric. 
At first the current rises sharply on the energy scale r 2 at £ = Cq and then in¬ 
creases monotonously and levels-olF on the scale Ec ^ T (in accordance with eq. 
(33)). As will be discussed in the Conclusion, these features might be relevant for 
interpretation of some aspects of experiments on resonant tunneling in quantum 
dots. 


V. FILLED BANDS 

As we saw in the previous sections, the main characteristic feature of the one- 
electron approximation is that the tunneling width of the resonant level is unaffected 
by interaction. As a result, this approximation is only valid when the energy band 
in at least one of the external electrodes is empty. In this section we turn to the 
case when the energy bands in both electrodes are filled, so that the one-electron 
approximation would give qualitatively incorrect results. 


17 








It will be shown below that renormahzation of the tunneling width of the res¬ 
onant level by interaction plays a crucial role in this case, and has two major 
consequences. Firstly, at small coupling constants g, the value of conductance at 
resonance is not suppressed by the interaction but the width of the resonance de¬ 
creases with increasing g. The second consequence is that at sufficiently large g the 
effective txinneling width F vanishes and a Coulomb gap is formed in the position 
of resonant level as a function of its bare energy £o- The Coulomb gap suppresses 
the conductance. 


We consider at &st the situation when bosonic modes are coupled only to the 
resonant level itself (i.e., Zv{<jj) = 0 and Ai = A 2 ). A qualitative understanding 
of this situation can be obtained from the relation between the zero-temperature 
linear conductance G of the resonant level and its occupation probability n [14]: 


G 


_£^r,r2 . 2 

2irh r2 


(38) 


Equation (38) shows that if n is a continuous function of energy Sq of the resonant 
level, then the conductance reaches at n = 1/2 the same maximum value as without 
interaction. The fact that the resonant occupation probability is fixed at 1/2 is a 
manifestation of the electron-hole symmetry of the problem. Conversely, eq. (38) 
implies that the maximum value of the conductance can be suppressed only if n 
becomes a discontinuous function of £o» i-e. if a Coulomb gap appears in the effective 
renormalized energy of the resonant level. 

Quantitatively, we cetn first analyze the problem at small g when a direct per- 
tmbation calculation is possible. This calculation shows that suppression of the 
linear conductance due to the suppression of the elastic transmission probability 
P(e, e), which is a hallmark of the one-electron approximation (elastic transmission 
probability is suppressed due to appezirance of the inelastic chaimels), is precisely 


18 






compensated for by corrections arising from deviations of the boson evolution from 
free evolution discussed in Sec. 3. 

To go beyond the perturbation calculations we discuss at first the one-mode 
environment with frequency f2 that is larger than all frequencies associated with 
electron tunneling: So, eV, T, F fl. In this case, the fast boson mode adjusts 
itself instantly to the slow electron tunneling and we can employ the adiabatic 
approximation and average the total Hamiltonian of the system over the fast motion 
of the boson mode to obtain an effective electron Hamiltonian. As a result of such 
averaging we get that the tunneling amplitudes (2) are renormalized as follows: 

f. = (t-e'^) = . (39) 

This means that at low temperatures T -C F*, conductance should exhibit the 
same Lorentzian peak as in the noninteracting case but with renormalized width 

F«: 

P ^^r;r; i 

2irft (e-€of+ (r-)’>’ 

where FJ = FjC"^. We see that in accordance with our qualitative discussion, the 

conductance reaches a maximum value at e = £o which is the same maximum as in 

the non-interacting case. 

Away from the resonance, when | £—eo 1^ T*, the conductance can be calculated 
by perturbation theory in tunneling. If the Fermi levels in external electrodes 
are below eo, then the perturbation expansion should be performed starting with 
the empty resonant level as the zero-order approximation. Following the standard 
calculation of the second-order txmneling (see, e.g., [15]), we get, naturally, the same 
eqs. (20)-(22) for the current, without the tuimeling width F in the definition of 
the transition amplitude A (22). However, when the Fermi levels axe above Eq, the 
perturbation expemsion should be performed around occupied resonant level. The 



19 








perturbation theory gives then the same expression for the current (without F), but 
with the reversed sign of the energy difference £ — £o in the expression for A (22). 
This change of sign reflects the fact that for £ > £o the tunneling is more naturally 
interpreted as a tunneling of holes and not electrons (see the discussion below). 

Expanding the exponent in eq. (26) and integrating each term of the series 
over t we get that only the first term contributes to the zero-temperature linear 
conductance G. Taking into account that for £ > £o we should change the sign of 
£ — £o we get for G: 




(41) 


where J(r) is given by eq. (27) (note that for Zviu>) = 0 and Aj = Aj all factors F 
coincide, Fn = F 22 = F 12 = F). Taking the small-r limit of J(t) in eq. (41) we get 
£in asymptotic expression for the conductance for | £ — £0 | +Ec > Q: 


G(£) = 


r.Fj 


(42) 


