V4
Quantum Transport in Nanostructures
Thomas Schäpers
Peter Grünberg Institute (PGI-9), and
JARA-Fundamentals of Future Information Technology
Research Center Jülich, 52425 Jülich, Germany
Contents
1
Introduction
2
2
Fabrication of nanostructures
2.1 Heterostructures and two-dimensional electron gases . . . . . . . . . . . . . .
2.2 Topological insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 One-dimensional structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
3
4
3
Characteristic length scales
3.1 Elastic mean free path .
3.2 Inelastic mean free path
3.3 Phase coherence length
3.4 Transport regimes . . .
3.5 Magnetic length . . . .
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Diffusive transport
4.1 Classical diffusive motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Einstein relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Typical transport parameters of a low-dimensional semiconductor structures
4.4 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Weak antilocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Universal conductance fluctuations . . . . . . . . . . . . . . . . . . . . . .
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V4.2
1
Thomas Schäpers
Introduction
If the dimensions of semiconductor structures shrink to only a few tenths of nanometers, very
interesting phenomena in the electron transport can be observed. Most of these phenomena rely
on the wave nature of the electrons and the quantization of energy levels. This implies that
mostly low temperatures are required for their observation. Here, a brief overview about the
transport phenomena in nanostructures is given. In the following section, it will be explained
how the electron transport can be restricted to two or one dimensions. As we will see later,
the dimensionality of the transport can only classified properly, if the relevant length scales are
known. Therefore in Sect. 3, the meaning of the elastic mean free path, the phase coherence
length, and other important length scales will be explained in detail. In Sec. 4 the diffusive
transport regime is discussed with an emphasis on electron interference effects, i.e. weak localization and universal conductance fluctuations.
Of course, here only a brief overview on the transport in nanostructures can be given. For
further studies the text books of Th. Ihn [1], Th. Heinzel [2] and S. Datta [3] are recommended.
More detailed and specific information on transport in mesoscopic structures can be found in
the review article of Beenakker and van Houten [4].
2
Fabrication of nanostructures
Nanoelectronic structures with reduced dimensionality can be fabricated by using different approaches. With respect to semiconductors very often a two-dimensional electron gas in a semiconductor heterostructure is used as a basis for the realization one- or two-dimensional structures. These structures are prepared by employing a so-called top-down approach, where the
samples are defined by lithographical means and various metallization and etching steps. Generally, one observes the trend that for structures being as small as a few tenth of nanometers
a very complex and cost extensive processing technology is required. As an alternative a different approach, the so-called bottom-up approach, has gained a lot of interest, recently. Here,
the nano-scaled structures are fabricated directly, without using the very advanced patterning
techniques mentioned above [5, 6, 7, 8].
In this section, first, the properties of two-dimensional electron gases (2DEG) will be introduced. Subsequently, one-dimensional structures will be described, which are prepared by a
top-down as well as by a bottom-up approach.
2.1
Heterostructures and two-dimensional electron gases
Many samples used for studies on semiconductor nanostructures are based on two-dimensional
electron gases (2DEG) in heterostructures [9]. Here, an n-doped semiconductor layer with a
larger bandgap, e.g. Al0.3 Ga0.7 As, is grown epitaxially on a semiconductor layer with a lower
Quantum Transport in Semiconductor Nanostructures
V4.3
Fig. 1: a) TBE:Band profile of a GaAs/AlGaAs and an InP/InGaAs heterostructure. In the
GaAs/AlGaAs heterostructure the two-dimensional electron gas is found in a triangular potential well. b) For the alternative material system InP/In0.77 Ga0.23 As/In0.53 Ga0.47 As the twodimensional electron gas is formed in the strained In0.77 Ga0.23 As layer. (Figure taken from [11])
band gap, e.g. GaAs [10]. Due to the adjustment of the Fermi level, electrons are transferred
from the n-doped layer into the other semiconductor with the lower band gap. The band bending
due to the band offset of both materials takes care that these electrons are only found at the
interface of both layers. The electrons are trapped in a potential well. This is situation is
depicted in Fig. 1 (a).
The electrons can only move freely along the interface, therefore these structures are called
two-dimensional electron gases. In order to suppress impurity scattering, the impurities are
separated by a spacer layer from the two-dimensional electron gas (modulation doping). Beside the AlGaAs/GaAs material system it is also possible to realize a 2DEG in a InGaAs/InP
heterostructure, as shown in Fig. 1 (b) [12]. At an indium content of 53% InGaAs is lattice
matched to InP. A considerable improvement with regard to higher mobility and electron concentration is achieved if the 2DEG is realized in a 10 nm thick strained InGaAs layer with an
indium content of 77%.
