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Quantum Transport in the Transient Regime and Unconventional Geometries
By
Bozidar Novakovic
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Electrical and Computer Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
2012
Date of final oral examination: 08/17/
The dissertation is approved by the following members of the Final Oral Committee: Irena Knezevic, Associate Professor, Electrical and Computer Engineering John H. Booske, Professor, Electrical and Computer Engineering Luke J. Mawst, Professor, Electrical and Computer Engineering Max G. Lagally, Professor, Material Science and Engineering Maxim G. Vavilov, Assistant Professor, Physics Zhenqiang Ma, Professor, Electrical and Computer Engineering
i
ACKNOWLEDGMENTS
I am grateful to my advisor Prof. Irena Knezevic. Her generous support made this work
possible and her advising helped me to significantly improve my technical skills, as well as my
presentation and writing skills. I would also like to thank her for always being very responsive and
for allowing me to pursue my interest in physics.
My thanks also go to my committee members, Prof. John Booske, Prof. Luke Mawst, Prof.
Max Lagally, Prof. Maxim Vavilov, and Prof. Zhenqiang Ma, for making my final examination
interesting and for their useful comments on both the technical part of my work and on my future
career. Special thanks go to Prof. John Booske, Prof. Luke Mawst, Prof. Mark Eriksson, Prof.
Leon McCaughan, and Prof. Robert Joynt for providing very useful feedback at the time of my
preliminary examination.
I would also like to thank my colleagues from the Nanoelectronics Theory Group with whom I
had many useful discussions over the years. They are in alphabetical order: Dr. Zlatan Aksamija,
Jie Chen, Amirhossein Davoody, Dr. Xujiao Gao, Dr. Edwin Ramayya, Yanbing Shi, Nishant Sule,
and Dr. Han Zhou. Some of them were my good friends outside our laboratory setting as well. My
thanks go to Dr. Minghuang Huang and Matt Kirley for their collaboration leading to interesting
journal publications and to Kiritkumar Makwana from the Physics Department for his friendship.
Finally, I would like to thank Prof. Vitomir Milanovic, Prof. Jelena Radovanovic, and Prof.
Dragan Indjin, for helping me make my first steps in science at the University of Belgrade, and for
taking care of my subsequent career.
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TABLE OF CONTENTS
Page
LIST OF FIGURES...................................... iv
iii
- 1 Introduction ABSTRACT xi
- 1.1 Theme 1: Quantum transport in the transient regime
- 1.1.1 Device model
- 1.1.2 Electronic Quantum Master Equations and the transient-regime transport
- 1.2 Theme 2: Quantum transport in unconventional geometries
- 1.3 Summary of completed tasks and contributions
- 2 Electronic quantum master equations
- 2.1 The Single-Particle Quantum Master Equation
- 2.1.1 The Density Matrix and The Liouville-von Neumann Equation
- 2.1.2 The BBGKY Chain and the Single-Particle QME
- 2.1.3 The Pauli Master Equation
- 2.1.4 A Single-Particle QME Beyond the Born-Markov Approximation
- 2.1.5 Monte Carlo Solution to the QME
- 2.2 Reduced Many-Particle QMEs
- 2.2.1 The Nakajima-Zwanzig Equation
- 2.2.2 The Born-Markov Approximation
- 2.2.3 The Conventional Time–Convolutionless Equation of Motion
- 2.2.4 The Eigenproblem of the Projection Operator
- 2.2.5 A Partial–Trace–Free Equation of Motion
- 2.2.6 Memory Dressing
- 2.3 Coarse-Graining for the Steady State Distribution Function
- 2.3.1 The Exact Dynamics and the Coarse-Graining Procedure
- 2.3.2 The Short-Time Expansion of Fτ
- 2.3.3 Steady State in a Two-Terminal Ballistic Nanostructure
- 3 Quantum transport in the transient regime Page
- 3.1 The time evolution of distribution functions
- 3.2 The transient regime modeling in nanoscale devices
- 3.2.1 The system of coupled equations
- 3.2.2 The initial conditions
- 3.2.3 Injection into the localized device states due to scattering
- 3.3 Transient in an nin diode
- 3.4 Conclusion
- 4 Quantum transport in unconventional geometries in a magnetic field
- 4.1 Theoretical and numerical framework
- 4.1.1 Curvilinear Laplacian in the tight-binding form
- 4.1.2 The tight-binding 2D Schr¨odinger equation
- 4.1.3 The Peierls phase approximation
- 4.1.4 Handling the magnetic field: the local Landau gauge
- 4.2 Examples
- 4.2.1 Cylindrical geometry
- 4.2.2 Toroidal geometry
- 4.3 Summary and Concluding Remarks
- 5 Summary and conclusions
- LIST OF REFERENCES
- Appendix A: The details of the algorithm for the transient-regime calculations APPENDICES
- Appendix B: The surface divergence of the magnetic vector potential
- Appendix C: Hermiticity of the exact and tight-binding Hamiltonians
- Appendix D: Stabilized Transfer Matrix Solution to Eq. (4.19)
- Appendix E: Local Landau Gauge
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LIST OF FIGURES
Figure Page
1.1 Cut-off frequency fT=604 GHz for an InP/InGaAs pseudomorfic heterojuction bipo- lar transistor (PHBT) that used to hold a record for speed in 2005. Reprinted with permission from [W. Hafez and M. Feng, Appl. Phys. Lett. 86 , 152101 (2005)]. ⃝c[2005], American Institute of Physics......................... 2
1.2 Electromagnetic spectrum with the THz frequency window indicated with vertical red lines (300 MHz - 3 THz). Current optoelectronic devices, like quantum cas- cade lasers, are the most efficient at the frequencies above, while solid state devices at the frequencies below the THz window. Therefore, there is a need for increas- ing the performances of both types of devices, in order to be able to cover the THz window with efficient sources, detectors, mixers, and other types of devices. This work focuses on the lower-frequency-side devices, specifically modeling the tran- sient response to a sudden turn on of bias in nanoscale ballistic devices. Source: http://www.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html........... 3