2irh (I £ - £0 I +jBc)* 

It is interesting to note that if we had directly applied one-electron approxima¬ 
tion (expressed by eqs. (20) - (22)) both for £ < £0 and £ > £0, we would get the 
same eq. (42) but with £0 — £ instead of | £ — £0 ]. Equation (42) would imply then 
that there is a resonant peak of the linear conductance at £ = £0 -I- Ec. Simple phys¬ 
ical reasoning demonstrate why this conclusion is incorrect. To see this, notice that 
one of the assumption behind the one-electron approximation is that the photon 
modes are in equilibrium when the resonant level is empty, so that electron tun¬ 
neling to the level displaces them out of equilibrium. Such a displacement results 
in the shift of the effective resonant level energy from £0 to £0 -f Ec. The correct 
picture for £ > £0 is, however, that the photon modes are in equilibrium when the 
resonant level is occupied, and their displacement is caused by electron tmmeling 
from the level or, in other words, tunneling of holes to the level. Thus, at £ > £0 


20 



the tunneling holes experience the same influence of the interaction with photons 
as electrons at £ < eo- 

To complete our analysis of the one-mode environment we note that eq. (41) is 
reduced in this case to; 


where z =| e — Eo | /fl, and 7* is another incomplete gaunma function [16]. At ^ » 1 
and I e — £0 [^ Ece~^^ the sum in eq. (43) is dominated by the terms around n 'zt g 
and eq. (43) coincides with eq. (42). At small energies | £ — £0 [■C Ece~^^ the sum 
is dominated by the first term (n = 0), so that G{e) = /2nh)riT2e~^^/{e — £o)^- 

Since this expression coincides with the result (40) of adiabatic approximation in the 
wide energy range, Cj £—£0 [■C we can conclude that the combination 

of eqs. (40) and (43) gives the G(£) dependence that is valid for all energies. 

These results show that in the limit g —» 00 eq. (42) becomes exact at all 
energies and describes the Coulomb gap appearing in the renormalized energy of 
the resonant level. Namely, eq. (42) implies that when the bare energy is above 
the Fermi levels in the electrodes, £0 —> +0 the renormalized energy is So + Ec, 
but as soon as it is below the Fermi level, £0 —» —0, the renormalized energy is 
—(£0 + Fc). The discontinuity of the renormalized energy at £0 = 0 is a result of the 
discontinuity of the occupation probability n of the resonant level at this energy. 

For Ohmic environment, the renormalization of the tunneling width can be writ¬ 
ten following (39) as F* = F(F/fl)®^^*“®) if F fl. It means that F is renormalized 
to zero at flr > 1. In this case the perturbation expression (41) will be valid at all 
energies. With the exponential cutoff this expression gives for conductance: 

(- 1 )" 


G = -g,z) = Goe2^[F(l - g)z^-^ - 


^0 l-g+n 


f- 


( 44 ) 


21 





Fbr Ohmic spectrum with Lorentziaui cutoff we can use the asymptotic form of 
7 (r) for T ^ Q"* to find the conductance at small energy (z <C 1 ), and g < 1: 

G = . (45) 

Equation (45) agrees with eq. (44) at 2 1, 3 < 1. At arbitrary energy and g, 

the conductance can be calculated numerically, and is shown in Fig. 4. This figure 
confirms that the evolution of the conductance with g has a transition point at 
5=1 (as indicated also by eqs. (44), (45)). For 5 < 1 the conductance has a peak 
at £ 2 ; £ 0 , while for 5 > 1 the peak is completely suppressed. At the transition 
point 5 = 1 we get from eq. (44): 

G = Go(ln 2 + 7)^ (46) 

For 5 > 1 the form of the conductance curves agrees qualitatively with (42) which 
gives G = Go/(g — 1)^ at small z. At large g this expression also agrees with eq. 
(42). 

These results imply that in the case of Ohmic environment the Coulomb gap is 
formed at finite 5 , namely, 5 = 1. The reason for this is the logarithmic divergence 
of the bosonic correlators J(r) in this case. Thus, formation of the Coulomb gap for 
Ohmic environment is a zero-temperature first-order phase transition. The physical 
picture of the Coulomb gap is the same as discussed above for the one-mode czise. 
The Coulomb gap provides a physical explanation of the low-energy singularity 
obtained recently [17] in tin impurity model similar to our model. 

We now turn to the case of the bias voltage fluctuations ^ 0, Zg{u}) = 0). 

In this case the photon modes are coupled to the charge of the external electrodes, 
and the picture of tunneling is much closer to the usual Coulomb blockade in small 
tunnel jimctions than in the case of the resonant level fluctuations. In the latter case 
electron interact with the photon modes only when it occupies the resonemt level 


22 











s 


diiring the tunneling, wliile the states of the modes before and after the tmineling 
are the same. In contrast to this, in the case of voltage fluctuations initial and 
final states of the photon modes are different because the tunneling changes the 
charge of external electrodes. This leads to stronger suppression of conductance, 
in particular, the linear conductance is completely suppressed at T = 0, and the 
cmrent-voltage characteristic has a power-law singularity at small voltages. 

To calculate the current in the small-voltage limit we notice that in this limit 
the relevant time scale for integration over t in eq. (26) is much larger than the 
time scales for r and t'. (The small-voltage limit implies that s' — e ■C] £o ~ ^ I-) 
Neglecting t and r' in comparison with t we obtain: 

P(£,e') = ^ I /“ P , (47) 

where J(t) is given by eq. (27) with F(u}) = Fv{u>). 