2.2
Topological insulators
In the previous section we discussed the formation of a two-dimensional electron gas at a heterointerface. Recently, a new class of materials emerged, so-called 3-dimensional topological
insulators, where a two-dimensional electron gas is formed at the surface [13, 14]. Typical
materials are Bi2 Te3 or Sb2 Te3 . Layers of topological insulators can be grown by means of
molecular beam epitaxy. As illustrated in Fig. 2, the material is insulating within the bulk. At
the surface helical states are present. Helical means that the spin orientation is locked to the
direction of propagation. The underlying reason for the formation of these surface state is the
band inversion, i.e. the p-type band, which is usually the valence band, is energetically located
V4.4
Thomas Schäpers
above the s-type band, being usually the conduction band. The band inversion is due to the
very strong spin-orbit coupling, leading to a large spin-orbit splitting of the p-type bands. The
surface states are formed because at the interface to a normal insulator (vacuum) or a semiconductor, the bands of a certain type, e.g. s- or p-type have to merge. Because the order of bands
in ”upside down” in the topological insulator, this is only possible by forming interface states
crossing the energy gap of both materials. As long as the Fermi level is located within the bulk
energy gap only the two-dimensional surface states conduct.
Fig. 2: Schematics of the band structure of a topological insulator. The surface states are
located within the band gap.
2.3
One-dimensional structures
The transport in a two-dimensional electron gas can be restricted to only one-dimension by
employing electron beam lithography and wet chemical or dry etching. A typical example of an
InGaAs/InP quantum wire is shown in Fig. 3. Here, first a metallic etching mask was defined by
electron beam lithography, while subsequently the semiconductor wire was shaped by means of
wet chemical etching. Alternatively, a one-dimensional channel can also be obtained by using
Fig. 3: Scanning electron micrograph of an InGaAs/InP quantum wire.
split-gates. Here, a negative gate voltage applied to a pair of adjacent gates, which are separated
Quantum Transport in Semiconductor Nanostructures
V4.5
by a small gap, is used to deplete the two-dimensional electron gas underneath. The carriers
only remain in the area between both gates.
As mentioned above, nanoscaled wire structures can also fabricated directly by means of a
bottom-up approach using epitaxial methods. In the vapor-liquid-solid growth mode, small
metallic droplets placed on a semiconductor substrate are taken as a seed to initiate the growth
of semiconductor nanowires [5, 6, 8]. Alternatively, nanowires can be grown by means of
selective-area metal-organic vapor phase epitaxy [15]. Here, a substrate consisting of a semiconductor wafer covered with a hole-patterned SiO2 layer is employed. In Fig. 4 (a) a scanning
electron micrograph of InAs nanowires grown by this method are shown [15]. In order to allow
transport measurements, the wires are detached from the substrate and placed on a SiO2 -covered
Si wafer. After determining the positions of the randomly placed wires by using a scanning electron microscope, the wires are contacted individually by employing electron beam lithography.
An example of a contacted InAs nanowire is shown in Fig. 4 (b) [15].
Fig. 4: Scanning electron micrograph of: (a) as-grown InAs nanowires, (b) of an InAs nanowire
with the individually defined Ti/Au contact fingers [15]
In InAs nanowires the Fermi-level pinning at the surface within the conduction band leads to a
downwards bending of the conduction band Ec at the surface (see Fig. 5). As a result, electrons
are accumulated here, forming a 2-dimensional electron gas. This mechanism ensures that even
at very small diameters of a few tenths of nanometers the nanowire is conductive.
3
Characteristic length scales
Nanoelectronic systems can be classified by relating its size to specific characteristic length
scales. These length scales determine in which fashion the carriers propagate through the conductor. Below we will introduce first the elastic and inelastic mean free path which result from
the scattering processes occurring in the sample. A length scale which gives information about
the loss of phase memory is the phase coherence length. In case of a magnetic field applied
the bending of the electron trajectories due to the Lorentz force defines the so-called magnetic
length.
V4.6
Thomas Schäpers
Fig. 5: Surface accumulation layer in an InAs nanowires forming a 2-dimensional electron
gas. Here, EF is the Fermi level while Ec and Ev are the conduction and valence band edges,
respectively.
3.1
Elastic mean free path
The elastic mean free path le is a measure for the distance between two elastic scattering events.
These scattering events occur due to the fact that the conductor is not an ideal conductor but
rather contains irregularities in the lattice, e.g. impurities or dislocations. The scattering events
are considered to be elastic which means that the electron does not alter its energy. A typical
example is the scattering of an electron at a charged impurity. Due to the large difference of the
masses of the scattering partners effectively no energy is transferred from the electron during
the scattering event, whereas its momentum can change largely. The elastic mean free path is
defined in that way that it is assumed that the movement of the electron in the initial direction
is suppressed completely. In some cases many scattering events are necessary to fulfill this
condition. The scattering mechanism are sometimes classified as large angle and small angle
scattering. The elastic mean free path le can be calculated from the scattering time τe between
successive scattering events
le = τe vF ,
(1)
where vF = ~kF /m∗ is the Fermi velocity, with kF the Fermi wave number and m∗ the effective
electron mass. For semiconductors τe can be obtained from the electron mobility
µe = eτe /m∗ .1
3.2
(2)
Inelastic mean free path
A further effect beside the above discussed static irregularities are the non-stationary scattering
events. A typical example are lattice vibrations, in the quantum picture phonons. An electron
1
The electron mobility is a measure for the increase of the drift velocity vd if an electric field E is a applied
vd = µe E.