1.3 A resonant tunneling diode (RTD) oscillator. The highest oscillating frequency achieved with this device is around 1 .1 THz, as indicated by the blue short line in the right panel. The photograph of the physical layout of the device is shown in the left panel Reprinted with permission from [M. Feiginov, C. Sydlo, O. Cojocari, and P. Meissner, Appl. Phys. Lett. 99 , 233506 (2011)]. c⃝[2011], American Institute of Physics..... 4
1.4 (a) Illustration of the curvature formation process based on selective underetching and exploitation of the lattice mismatch. (b) Various curved geometries fabricated. (c) Hall bar measurement setup. (d) Tube formed and (e) its scanning electron micrograph. (f) Two point measurement setup. Reprinted with permission from [N. Shaji, H. Qin, R. H. Blick, L. J. Klein, C. Deneke, and O. G. Schmidt, Appl. Phys. Lett. 90 , 042101 (2007)]. c⃝[2007], American Institute of Physics..................... 5
1.5 Two-terminal configuration displaying a quantum-mechanical, nanoscale active re- gion connected to two large semiclassical reservoirs of charge............. 6
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Figure Page
2.1 (a) The electron charge density and potential energy for an nin Si diode at 77 K, biased at 0.25 V, where the solid lines are results of using the master equation (2.13), while the dashed lines are results of using the Monte Carlo BTE. (b) Similar as (a), but with results for the average kinetic energy and drift velocity. Reprinted with permission from [M. V. Fischetti, Phys. Rev. B 59 , 4901 (1999)]. ⃝c(1999) by the American Physical Society...................................... 22
2.2 Drift velocity overshoot in silicon. The result of the quantum Monte Carlo technique is shown with the solid line, while the semiclassical result is shown with the dashed line. Reprinted with permission from [C. Jacoboni, Semicond. Sci. Technol. 7 , B (1992)]. c⃝(1992) IOP Publishing Ltd.......................... 30
2.3 Decomposition of the total Liouville space H^2 S+E into the subspaces of the projection operator P and the isomorphism between the unit subspace
H^2 S+E
P=1 and^ H
2 S for an operator x acting on HS+E. Reprinted with permission from [I. Knezevic and D. K. Ferry, Phys. Rev. A 69 , 012104 (2004)]. c⃝(2004) by the American Physical Society. 40
2.4 Schematic of the two-terminal ballistic nanostructure, negatively biased at the left contact, with the boundaries between the open system and contacts shown at xL and xR, and with the graphical representation of the wave function injected from and the hoping type interaction with the left contact. It is similar for the wave function and interaction for the right contact.............................. 51
2.5 Schematic of the two-terminal ballistic nanostructure with non-ideal contacts, neg- atively biased at the left contact, with the boundaries between the open system and contacts shown at xL and xR, and with the graphical representation of the wave func- tion injected from the left contact and the hoping type interaction between the left contact and the active region due to injection and between the active region and both contacts due to extraction. It is similar for the wave function and interaction for the right contact. Reprinted with permission from [I. Knezevic, Phys. Rev. B 77 , 125301 (2008)]. c⃝(2008) by the American Physical Society................... 55
3.1 The difference between the right-propagating device and left contact distribution func- tions, fk(t) − f (^) kL , normalized by fk(0) − f (^) kL. The difference should decay to zero in the steady state, showing that a longer contact relaxation time τ leads to a shorter transient regime. It is similar for the left-propagating device states. The injection rate for this specific k-state is assumed to be ∆k = 1 ps.................... 61
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Figure Page
3.6 (a) Potential and (b) charge density in the nin diode as a function of time upon the application of -25 mV to the left contact. The n-type regions are doped to 1017 cm−^3 , while the middle region to 1013 cm−^3. The contact momentum relaxation time is τ =120 fs, as calculated from the textbook mobility value corresponding to the contact doping density. The temperature is T = 300 K and the device size is 100 nm − 50 nm − 100 nm, for the two n-doped regions enclosing the middle one. The scattering-enhanced injection rate is chosen to be ∆scatt = 1/τ , and therefore the quantum well bottom rises quickly on the time scale comparable to the contact relaxation time. In the electron density plot, one can observe the initial depletion near the cathode side, and its subsequent population during the transient........... 69
3.7 Current density versus time for the nin diode from Fig. 3.6 upon the application of - mV to the left contact. Jdev plotted here refers to the device current from Eq. (3.5d). The transient time is relatively long, owing to small τ.................. 69