We see that expression (47) for the transition rate factorizes at small voltages 
in the two terms. One describes its voltage dependence (dependence on s — s' in 
eq. (47)) smd coincides with the similar term for the first-order tunneling in small 
tunnel junctions [6]. The second term has the meaning of the transition matrix 
element and coincides with the transition element (41) in the case of the resonant 
level fluctuations. Combining these two terms we get, for example, for Ohmic 
environment: 

enT,T2e-'^^^-^-^^r^{l+g) ] eF _ 

27r «(3-2a)r(l-I-g) |e-SoPri+5)’ ^ 

We see that the term responsible for the voltage dependence of the current sup¬ 
presses the cmrent imiformly for all s, while the second term sharpens the reso¬ 
nance suppressing the current for s away from Sq. Thus, in contrast to the resonant 
level fluctuations which suppress the resonance, fluctuations of the voltage make 
the resonance sharper with increasing interaction strength. 




23 







VI. CONCLUSION 


In conclusion, we have considered resonant txuineling of electrons interacting 
with fluctuations of electromagnetic field or some other dissipative environment. 
Broadly speaking, interaction changes position and shape of the resonance. Our 
calculations might be relevant for experiments on electron transport in quant inn 
dots. In particular, the one-electron approximation used in Sec. 3, is adequate for 
vertical double-barrier heterostructures similar, e.g., to those studied in (1], [2], [4]. 
In such structures the current appears only at large bias voltages, at which both 
conditions of applicability of the one-electron approximation are satisfied; first, 
the relevant electron states in the collector electrode are empty, and second, trans¬ 
parency of the collector barrier is much larger than that of the emitter barrier. 

One interesting, and so far unexplained result of experiment [1] is the large low- 
temperature width of the current step that corresponds to the tunneling through 
the lowest OD state in the quantum dot. In this experiment conditions (24) of 
applicability of the one-electron approximation were not, strictly speaking, satisfied; 
the rate of tunneling through the emitter barrier F i was not always smaller them 
the collector timneling rate r 2 . However, the ratio Fi/Fj was at most on the 
order of one, so that one-electron approximation should be at least qualitatively 
valid. Then the results obtained above in Sec. 3 are applicable and suggest that a 
possible explanation of the extra width of the current step could lie in a moderately 
strong {g ~ 1) interaction of electrons with some dissipative environment, since, 
as was shown in Sec. 3, at y 1 the width of the current step is expected to 
be roughly equal to Ec (i.e., much larger than the tunneling width F). In the 
vertical double-barrier heterostructures, the most probable candidate for dissipative 
environment interacting strongly with tunneling electrons are the charged dopmits 


24 









s 


in the depletion layer of the collector electrode. For interaction with dopants in 
the depletion layer the energy Ec is the difference between resonant level energy 
under the two conditions, when the charges in the depletion layer are frozen, zuid 
when they are allowed to rearrange to screen the electron on the resonant level. 
Qualitatively, the reason for the widening of the current step in this situation can 
be understood as follows. The current appears at voltages at which electron can 
tunnel through the resonant level when the level energy is lowered to Eq by the 
rearrangement of the environment in response to the transfer of electron to the 
level. However, the current saturates only at voltages when electron can tunnel 
even if the environment is not responding to electron transfer so that the effective 
energy ci *;he resonant level is eo + Ec. An additional argument in favor of such an 
explanation is the asymmetric shape of the current step, with steeper rise of the 
current at small voltage and more gradual leveling-off at larger voltages [1]. As was 
discussed in Sec. 3, a similar shape of the current step results from the interaction 
with environment with Ohmic spectral density of excitations. 

Another aspect of our results is that they provide a specific example of the 
situation when the calculation of conductance is a multi-electron problem despite 
the fact that there is no direct electron-electron interaction in the Hamiltonian (3). 
This imphes that the frequently used ideology of the “generalized Landauer formula” 
based on the solution of the one-electron problem for a given bosonic field [18], [19], 
is not V2ilid zis a general approach, and establishing conditions of its validity is an 
important problem. 

ACKNOWLEDGMENTS 

Useful discussions with V.J. Goldman and K.K. Likharev are gratefully acknowl¬ 
edged. One of us (V.P.) appreciate useful discussions with R. Suris at the beginning 


25 







of this work and with B. Spivak at the completion of it. This work was supported 
by ONR Grant # N00014-93-1-0880. 


26 







REFERENCES 


[1] B. Su, V.J. Goldman, and J.E. Cunningham, Science 255, 313 (1992); Phys. 
Rev. B 46, 7644 (1992); and private communication. 

[2] P. Gueret, N. Blank, R. Germann, and H. Rothuizen, Phys. Rev. Lett. 68, 1896 
(1992). 

[3] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. 
Harmans, and C.T. Foxon, Phys. Rev. Lett. 69, 1592 (1992). 

[4] M. Tewordt, L. Martin-Moreno, J.T. Nicholls, M. Pepper, M.J. Kelly, V.J. Law, 
D.A. Ritchie, J.E.F. Frost, and G.A.C. Jones, Phys. Rev. B 45, 14407 (1992). 

[5] E.B. Foxman, P.L. McEuen, U. Meirav, N.S. Wingreen, Y. Meir, P.A. Belk, 
N.R. Belk, M.A. Kastner, and S.J. Wind, Phys. Rev. B 47, 10020 (1993). 

[6] G.-L. Ingold and Yu.V. Nazarov, in: Single Charge Tunneling, ed. by H. Grabert 
and M. Devoret (Plenum, New York, 1992). 

[7] L.I. Glazman and R.I. Shekhter, Sov. Phys. JETP 67, 163 (1988). 

