ABSTRACT A new approach for modeling quantum transport that explicitly takes into account the electronelectron interaction is presented. The approach is based on the computation of transport properties of many-particle systems using quantum trajectories

X Oriols

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A Many-Particle Quantum-Trajectory Approach 
for Modeling Electron Transport and its 
Correlations in Nanoscale Devices 
X. Oriols 
Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Spain 
e-mail: Xavier.Oriols@uab.es 
 
ABSTRACT 
A new approach for modeling quantum transport 
that explicitly takes into account the electron-
electron interaction is presented. The approach is 
based on the computation of transport properties of 
many-particle systems using quantum trajectories. 
INTRODUCTION 
The coulomb interaction (due to the charge of 
the electron) and the exchange interaction (due to its 
fermion nature) are always present in any electron 
device. The role of both interactions can be obtained 
by directly solving the many-particle Schrödinger 
equation. However, due to computational 
limitations, its direct solution is only possible for a 
very limited number (N<5) of electrons. Therefore, 
much of our understanding of electron transport is 
based on a single-particle (“mean field”) 
approximations [1]. In this conference, we will 
present a new approach for the modeling of 
quantum transport that overcomes this assumption 
by dealing with many-particle effects using 
quantum (Bohm) trajectories [2]. 
MANY-PARTICLE TRANSPORT PROPERTIES IN TERMS                                  
OF QUANTUM TRAJECTORIES 
  It is well-known that Bohm trajectories exactly 
reproduce the results obtained from, either the 
single- or many-particle Schrödinger equation [3,4]. 
However, the application of such trajectories to 
transport has been quite limited because the 
computation of these trajectories needs, in general, 
the earlier knowledge of the full wave-function.   
In this conference, we show that a Bohm 
trajectory associated to a many-particle Schrödinger 
equation can be computed from a wave-function, 
),( trbb
r
Ψ , solution of the single-particle Schrödinger 
equation (with a complex unknown potential) [2]. 
Thus, the practical computation of Bohm 
trajectories in a system of N-interacting electrons 
can be greatly simplified. The electron-electron 
interaction is introduced in the computation of 
Bohm trajectories by taking into account the exact 
Coulomb force between pairs of electrons and the 
applied bias at the boundaries of the device. Thus, 
Identical bias with different lateral areas (Ly·Lz), 
provide different strength of the interaction. The 
exchange-interaction needs the simulation of N2 
Bohm trajectories to assure the antisymmetrical 
behavior of the wave-function. Figs. 1, 2 and 3, 
show the accuracy of our approach (blue □) when 
compared to the exact two-particle Schrödinger 
solutions (red ∆).  
CURRENT AND NOISE  OF INTERACTING-SYSTEMS 
     The computational viability of the previous 
many-particle Bohm-trajectories formalism for is 
shown in Figs. 4 and 5 where 50≈N interacting-
electrons are simulated. Since we are dealing with 
electron trajectories, most of the tools used in 
Monte-Carlo simulator can be directly adapted. In 
Fig. 4 and 5 we show how the (average) current and 
the (Fano factor) noise are sensible to electron-
electron interactions. Such interactions provide 
correlations among electron dynamics that can not 
be simulated with independent-electron formalisms.   
CONCLUSION 
A new approach for modeling quantum transport 
that explicitly takes into account the Coulomb and 
exchange interactions is presented. The approach 
opens a new path to provide a deeper understanding 
of nanoscale devices, since it can directly provides 
information on DC, AC and noise performances of 
interacting-electrons phase-coherent systems [5,6].  
REFERENCES 
[1] Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000). 
[2] X. Oriols (submitted). 
[3] J.Suñé, X.Oriols, Physical Review Letters 85,  894 (2000). 
 [4] X.Oriols, Physical Review A ,71, 017801 (2005). 
[5] X. Oriols, et al. Appl. Phys. Lett., 85 3596 (2004). 
[6] X.Oriols et al. Physical Review B, 71, 245322  (2005). 
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Lateral Area=L
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X1 transmitted
X2 reflected
Ti
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e 
(p
s)
Po
sit
ion
 X2
 (n
m)
Position X1 (nm)
X1 transmitted
X2 transmitted
Lateral Area=L
y ·L
z =45 nm 2
Lx
Lz
Ly
 