3.8 The influence of the injection into the localized device states due to scattering on the potential and device current transient evolution. The upper left panel, the upper right panel, and the lower left panel show the initial time-dependence of the device potential profile for three different scattering enhanced injection rates, ∆scatt = 1/τ , ∆scatt = 10^12 s−^1 , and ∆scatt = 0, respectively. The nin diode used here has the same parameters as before. We see that with larger localized states injection rates the quan- tum well bottom is rising faster. However this initial difference in the quantum well bottom height does not affect the barrier height too much and therefore the transient current is very similar for different ∆scatt as shown in the lower right panel. What does get affected is the initial difference between Jcon and Jdev, equal to the change of electron charge over time dQ/dt............................. 71
3.9 Comparison of current densities for the same nin diode used in the previous figures and for variable contact relaxation time τ. Each curve is shown just until the moment it saturates, to save the computational time. Having less scattering in the contacts helps to reach steady state faster. This phenomenon was discussed in connection to Fig. 3.1, where the same conclusion was reached using the low current limit and solving Eqs. (3.5c) analytically. If the current were limited by the scattering in the contacts, then one would expect that more scattering helps to establish steady state faster. However, in our model, the current is controlled by the ballistic device’s potential profile. The device requests some current from the contacts, which act as charge reservoirs, but the contact ability to supply the charge of a given momentum will be limited due to momentum randomization in the presence of scattering................. 72
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Figure Page
4.1 (a) The 4-point scheme for discretization of the first partial derivatives by using Eq. (4.9). (b) An example of a curved surface parametrized by curvilinear coordinates u^1 and u^2. Leads are planar continuations of the curved surface in both directions along the longitudinal coordinate u^1. In order to calculate the wave function value at the grid point denoted by the large solid circle (central point) from the 4-point scheme, we have to know the metric tensor values at all the surrounding points, denoted by the small solid circles, as well as at the central point..................... 80
4.2 An example of a curved nanoribbon for calculating the magnetic vector potential: the nanoribbon (dark blue) on top of a half-torus (light blue), with the leads extending downward at both ends. The Cartesian coordinate system in the left lead (shown in the lower left corner) is taken as a reference system, according to which the magnetic field components Bx, By, and Bz are specified. Gauge transformations (GT), equiva- lent to coordinate system translations and rotations in the plane perpendicular to each component, are also shown................................ 87
4.3 (a) Normalized conductance vs. magnetic field for a nanoribbon with helicity in cylin- drical geometry. The leads are assumed cylindrical to avoid the need for transitional regions and as a consequence have the Landau levels energies affected by the mag- netic field flux in the leads (Landau level quantization), albeit weakly. The lead on the left-hand side is the injecting lead. Magnetic field direction with respect to the structure is shown in (b) and would be parallel to the leads if they were exactly planar. Even at zero field, not all of the three injected modes propagate through, due to the ribbon helicity. (b) Electron density at the Fermi level shown for By=2.663 T on the log 10 scale, where there is resonant transmission, and for By=4.5 T, where the normal- ized conductance is zero (transmission suppressed). A quasibound state [right panel of (b)] is associated with the resonant transmission feature from (a)........... 96
x
Figure Page
4.7 (a) Normalized conductance vs. Fermi level in the leads for a nanoribbon in toroidal geometry with and without helicity, in the absence of magnetic field. The leads are assumed cylindrical to avoid the need for transitional regions. The number of injected modes is changing due to the change in the Fermi energy. Comparison of the con- ductance for these two nanoribbons shows that helicity is the reason for the observed difference between the actual normalized conductance and the number of propagating modes. (b) Normalized electron density at the Fermi level, represented through color (white – high, black – low) for the nanoribbons with (right panel) and without helicity (left panel) when the Fermi level energy is equal to 13.5 meV.............. 100
A.1 Flowchart of the numerical algorithm for the calculation of the electronic response of a biased two-terminal nanostructure during a transient.................. 118
xi
ABSTRACT
This thesis covers aspects of the theory and simulation of small electronic semiconductor
nanostructures. It addresses the transient regime and unconventional geometries, and the devel-
opment of simulation algorithms and methodology to study them computationally. The thesis is
divided into two themes: in Theme 1, we investigate the quantum electronic transport properties
in the transient regime, while in Theme 2 the quantum electronic transport properties in nanostruc-
tures with unconventional geometries in a magnetic field. Both of these topics can lead to potential
useful applications. Knowing the properties of the transient regime enables such device modeling
that can lead to faster devices. Faster devices are needed not only in digital electronics, where
the speed today is mostly limited by the dissipation, but also in high frequency signal generation,
detection and manipulation in the area of sensing and telecommunications. On the other hand,
nanostructures with unconventional, curved geometries are gaining popularity today, with various
experimental groups producing one-dimensional and two-dimensional nonplanar nanostructures.