[8] N.S. Wingreen, K.W. Jacobsen, jind J.W. Wilkins, Phys. Rev. Lett. 61, 1396 
(1988). 

[9] M. Jonson, Phys. Rev. B 39, 5924 (1989). 

[10] V.V. Ponomarenko, Europhys. Lett. 22, 293 (1993). 

[11] K.A. Matveev and A.I. Larkin, Phys. Rev. B 46, 15337 (1992). 

[12] Y. Meir and N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992). 


27 









[13] G.D. Mahan, Many-Particle Physics (Plenum, New York, 1991). 

[14] V.V. Ponomarenko, Phys. Rev. B 48, 5265 (1993). 

[15] D.V. Averin and Yu.V. Nazarov, in: Single Charge tunneling, ed. by H. Grabert 
and M. Devoret (Plenum, New York, 1992). 

[16] Handbook of Mathematical Functions, ed. by M. Abramowitz and I.A. Stegun 
(Dover, New York, 1972), Ch. 6. 

[17] T. Giamarchi, C.M. Varma, A.E. Ruckenstein, P. Nozieres, Phys. Rev. Lett. 
70, 3967 (1993). 

[18] S. Feng, Phys. Lett. A 143, 400 (1990). 

[19] F. Hekking, Yu.V. Nazarov, and G. Schon, Europhys. Lett. 20, 255 (1992). 


28 






FIGURES 


FIG. 1. Schematic diagram of the resonant tunneling structure considered in this work. 
Zg{u)) and Z((j) are effective impedances of the structure representing the density of the 
photon modes responsible for fluctuations of the resonant level energy, and fluctuations of 
the bias voltage V, respectively. 


FIG. 2. Differential conductance of the resonant tunneling structure with the Ohmic 
spectrum of photon modes in the one-electron approximation. Gmax denotes the maximum 
conductance in the non-interacting case, Gmax = (2e^Arj/ffhr2). The curves illustrate he 
transition between low-energy and high-energy conductance peaks with increasing strength 
of the coupling to photon modes. 


FIG. 3. DC current through the resonant tunneling structure with the Ohmic spec¬ 
trum of photon modes for several values of the temperature T and coupling constant g — I 
- see eq. (37). In the low-temperature limit the width of the current step is determined by 
the charging energy Ec and is much larger than the tunneling width of the resonant level. 


FIG. 4. Linear conductance of the resonant tunneling structure with Ohmic spectrum 
of the photon modes coupled to the resonant level. The plot shows suppression of the 
resonance at £ = £o when the coupling strength is increased beyond 5=1. The conductance 
scale Go is defined as in eq. (43). 


29 


































41 


I 

i 


Submitted to Phys. Rev. B 


Effect of inage charge on single-electron tunneling 

A. N. Korotkov 


Department of Physics, State University of New York, 
Stony Brook, NY 11794-3800, USA, 
and Institute of Nuclear Physics, Moscow State University, 

Moscow 119899 GSP, Russia. 


The existence of the image charge causes a modification of the 
tunnel barrier shape which depends on the effective junction 
capacitance C. The tunneling rate can be calculated using the 
expression of the "orthodox" theory of single-electron tunneling, 
with an additional prefactor of the order of exp(TeVhC) where x 
is the traversal time of tunneling. 


1 








Correlated tunneling in systems of ultrasmall tunnel junctions 
is a rapidly developing field of mesoscopic physics.The simple 
"orthodox" theory of single-electron tunneling^ provides the basis 
for theoretical analysis of these processes. There are, however, 
several effects not taken into account by "orthodox" theory.^ In 
particular, in Kefs. 4-6 the influence of the finite traversal time 
of tunneling t was considered (in the "orthodox" theory, t is 
assumed to be infinitesimal). It was shown that the effect becomes 
important when t is of the order of hC/e^, where C is the Junction 
capacitance. 

The main focus of Refs. 4-6 was on the shape of the dc I-V 
curve. In contrast, in the present paper we calculate the 
multiplicative correction of the order of expCxeVhC) to the 
tunneling rate, which is independent (in our approximation) of the 
dc voltage applied to the system. The origin of this correction is 
the variation of the image charge at the edges of the tunnel 
junction. 

First, consider a tunnel junction biased by a fixed dc voltage 
V>0. Assume that the temperature T is zero, then tunneling is 
possible only in one direction (say, from left to right). The 
effect of the image charge on the tunneling was considered in a 
number of papers (see, e.g.. Ref. 7 and references therein). In the 
simplest model we assume that the image charge follows the position 
of the electron inside the barrier as in the static case. This 
so-called "static" image^ model is valid when the frequencies of 
the surface plasroons are much larger than l/x. If we also assume 
that the Thomas-Fermi screening length in electrodes is much less 
than the thickness of the barrier, the image charge can be 





calculated by a simple multiple reflection procedure (Fig. la), and 
the effective barrier shape is the sum of the initial shape Uq(x) 
and the correction U,^(x) due to image charges. 

Note that in this case the total image charges at the left 
side of the junction and at the right side depend linearly on 
the position x of the electron inside the barrier (measured, say, 
from the surface of the left electrode) 

Q=(x/L-l)e. Q =(-x/L)e, (1) 

1 r 

where L is the barrier thickness. In reality these charges are 
located at the electrode surfaces and supplied by the voltage 
source. 