Fig. 1.  Two Bohm trajectories in a two-particle configuration 
space (X1,X2,t) with identical initial positions but different 
lateral area. Only the Coulomb interaction determines 
(X1=trans., X2=trans.) or (X1=trans.,X2=reflc). In red (∆), 
exact Bohm trajectories from a two-particle Schrödinger 
equation and, in blue (□), our approach.  Inset: 3D device 
dimensions (transport in x direction). The triple-barrier region is 
indicated. 
0,05 0,10 0,15 0,20
0,5
1,0
1,5
2,0
2,5
Ramo-Schokley
Visual guided line
 
Fa
no
 F
ac
to
r 
Applied bias (Volts)
0,05 0,10 0,15 0,20
0,00
0,10
0,20
0,30
0,40
0,50
Visual guided line
# Particles right contact
# Particles left contact
Ramo-Schokley
 C
ur
re
nt
 (µ
A)
 
Fig. 4: Simulated (a) current and (b) noise (Fano factor) for a 
three barrier diode using our approach with the explicit 
consideration of interacting electrons. See Fig. 5. The results 
from our approach are obtained through the Ramo-Shockley 
theorem [6].  
0,00 0,05 0,10 0,15 0,20
0,0
0,2
0,4
0,6
0,8
1,0
X1 transmitted
X2 transmitted
X1 transmitted
X2 reflected
X1 reflected
X2 transmitted
X1 in barrier region
X2 in barrier region
R
at
io
 o
f p
ar
tic
le
s 
 , 
  P
re
se
nc
e 
Pr
ob
ab
ili
ty
Time (ps)
X1 reflected
X2 reflected
     Lateral  Area = Ly·Lz=81 nm
2 
Coulomb and exchange interactions
 
Fig. 2.  In red (∆), exact time-evolution of probability presence 
of a two-particle Schrödinger equation impinging in the double 
barrier where Coulomb and exchange interaction are 
considered. In blue (□), same scenario simulated with our 
approach. 
 
0,00 0,05 0,10 0,15 0,20
0,0
0,2
0,4
0,6
0,8
1,0
X1 transmitted
X2 transmitted
X1 transmitted
X2 reflected
X1 reflected
X2 transmitted
X1 in barrier region
X2 in barrier region
R
at
io
 o
f p
ar
tic
le
s 
 , 
  P
re
se
nc
e 
Pr
ob
ab
ili
ty
Time (ps)
X1 reflected
X2 reflected
     Lateral  Area = Ly·Lz=324 nm
2 
      Only Coulomb interaction
 
Fig. 3. Identical results of figure 2 without exchange 
interaction. Excellent agreement between our approach (in blue 
□) and two-particle Schrödinger equation (in red ∆).  
0,00
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 Bi
as
 (m
V)
Tr
an
sm
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 c
oe
ffi
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Electron energy (eV)
Fe
rm
i L
ev
el
 
Fig. 5.  The triple-barreir diode transmission coefficient of the 
device simulated in Fig. 4, showing the presence of two 
resonances ( ) at V=0.12 V and V=0.15 V (a richer 
phenomenology in three-barrier than in two-barrier diodes). The 
electron correlations explain the different noise behavior after 
such resonances (NDC1 and NDC2) in Fig. 4(b).    
(a) 
(b) 
Our approach 
Our approach 
∆ ∆ 
∆ ∆
∆
∆
∆
∆ ∆ ∆ 
□ □ □ □ 
□ 
□ 
□ 
□ 
□ 
□ □ 
∆ 
NDC1 NDC2 
NDC1 NDC2 


A Many-Particle Quantum-Trajectory Approach for Modeling Electron Transport and its Correlations in Nanoscale Devices

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A Many-Particle Quantum-Trajectory Approach for Modeling Electron Transport and its Correlations in Nanoscale Devices

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