It is reasonable to expect that curvature, potentially coupled with an external magnetic field, can
have large effects on the coherent transport properties. Therefore, it may be possible to control and
tune the conductance using mechanic or mechano-magnetic means.
Theme 1 starts by overviewing the most important quantum master equations, which give the
time-dependence of the distribution functions fk in the nanostructure. Since every nanostructure
with current flow is coupled to and depend on its environment, it is treated by using an open system
formalism. The major objective is to reduce the extremely complicated equations involving many
particles/variables to a system which is analytically or numerically managable. The reduction is
xiii
the so called Usuki stabilized transfer matrix method, to obtain the conductance and electron den-
sity in the linear transport regime. This method of solving the tight-binding Schr¨odinger equation,
which starts from one contact and integrates the equation through the curved nanoribbon to the
other contact, introduces a preferable direction which affects the choice of gauge that gives rea-
sonable physical results. We devise a type of a local Landau gauge in order to compute A from the
given magnetic field B. By applying this method to several curved nanoribbons in cylindrical and
toroidal geometries, with or without helicity, we observe several interesting features in the con-
ductance, like conduction supression, similar to the quantum point contact case when the gate bias
is varied, and resonant reflections, previously observed in non-uniform, planar 2D nanostructures,
such as various cavities.
Chapter 1
Introduction
This thesis covers two themes, each dealing with novel aspects of quantum electronic transport.
Theme 1 is the study of the transient behavior in nanoscale electronic devices. Devices, considered
here to be ballistic due to small dimensions, are connected to macroscopic contacts (reservoirs
of charge) that have efficient scattering mechanisms. The novelty in this work is that the tran-
sient regime is studied using a quantum-mechanical open system formalism, which is necessary to
address the mainly quantum-mechanical nature of transport in nanoscale electronic devices. The
main results of this part of the thesis are the calculations of the transient behavior of charge den-
sity, current density, and potential, in response to a sudden change in bias applied to the contacts.
Theme 2 is the study of the influence of curvature on the steady-state transport in nanoscale ballis-
tic electronic devices in a magnetic field. In particular, we are interested in the charge density and
conductance in the linear transport regime in curved devices, where the curvature can be cylindri-
cal, toroidal, helical, or in principle arbitrary. The curvature can have strong effects on the coherent
conduction properties of non-planar nanoribbons, especially in the presence of magnetic fields.
solid state optoelectronics
electronics THz
Figure 1.2 Electromagnetic spectrum with the THz frequency window indicated with vertical red
lines (300 MHz - 3 THz). Current optoelectronic devices, like quantum cascade lasers, are the
most efficient at the frequencies above, while solid state devices at the frequencies below the THz
window. Therefore, there is a need for increasing the performances of both types of devices, in
order to be able to cover the THz window with efficient sources, detectors, mixers, and other
types of devices. This work focuses on the lower-frequency-side devices, specifically modeling
the transient response to a sudden turn on of bias in nanoscale ballistic devices. Source:
http://www.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html.
Figure 1.3 A resonant tunneling diode (RTD) oscillator. The highest oscillating frequency
achieved with this device is around 1 .1 THz, as indicated by the blue short line in the right panel.
The photograph of the physical layout of the device is shown in the left panel Reprinted with
permission from [M. Feiginov, C. Sydlo, O. Cojocari, and P. Meissner, Appl. Phys. Lett. 99 ,
233506 (2011)]. c⃝[2011], American Institute of Physics.
The motivation to study curved devices is based on the fact that curvature alone, or coupled
with a magnetic field, is a new and potentially useful mechanical degree of freedom in electronic
device design. Such curved nanostructures can strongly affect the coherent electronic transport
through them. In addition, various curved nanostructures are readily produced today, with one
interesting example shown in Fig. 1.4.