Now consider the same tunnel junction separated from the 
voltage source. Let the initial voltage V be greater than e/2C 
(after the tunneling of one electron this voltage becomes V-e/C). 
In contrast to the fixed voltage case, now the total charge of each 
electrode is fixed. Hence, in comparison with the previous case, 
there are additional charges -Qj-e, -Q^ uniformly distributed along 
the electrode surfaces (Fig. lb). This leads to the additional 
electric field E=(Q -Q -e)/2CL=~xe/(CL^) which depends on the 

r 1 

position of the electron. Hence, the effective barrier becomes 
UQ(x)+Uj^(x)+Ug^(x) where 

U„.,(x) = -e fE(x’) dx’ = (xVL2)(eV2C). (2) 

SET Jq 

Let us emphasize that at the point x=L this additional energy 
coincides with the change e^/2C of the electrostatic energy of the 








system after the tunneling. Because of this fact (which is valid 

only in the "static" image model), taking into account only the 

linear part of 

Uli^{x) = (x/LXe^/BC), (3) 

we would exactly reduce the case of the separated junction to the 

case of the tunnel junction biased by the fixed voltage V-e/2C. 

Then the tunneling rate T could be calculated using the expression 

P ^ I(V-e/2C) (V>e/2C, T=0) (4) 

where I(V) is the I-V curve of the junction biased by a fixed 
voltage. Equation 4 exactly coincides with the equation used in 
the "orthodox" theory of single-electron tunneling. 

The correction to Eq. 4 is caused by the remaining part of the 
barrier modification 

W*' = “set - "lin = -(x/L)(l-x/L)(eV2C). (5) 

Assuming ^corr^'^^o'^^im using the WKB approximation it is easy to 
calculate the tunneling rate: 

r = K 

€ 

L 

K = exp(- ^ J <U-{x?!‘u”(x)^^^^ ^coRR^^^ 

0 0 IM 

The expression for the correction factor K depends on the 
barrier shape and in the general case cannot be exactly expressed 


( 6 ) 

(7) 


4 






s 


in terms of capacitance C and traversal time x. However, for an 
estimate let us assume UQ(x)+Ujj^(x)=const. This gives a simple 
expression 

K = exp(|^ ^). T = L((UQ+Uj^)/2m)'^^2. (8) 


Thus, similar to the nonlinear effects considered in Refs. 
4-6, the correction is essential when the traversal time x is not 
too small in comparison with e^/hC. In our approximation this 
correction does not depend on the voltage. 

Equations (5-8) can be easily extended to the case of a tunnel 
junction inside an arbitrary single-electron circuit containing 
other tunnel junctions, capacitances, voltage sources and 
resistances, with the only restriction (usual for the "orthodox" 
theory) that any resistance should be either much smaller or much 
greater than fi/e^ and x/C (in the most interesting case the last 
two values are of the same order). Then it is straightforward^ to 
introduce the effective capacitance of the tunnel junction and 
the only change in Eqs. (5)-(8) is the substitution The 
effective capacitance is defined via the difference between the 
voltage Vj=V before tunneling and the voltage V^. after tunneling. 



= e/(Vj-V^). 


( 9 ) 


For example, in the system of two junctions connected in series 
(the "single-electron transistor") the effective capacitance is the 
sum of the junction capacitances, C^ff^Cj+Cg. 

The simple substitution in Eqs. (5)-(8) is possible 


5 








only if the circuit size is much less than tc (c is the speed of 
light), so it will not be valid for too large circuits. In this 
case as well as for arbitrary resistances in the circuit, a more 
complicated theory based on the equations of Refs. 4-5 is 
necessary. 

Generalization to the case of finite temperature T is also 
quite simple because the barrier change Uj.qj^(x) does not depend on 
the temperature. The general expression 

r = K I{V*)/(l-exp(-eVVT)), (10) 
V' = V-e/2C.„ = K = K(C,„) 

is similar to that of the "orthodox" theory; the only difference is 
the prefactor K. The existence of this prefactor depending on the 
effective capacitance of the junction is the main point of the 
present paper. 

Now let us discuss the possibility of observing the considered 
effect in experiment. The simplest way is to compare the dc I-V 
curve of the single tunnel junction biased by a fixed voltage and 
the dc I-V curve of the double-junction system. At T~h(Cj+C 2 )/e^ 
the current in the double-junction system should be larger than 
that predicted by "orthodox" theory. The simplest check is to 
compare the low-voltage resistance of one junction with the 
differential resistance of the double-junction system at the 
voltage just above the Coulomb blockade threshold. In "orthodox" 
theory these two values coincide, if the background charge is not 
close to zero and the temperature is low. 


6 







Let us estimate possible experimental parameters. For the 
tunnel junctions metal-insulator-metal the typical traversal time 
T is about 3*10"'® s. The correction factor K in this case is 
essential for e/OO.3 V. Hence, in principle, the effect can be 
observed using the scanning tunneling microscope.'"^ However, in 
this case it is practically impossible to prepare identical tunnel 
junctions for single-junction and double-junction experiments. 

The traversal time in semiconductor tunnel junctions can be 
made much longer by the use of low tunnel barriers. For x long 
enough the model of "static" image charge can be applicable® in 
spite of the fact that the plasmon frequencies in semiconductors 
are much lower than those in metals. In Ref. 8 tunnel junctions 
having traversal time up to 3*10"'® s were used. For our estimate, 
let us take the more moderate value t=10"'® s. Then for observation 
of the effect considered in the present paper, the typical voltage 
e/C should be of the order of h/Te~7 mV. Hence, the typical 
capacitance may be about 2*10"'^ F. Note that the condition 
T~hC^^j./e® means that the typical voltage of the exponential 
nonlinearity of the I-V curve should be of the order of Coulomb 
blockade threshold. 

