Download The non-equilibrium Green`s function method

The non-equilibrium Green’s function method: an introduction
P. Vogl
Walter Schottky Institute, Technische Universität München, Am Coulombwall 3, 85748 Garching,
Germany
and T. Kubis
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Abstract
We present an elementary introduction of the non-equilibrium Green’s function method, applied to
stationary quantum transport in semiconductor nanostructures. This article provides an overview of the
strengths and weaknesses of the method.
1. Introduction
A proper treatment of quantum transport is one of the most difficult problems to deal with in solid state
theory. While there have been many models and concepts developed to deal with particular aspects of
quantum transport, the most general and rigorous theoretical framework is provided by the so-called
non-equilibrium Green’s function theory (NEGF) developed by Keldysh 1964 and,1 in a slightly different
form, independently by Kadanoff and Baym.2 It took an amazing 40 years before this method became
recognized and employed as the framework of choice for a quantitative and predictive analysis of carrier
dynamics in semiconductor based nanostructures.3 -8 There are probably several reasons why it took so
long. First if all, there are only few devices where quantum mechanical effects and incoherent scattering
effects play an equally important role and call for a fully quantum mechanical treatment. The most
prominent examples are quantum cascade lasers (QCL) invented by Capasso and coworkers in 2004.9
Particularly in the THz regime, the physics of QCLs is controlled by a carefully balanced competition
between coherent tunneling and incoherent phonon emission processes.10-13 Secondly, NEGF
calculations for realistic devices are extremely demanding computationally and have only become
feasible recently.10-16
One of the first detailed implementations of NEGF to semiconductors and semiconductor devices has
been developed by Lake and coworkers and applied to NEMO-1D, a sophisticated simulation tool for
multi-quantum well devices such as resonant tunneling diodes.8 In this paper, the authors laid down the
framework for applying NEGF to semiconductors by deriving most scattering self-energies that are
relevant for semiconductors. Another major step forward was provided by Wacker et al, who provided
several in-depth NEGF studies of the carrier dynamics and optical properties of QCLs.7
Currently, an increasing number of groups are employing and expanding the NEGF formalism to study
quantum transport aspects of modern semiconductor nanostructures.10-16 Nevertheless, the formalism
still suffers from being considered somewhat obscure and difficult to grasp and handle. In this paper,
we will attempt to give an overview of the basic elements of the formalism for nonexperts that may help
to develop a better feeling for strengths and weaknesses of NEGF.
Let us emphasize that NEGF is a solution framework rather than a concrete method for calculating
properties of open quantum devices. It ensures that the non-equilibrium carrier distribution in a device
is consistently calculated with the energy, width and occupancy of its quantum mechanical eigenstates
(scattering states, to be precise). Conservation of charge, momentum and energy is guaranteed only if
the scattering self energies are calculated exactly and self-consistently, i.e. using fully dressed Green’s
functions and vertices in many-body terms.13,17,18 Thus, the unavoidable use of approximations requires
much more effort and care than in other methods just to avoid artifacts such as a violation of current
conservation within the device.
The NEGF method is able to deal with explicitly time-dependent as well as with stationary problems.
However, the calculation of the required set of 4 types of Green’s functions for time-dependent carrier
dynamics is still pretty much out of reach for a quantitative prediction of realistic device structures.
Therefore, we limit the present discussion to stationary problems where 2 types of Green’s functions
suffice.
2. The basic quantities in stationary transport
To keep things simple, we consider here laterally homogeneous, and therefore effectively onedimensional, open quantum nano-devices under stationary bias. We envision the system to be in
contact with very large equilibrium carrier reservoirs via semi-infinite leads to the left and to the right of
the active device that is characterized by a stationary carrier density
and current density
. Since
there is a current flowing across the open device, the carrier distribution will deviate from equilibrium
and cannot be characterized by local or global Fermi levels. Technically, the NEGF formalism captures
this situation by 2 Green’s functions.3-8 The poles of one of them, the so-called retarded Green’s function
,
(1)
yield the energy and width of the electronic scattering states of the system. They may be resonances of
finite width or bound states. The open quantum system is characterized by a Hermitian part
that
contains the kinetic energy and the static potential containing band offsets and the electrostatic Hartree
potential, and a non-Hermitian part that provides energy and momentum dissipation and coupling to
the leads. The non-equilibrium occupancy of these scattering states that results from the incoming and
outgoing carrier states and from scattering processes is described by the so-called lesser Green’s
function
that determines the carrier density and current density,
∫
(2)
∫
(
)
When no scattering processes within the device take place, the last equation reduces to the well-known
Landauer formula,4
∫
(3)
To make this a bit more concrete, consider a laterally and vertically homogeneous, infinitely extended
scattering-free (i.e. ballistic) semiconductor device in a one-band approximation, characterized by an
effective mass
. In this case,
(
)
where the energy is decomposed into a parallel and vertical part,
(4)
(
)
and
is
the Fermi distribution function.