In conclusion, we have found a correction to the tunneling 
rate which is used in the "orthodox" theory of single-electron 
tunneling. The correction is essential when xeVhOl. It leads, in 
particular, to an increase of the current through the 
double-junction system in comparison with that calculated using the 
"orthodox" theory. 


7 








ACKNOWLEDGEMENTS 


Fruitful discussions with D. V. Averin, K. K. Likharev and 
Yu. V. Nazarov are gratefully acknowledged. The work was supported 
in part by Russian Fund for Fundamental Research, Grant 
#93-02-14136 and ONR Grant #N00014-93-l-0880. 


8 









REFERENCES 


^ K.K. Likharev, IBM J. Res. Dev. 32, 144 (1988). 

^ D.V. Averin and K.K. Likharev, in: Mesoscopic Phenomena in 
Solids, ed. by B. Altshuler et al. (Elsevier, Amsterdam, 1991), 
p. 173. 

^ Single Charge Tunneling, ed. by H. Grabert and M.H. Devoret 
(Plenum, New York, 1992). 

^ Yu.V. Nazarov, Solid State Commun. 75, 669 (1990). 

® Yu.V. Nazarov, Phys. Rev. B 43, 6220 (1991). 

® A.N. Korotkov and Yu.V. Nazarov, Physica B 173, 217 (1991). 

^ B.N. Persson and A. Baratoff, Phys. Rev. B 38, 9616 (1998). 

® P. Gueret, E. Marclay, and H. Meier, Appl. Phys. Lett. 53, 1617 
(1988). 


9 








Figure caption 


Fig. 1 


(a) Image charges for voltage biased tunnel junction and 

(b) additional charges in the case of the separated 
Junction. 


10 












Quantum and Many-Body Effects on the 
Capacitance of a Quantum Dot 


Lotfi Belkhir 

Department of Physics, 

State University of New York, 
Stony Brook, New York 11794-3800 


Abstract 


We calculate exactly, using finite size techniques, the quantum mechanical 
and many-body effects to the self-capacitance of a spherical quantum dot in 
the regime of extreme confinement, where the radius of the sphere is much 
smaller than the effective Bohr radius. We find that the self capacitance os¬ 
cillates as a function of the number of electrons close to its classical value. 
We also find that the electrostatic energy as a function of the number of elec¬ 
trons extrapolates to zero when N = 1, suggesting that the energy scales like 
e^N{N - 1) instead of {N e)^. We also provide evidence that the main devi¬ 
ations from the semiclassical description are due to the exchange interaction 
between electrons. This establishes, at least for this configuration, that the 
semiclassical description of Coulomb charging effects in terms of capacitances 
holds to a good approximation even at very small scales. 

PACS numbers: 73.20.Dx, 73.40.Gk, 72.15.Qm 


Typeset using REVT^ 


1 




With the rapid advances in the fabrication of increasingly smaller quantum dots, ap¬ 
proaching the atomic scale, the question of quantum and many-body effects on the essential 
characteristics of these objects has become a central issue. The current stage of theoretical 
understanding of quantum dots relies essentially on a semi-classical picture*’^, which malces 
the assumption that the Coulomb charging effects can be described in terms of classical 
capacitances, where the resonance energies can be separated in a single-particle confinement 
energy, and a constant Coulomb charging energy terms. The subject has been further stud¬ 
ied in many recent theoretical^"* and experimental investigations*"**. The classical picture 
of Coulomb blockade, however, has recently been questioned by Johnson and Payne*^ who, 
using a harmonic model interaction that is exactly solvable*®, showed that, in presence of 
magnetic field, the resonance energies could not be written as the sum of single-particle and 
charging energies terms. They argued that the model interaction shows a behavior similar 
to a Coulomb interaction with a cutoff** for a certain rzmge of electron-electron separation. 
On the other hand, however, the semiclassical description seems so far to provide a qual¬ 
itatively correct picture, given that some experiments*^’**, performed in the regime where 
confinement and charging energies are of equal importance, can be well explained by this 
model. 

We investigate this issue further, using exaw:t finite size calculations techniques. We consider 
an isolated spherical quantum dot in zero magnetic field, and solve numerically the full 
Coulomb interaction problem, for up to 30 electrons, in the regime where the quantum 
effects are expected to be maximal, i.e when the radius of the dot is much smaller th 2 ui the 
effective Bohr radius (i.e i2 C oq). It is assumed that the added electrons move on the surface 
of the sphere, which could be a reasonable model for a metallic sphere. It has recently been 
shown*®, in the framework of density-functional theory, that in most experimental situations 
the main contribution to the capacitance of a quantum dot, in the presence of leads and 
backgates, comes from the self-capacitance. The contributions of the leads and backgates 
was found to be 30% at most. The results for an isolated dot are therefore not irrelevant to 
actual experiments. 

We do the calculation for both spin unpolarized and spin polarized cases. We find that: 
(i) the interaction energy spectrum scales like N{N — l)e*/2C, where N is the number of 
electrons at the surface of the dot, and R its radius. This corroborates the semi-classical ex¬ 
pression for the charging energy suggested recently by some authors*®’**, instead of the more 
widely used expression {Ney/2C. (ii) the resonance energies do sep 2 U'ate into confinement 
and charging energies to a good approximation; (iii) the self-capacitance of the isolated dot, 
defined from the charging energy, oscillates as a function of N, around its classical value, 
i.e R, and gets closer to R as N increases, (iv) The mun deviation of the self-capacitance 
from the classical value are due to the exchange interaction between electrons, which, in 
agreement with Hund’s first rule, tend to lower the ground state energy of the system, and 
thus increase the value of the capacitance. This increase is however never greater than 25% 
for N >2. 

Our Hamiltonian for an isolated spherical dot, is given by 


H = 


1 

2m* R? 


N 


E li'.P + E 


»<j 


cr, 



( 1 ) 


where Lj = —fftR,- x Vi and Fj are the angular momentum and the position of the i-th 


2 





particle, and e the dielectric constant. The eigenvalues of |L|^ are equal to /(/ + 1) with / 
an integer. The kinetic energy of an electron in the shell / is £/ = 1(1 + 1). The 

maximum number of electrons in a shell of angul 2 Lr momentum / is 2/ + 1 and 2(2/ + 1) for 
a spin polarized and spin unpolarized cases respectively. 

The dot contains N electrons, with an effective mass m*, and charge —|c|. Notice the 
absence of a confinement term, due to the use of a spherical geometry, where the electrons 
are constrained to move on the surface of the sphere of constant radius R. Previous quantum 
mechanical calculations of the capacitance of quantum dots*’** involved disc geometries with 
parabolic confinement potential. It was found in those calculations that the effective size of 
the dot increases with the number of electrons, in our case however, the size of the dot is 
absolutely rigid, which will facilitate considerably the comparison of our calculations with 
the classical results. 

The Coulomb interaction between two electrons moving on the surface of a sphere with 
radius R can be rewritten as 




1 

R |n, - tiii 



2/+1 




( 2 ) 


In second quantization form, the interaction operator is given by 


where the Yi'^({l) are the usual spherical harmonics, and oJmv 

To carry out our numerical calculations, we choose the convenient single-particle basis states 
defined by 






• SI** " / /vv - 


In this hash the two body interaction operator is given by 

V — \ C C 

2 l2m2ff Igmgv' 

allindiee* 


allindiee* 

where the matrix elements Me given by 


(4) 


(5) 


— V* Vi+*4-/a-<3 (2/i + 1)(2/4 4-1) .- , T wi. 

**l,m,«2m2J3mjl4m« ^ ( ) ^ 2/3 -f- l)( 2/3 -|- 1 ) ’ > ’ I 3» 3 

< /i,0;L, 01 / 3,0 >< /4,m4;X,M|/2,m2 > < /4,0; L,0|/2,0 > 


( 6 ) 

(7) 


where the terms < /,m;/i,mi|/ 2 ,m 2 > are the usual Clebsh-Gordon coefficients, which are 
non zero only when m -f mi — m 2 = 0. 

We also assume the limit of R—* 0, when the energy separation between successive angular 
momentum shells is large compared to the Coulomb interaction energy, so that mixing 
between shells can be neglected. This is equivalent to assuming that R <C ao> where 


3 






Oo = is the effective Bohr radius, determined solely by the material’s properties. In 
GaAs quantum dots, Oo = lOnm, and the confinement energy is typically about 

15/,meV, which for a spherical dot yields a radius R = 8nm. The strong confinement regime 
could be attained by either reducing the size of the dots, or using materials with a higher 
dielectric constamt e, which would increase the effective Bohr radius oq. The assumption of 
i2 —» 0 drastically reduces the Hilbert size of the quantum system, since intershell transitions 
can be completely ignored, which allows us to do the calculations for up to 30 electrons. 
The calculation done in this limit is similar to the finite size calculations done in the context 
of the fractional quantum Hall effect^*, where the limit of infinite magnetic field is assumed 
in order to ignore transitions to higher Landau levels. 

We calculate the energy spectrum by exact diagonalization of the Hamiltonian. The calcu¬ 
lation is done by filling the angular momentum shells by adding electrons one by one, and 
calculating the ground state of the whole many-body system. In less than half-filled shell, 
all electrons tend to have the same spin in order to gain the exchange energy in accordance 
with Hunds’ rule. As half-filling is reached, there is a sudden increase in the interaction 
energy due to the fact that the additional electrons must have opposite spins, in order to 
satisfy the Pauli principle, and thus lose the exchange energy. Our results for the ground 
state interaction energy Ee{N) as a function of N, spin unpolarized case, are shown in figure 
1. It shows an almost linear behavior, and extrapolates to 0 when = 1, which is consistent 
with the semiclassical expression N{N — l)c^//2. 

We first define the chemical potential as 

p(iV) = Ec{N) - EciN - 1) (8) 

We define the self-capacitance of our dot as C = AQ/AV which can be readily obtained 
from the chemical potential. For a single electron A Q = e and A V = [ti{N + 1) — tt{N)]/e. 
Thus 


C^.,(JV) = eVWJV + l)-/<(W)l (9) 

In the case where tunneling occurs through the channel where the dot is at its ground state 
before and after the event, the resonance energies are identical to the chemical potential. 
Notice that the definition of the chemical potential is usually taken as the difference between 
the total energies, rather than the interaction energies. In our case, both definitions are 
identical for electrons in the same shell, since the kinetic energy within one shell is constant. 
The only difference occurs when N electrons correspond to a state of completely filled shells, 
and the {N -1- l)th electron must occupy the next empty shell. The jump in the chemical 
potential will then be incremented by the confinement energy, in addition to the charging 
energy. Since we are choosing the confinement energy to be extremely large, we always 
subtract it from the total energy to keep only the interaction part. 