3. Boundary conditions
In quantum transport, the treatment of boundary conditions requires significantly more care than in
classical physics due to the nonlocality of quantum mechanics.19-22 A common problem is to define
Ohmic leads. In the context of NEGF, we may define leads as Ohmic if the current is controlled by the
interior of the device rather than by serial resistances or interface states. This definition has several
implications. First, there must be a smooth transition in the density of states of the leads and the device.
Secondly, the same scattering mechanisms must act within the leads and the device. Thirdly, the carrier
distribution within the leads must be a suitably accelerated Fermi distribution to reflect current
conservation. All of these conditions are necessary to avoid quantum mechanical reflections and pile up
of charge at the interface.23,24 This is illustrated in Figure 1 (a) and (b). Figure 1 (a) shows a full NEGF
calculation of the carrier density in a homogeneously n-doped piece of GaAs for zero bias as a function
of position. Zero bias implies the quantum mechanical current from left to right to equal the current
from right to left. If the leads are treated ballistically, the carriers accumulate near the device
boundaries due to the resistance they meet within the medium. Only by employing the same scattering
mechanisms and self-energies everywhere, the formalism yields a constant carrier density throughout
the system. Figure 1 (b) illustrates the calculated carrier density in a biased n++-i-n+ GaAs structure. If
the carrier distribution within the leads is assumed to be in equilibrium, one obtains an artificial pinchoff behavior; there is a depletion of carriers near the source side and an accumulation near the drain
side of the device.23,24
4. The semiclassical limit
Does the NEGF formalism reduce to the semiclassical limit when quantum effects play no role? 25,26 For a
realistic device structure, this is difficult to prove rigorously but the following example provides practical
evidence that, indeed, the answer is affirmative.
Figure 1: Left: Calculated carrier density for a homogeneously doped semiconductor at zero bias, attached to leads
with an equilibrium distribution. The expected homogeneous density is obtained only if the leads include the same
type of scattering self energies as the device (red curve), otherwise one obtains artificial charge accumulation (blue
curve).Right: GaAs n-i-n structure at room temperature under bias with asymmetric doping profile as indicated by
the grey lines. Once a current is flowing, the charge distribution within the leads must be a suitably shifted Fermi
distribution that reflects global current conservation (results shown by the red curve). Otherwise, NEGF
calculations yield artificial pinch-off effects (blue curve).
Consider a symmetric n-i-n GaAs diode at room temperature and for zero bias. Since such a device
contains neither quantum wells nor barriers, one expects the semi-classical Boltzmann equation to
adequately describe the carrier density, assuming one includes impurity and phonon scattering in the
standard fashion and includes electron-electron scattering at least within the Hartree approximation via
the Poisson equation. We have performed a charge self-consistent NEGF calculation that takes into
account the same type of scattering mechanisms. As shown in Figure 2, the NEGF electron density
mimics faithfully the semiclassical results.
Figure 2: Comparison of fully self-consistent NEGF and
semiclassical carrier dynamics calculations for a standard
GaAs resistor (50 nm n-i-n structure) at zero bias and
room temperature.
5. Quantum well structures: Why NEGF is essential
A nice illustration of the power of NEGF can be obtained by calculating the local carrier distribution in a
biased n-i-n structure that contains a quantum well. This is illustrated in Figure 3 for a n-i-n structure
with a 12 nm InGaAs quantum well in the intrinsic zone. The applied voltage across the 50 nm structure
is 150 mV and we show 3 results. The red curve shows the semiclassical Boltzmann results and yields the
well-known accumulation of charge near the quantum well barriers. This method completely ignores the
existence of quantum mechanical bound states and reflects the classical Thomas-Fermi density. The blue
curve illustrates another limiting case, namely the solution of the Schrödinger equation in the absence
of any scattering. This reflects a strictly coherent, energy conserving, ballistic transport. In this case, the
carrier density within the device is fully determined by the overlap of the lead wave functions with the
device. Since there are no lead carriers below the GaAs band edge, the quantum well states in the
intrinsic region remain unoccupied. Thus, the electron density within the quantum well only stems from
(continuum state type) lead electrons which explains the oscillatory density in the intrinsic region.