Figure (2a) and (2b) show the numerical results for the chemical potential, and the self¬ 
capacitance as a function of N in units of R for a spin unpolarized electron system. We 
see clearly that the capacitance is quite close to its classical value, modulo some quantum 
fluctuations. The peaks observed are due to the sudden increase in energy that occurs at 
half filling of each angular momentum shell. Notice also that, except for half filled shells, the 
quantum and many-body effects tend to increase the capacitance only slightly . The average 


4 




over all the 30 electrons, including the values at the peaks is 1.117/Z. Figure (3a) and (3b) 
show the results for the chemical potential and the self-capa^ritance of the spin polarized case. 
Observe now how the capacitance has became consistently greater than its classical value. 
However the difference is not greater than 25% for all N < 30. The average capacitance for 
the spin polarized case over Nmax = 29 is 1.123R. Also notice that the deviation tends to 
decrease asymptotically as N increases. This deviation is clearly due to the exchange energy 
between electrons of same spin. However this exchange energy becomes smaller in higher 
angul 2 u: momentum shells, which explains the asymptotic behavior of the self-capacitance. 
We sJso did the calculation for a system of spinless electrons (i.e no exchange interaction), 
and found that the capacitance become then precisely centered around its classical value R. 
In conclusion we have exactly calculated the energy spectrum and self-capacitance of a 
spherical quantum dot in the strong confinement limit where quantum effects are expected 
to be predominant. Remarkably, we have found that the semi-classical theory remains 
valid on average in this regime. We have also found that the main deviations from the 
semi-cla • al result are due to the exchange interaction between electrons, but that theses 
deviaticns do not exceed 25%. 

We gratefully acknowledge insighteful conversations with J. K. Jain, and K. K. Likharev, 
and T. Kawamura. This work was supported in part by the office of Naval Research under 
grant No. N00014-93-1-0880 


5 





REFERENCES 


* D. V. Averin and K. K. Likharev in ^Mesoscopic PbeDomena in Solids', edited by B. L. 
Altshuler, P. A. Lee and R. A. Webb, (Elsevier Science Publishers B.V., 1991). 

van Houten, C. W. Beenaker, and A. A. M. Staring, in ^Single Charge Tunneling', 
edited by H. Grabert and M. H. Devoret (Plenum Press, New York, 1992). 

®G. W. Bryant, Phys. Rev. Lett., 59, 1140 (1987) 

^L. I. Glazman and R. 1. Shekhter, J. Phys. Cond. Matt. 1, 5811 (1989). 

* A. Kumar, S. E. Laux, and F. Stem, Phys. Rev. B., 42, 1566 (1990). 

* Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett., 66, 3048 (1991). 

^ A. Grochev, T. Ivanov, and V. Valtchinov, Phys. Rev. Lett., 66, 1082 (1991) 

*M. Stopa, Phys. Rev. B., 48,18340 (1993). 

*U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett., 65, 771 (1990). 

*®L. P. Kouwenhoven et al., Z. Phys. B., 85, 367 (1991). 

U. Meirav et al, Z. Phys. B., 85, 357 (1991). 

**R. C. Ashoori et al, Phys. Rev. Lett., 68, 3088 (1992). 

J. Goldman, B. Su and J. E. Cunningham, Phys. Rev. B., 46, 7644 (1993). 

N. F. Johnson and M. C. Payne, Phys. Rev. B., 45, 3819, 1992. 

F. Johnson and M. C. Payne, Phys. Rev. Lett., 67, 1157, 1991. 

L. Banyai, et al, Phys. Rev. B., 36, 6099, 1987. 

*^P. L. McEuen, E. B. Foxman, U. Meirav, M. A. Kastner, Y. Meir, N.S. Wingreen and S. 

J. Wind, Phys. Rev. Lett., 66, 1926, 1991. 

**M. Macucci, K. Hess, G. J. lafrate, Phys. Rev. B., 48, 17354, 1993. 

** C. S. Lent, in * Proceedings of the International Symposium on Nanostructures and Meso¬ 
scopic Systems, Santa Fe ', edited by W. P. Kirk and M. A. Reed (Academic, Boston, 
1991). 

™D. V. Averin, A. N. Korotov, and K. K. Likharev, Phys. Rev. B., 44, 6199, 1991. 

F.D.M. Haldane in The Quantum Hall Effect, Eds. R.E. Prange and S.M. Girvin (Springer 
Verlag, New York, second edition). 


6 







FIGURES 


FIG. 1. Ground state interaction energy E{N)/N as a function of N for the spin unpolarized 
case. Notice that it extrapolates to 0 when iV = 1. 

FIG. 2. (a)The chemical potential and (b)the Self Capacitance of a spherical quantum dot as 
a function of N for the spin unpolarized electron system. 

FIG. 3. (a)The chemical potential and (b)the Self Capacitance of a spherical quantum dot as 
a function of N for the spin polarized electron system. 









8 


I 








9