Finally, the black curve represents the full NEGF calculation. The inelastic scattering processes lead to a
capture of carriers into the quantum well states and lead to a carrier density in the intrinsic zone that
lies in between the semiclassical and the strictly ballistic quantum mechanical calculations.
Figure 3: Carrier dynamics calculation for 50 nm n-i-n
structure at room temperature with a 12 nm InGaAs
quantum well as intrinsic zone attached to field-free
GaAs leads of the same n-density. The applied voltage is
150 mV across the structure. Red curve: calculation in
terms of charge-self-consistent semiclassical Boltzmann
equation. Blue curve: Calculation in terms of strictly
ballistic NEGF, equivalent to the solution of Schrödinger
equation of open system. Black curve: Fully selfconsistent NEGF calculation.
This result can be further illustrated by plotting the energy resolved density
∫
(
)
(5)
for the case of zero bias. Figure 4(a) depicts this density for a strictly ballistic calculation that actually
represents a NEGF calculation in the limit of all impurity, phonon, and interface scattering self energies
set equal to zero. Note that the Poisson equation is still solved self-consistently with the Schrödinger
equation even in this case. Figure 4(b), on the other hand, shows a complete NEGF calculation that
clearly illustrates the carrier capture into the first and second bound state of the quantum well. Due to
the charging effects caused by the capture, the bottom of the quantum well raises in energy so that
both states actually form resonances that slightly overlap with the lead states.
Figure 4: Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room
temperature with a 12 nm InGaAs quantum well as intrinsic zone for zero applied bias. The density scale is the
analogous to the one in Fig. 5, but for lower doping. (a) Strictly ballistic NEGF calculation (no scattering included).
(b) Fully self-consistent NEGF calculation.
Figure 5: Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room
temperature with a 12 nm InGaAs quantum well as intrinsic zone for zero applied bias. (a) Fully self-consistent
NEGF calculation. (b) NEGF calculation with coupling between the retarded and lesser Green’s functions artificially
turned off. This leads to an occupancy f(E) of the lowest bound state exceeding the Pauli limit f=1.
6. Limits and simplifications in NEGF
As mentioned in the introduction, the NEGF formalism guarantees conservation of important physical
principles only if all Green’s functions are evaluated exactly from a many-body standpoint. It is a very
significant weakness of the method that even plausible approximations can fail badly. Generally, it is
difficult to introduce simplications that do not violate basic conservation laws. As an example, we
discuss the so-called decoupling approximation.8 The NEGF formalism couples the energy of states with
their occupation via 4 coupled integro-differential equations. If the carrier density is not too high, it
seems plausible to decouple the equations of
and . This approximation can lead to a violation of
Pauli’s principle, however, since there is no mechanism that prevents the occurrence of over-occupied
states (i.e. states with occupancy higher than permitted by the Pauli principle).8 To exemplify this
situation, we consider the same GaAs-InGaAs-GaAs n-i-n structure as before, but with a slightly higher
carrier concentration in the n-region. Figure 5(a) shows the energy resolved carrier density of the n-i-n
structure for zero bias, as calculated by the full NEGF approach. By contrast, Figure 5(b) shows the result
of the decoupling of Green’s functions, leading to unphysically high occupation of the lowest bound
state.
7. RTD’s and inelastic scattering
We now turn to applications that illustrate the insight one can gain with NEGF calculations of quantum
devices.
We consider a 40 nm n-i-n structure where the central intrinsic zone is a resonant tunneling diode (RTD)
with two 5 nm thick AlGaAs barriers and a 5 nm GaAs well in between (see Figure 6). The two GaAs nregions are lightly doped (n = 2×1017 cm-3). In Figure 6, we show the typical N-shaped current-voltage
characteristics of this RTD, based on three different calculations.
Figure 6: Calculated current-voltage characteristics of
a resonant tunneling diode as shown in the inset.
Green curve: Strictly ballistic calculation (no
scattering). Red curve: NEGF calculation with
inelastic scattering turned off. Black curve: Fully selfconsistent NEGF calculation.
The green curve is a strictly ballistic calculation that neglects any kind of incoherent scattering. The
corresponding contour plot of the energy resolved electron density in Figure 7(a) reveals that, in this
case, the carriers shoot ballistically out of the left contact and hit the barriers. The current density peaks
near a voltage of 1.8 V when the Fermi level is aligned with the lowest quantum well resonance. Note
that this current density is controlled by density of states of the left contact. The red curve in Figure 6
corresponds to a NEGF calculation that does include elastic scattering, but neglects inelastic phonon
scattering. The corresponding I-V characteristics is very similar to the ballistic case. As we will show now,
both the green and red results are physically incorrect.
The black curve in Figure 6 depicts the full NEGF result where both elastic and inelastic scattering is
taken into account. The I-V curve peaks at a higher voltage this time, near 2.1 V that is, and the
maximum is much broader than in the elastic case. The reasons for this result can be deduced from
Figure 7(b). The carriers do not fly ballistically from the contact to the barriers but get captured by
inelastic scattering into the bound state formed by the triangular shaped potential in front of the left
barrier. Since the energy of this state is lower than the contact Fermi energy, it takes a higher voltage
before this bound state gets aligned with the quantum well resonance. This explains the shift of the
maximum current towards a higher voltage. Importantly, the shape and magnitude of the current
maximum is controlled by the density of states of this bound state which provides the initial state for
resonant tunneling.
Figure 7: Contour graph of the energy and position resolved carrier density in the resonant tunneling diode of
Figure 6 for an applied bias of 200 mV across the 40 nm structure. (a) Strictly ballistic NEGF calculation (no
scattering). (b) Fully self-consistent NEGF calculation.
This example shows that a resonant tunneling diode cannot be properly understood in terms of a model
that neglects inelastic scattering. While inelastic scattering is unimportant within the quantum well, it
determines the initial state and therefore the shape of the resonant tunneling current-voltage
characteristics.
8. THz quantum cascade lasers: a classics for NEGF
An important question that we have not addressed so far is whether fully self-consistent NEGF
calculations actually agree with experiment. We have applied this formalism to GaAs/AlGaAs THz QCLs
and included impurity, phonon, interface roughness scattering in the self-consistent Born
approximation. In addition, the electron-electron scattering has been included both in the Hartree
approximation as well as within the so-called GW approximation. For details, we refer to Ref. 13.
Importantly, the calculations contain no fitting parameter. Figure 8 depicts the calculated currentvoltage characteristics of such a QCL for a particular sheet doping density, together with experimental
data.27 As one see, theory and experiment agree very well with one another up to the voltage where
lasing starts and both thermal as well as hot electron effects become relevant that have not been
included in the calculations.
Figure 8: Comparison between experimental (Ref. 27) and
calculated (Ref. 13) current-voltage characteristics for
AlGaAs/GaAs quantum cascade structure.
9. Conclusions
The NEGF formalism provides the framework of choice for consistent carrier dynamics calculations of
open nanosystems where quantum effects and incoherent scattering play a comparable role. When
implemented with care, it reproduces the results of the semiclassical Boltzmann equation in the limit
where quantum effects such as resonant tunneling and interference are unimportant. By definition, it
also reproduces the solutions of the Schrödinger or Lippmann-Schwinger equation when inelastic
scattering is turned off.
A disadvantage of the method is the fact that charge and current conservation, and even Pauli’s
principle are not automatically satisfied once seemingly reasonable approximations are introduced.
Scattering vertices must necessarily be taken into account to infinite order, for example, to strictly obey
current conservation and it is difficult to achieve a numerically satisfactory convergence.
Approximations are unavoidable, though, once one seeks predictions for realistic nano-devices, simply
due to the excessive numerical effort required to solve the full set of Keldysh equations self-consistently.
In fact, it will take some time before quantitative NEGF solutions for time-dependent quantum transport
calculations become numerically feasible.
In comparison with semiclassical calculations, much more effort is required to properly take into
account the physics of contacts and the contact-device coupling. This is a consequence of the nonlocal
nature of quantum mechanics and the nature of scattering solutions in open quantum systems.
Acknowledgements
This work has been supported by the Austrian Scientific Fund FWF (SFB-IRON), the Deutsche
Forschungsgemeinschaft (SFB 631 and SPP 1285), and the Excellence Cluster Nanosystems Initiative
Munich.
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