Quantum Transport in Semiconductor Nanostructures
V4.7
moving within a crystal will be scattered by these lattice vibrations. On the other hand, a
moving electron can excite lattice vibrations and loose a certain amount of its energy. Since
an energy transfer occurs, these scattering events are considered to be inelastic. Again we can
define an inelastic scattering length lin as a measure for the length between scattering events.
Beside electron-phonon scattering, electron–electron interaction is another process, where a
considerable amount of energy is exchanged between both scattering partners.
3.3
Phase coherence length
Another length scale which is of importance for mesoscopic systems is the phase coherence
length lϕ . This parameter is a measure for the distance the electron travels before its phase is
randomized. By elastic scattering events, with a static scattering center, the phase of an electron
is usually not randomized. This does not mean that the electron phase is not modified by the
scattering event. The crucial point is that the phase is shifted by exactly the same amount if the
electron would travel the same path a second time. This is in strong contrast to inelastic scatting
events, e.g. electron-phonon scattering, where the scattering target changes with time. Here,
the phase shift an electron would acquire is different each time, since the scattering mechanism
is statistically in space and time. However, one must be careful to directly identify lϕ with lin
since they are not always identical, e.g. spin-flip scattering is considered to be phase-breaking
while it can be elastic at the same time.
3.4
Transport regimes
By comparing the definitions given above with the dimension L of the sample and the Fermi
wavelength λF different transport regimes can be classified. For the case that the elastic mean
free path le is smaller than the dimensions L of the sample (le < L), many elastic scattering
events occur. The carriers are travelling randomly, diffusively through the crystal (Fig. 6). If the
phase coherence length lϕ is shorter than the elastic mean free path (lϕ < le ), the transport can
be classified as classically. In contrast, if lϕ > le , quantum effects due to the wave nature of the
electrons can be expected (Table 1). This diffusive regime is thus called quantum regime. In
case that le is larger than the dimensions of the sample the electrons can transverse the system
without any scattering. This regime is called ballistic (Fig. 6). Depending on the magnitude of
the Fermi wavelength λF = 2π/kF in comparison to the dimension of the sample the transport
can either be regarded as classical ballistic or quantum ballistic (Table 1).
3.5
Magnetic length
In a magnetic field electrons are deflected by the Lorentz force, which is perpendicular to the
magnetic field B and the velocity of the electrons. Due to the magnetic field free electrons
will travel along a circle. The radius of this circle, the so-called cyclotron radius rc , can be
V4.8
Thomas Schäpers
Classical
le L, lϕ < le
Quantum
le L, lϕ > le
Classical
λF L < lϕ , le
Quantum
λF , L < le < lϕ
Diffusive
Ballistic
Table 1: Comparison of the different transport regimes.
Fig. 6: Illustration of the diffusive and ballistic transport regime.
Quantum Transport in Semiconductor Nanostructures
V4.9
calculated from the balance between the Lorentz force and the centrifugal force resulting in
rc =
m∗ vF
.
eB
(3)
The transport properties of a mesoscopic system can change significantly if rc approaches the
dimensions of the sample.
Especially, if quantum effects are considered it is sometimes convenient to define the magnetic
length
r
~
lm =
(4)
eB
2
so that rc = kF lm
. From the cyclotron radius we can further deduce the cyclotron frequency
ωc =
4
eB
vF
= ∗ .
rc
m
(5)
Diffusive transport
As introduced above, transport is considered to be diffusive if the elastic mean free path le
is much smaller than the dimensions of the structure. The carriers are propagating randomly
through the structure. In the classical diffusive transport regime, which will be discussed in the
first two sections the phase coherence length lϕ is also much smaller than the dimensions of the
sample, so that any interference effects can be neglected. In the quantum limit the phase coherence length exceeds the elastic mean free path. Here, even in large scale samples the electron
interference effects can lead to additional contributions to the resistance, i.e. localization effects
or conductance fluctuations.
4.1
Classical diffusive motion
First, we will discuss the classical transport, where the contribution of the electron phase is
neglected. In the presence of an electric field the electrons acquire a drift velocity
< v(t)> = −eEt/m∗ .
(6)
On average, after a time τe the electrons will be scattered so that the resulting average velocity,
the drift velocity, can be written as
vdrif t ≡< v(τe )> = −
eτe
E.
m∗
(7)
By using this expression the electron mobility µe of the electrons can be defined
µe =
eτe
,
m∗
(8)
V4.10
Thomas Schäpers
which quantifies how the average velocity of the electrons depend on the applied electric field
E. The electrical current density j connected to the drift velocity is given by
j = −ene vdrif t ,
(9)
where ne is the electron concentration.
The conductivity σ is a measure how large the current density is if an electric field is applied
j = σE .
(10)
From Eqs. (7) and (8) we can infer that the conductivity is given by
e2 ne τ
.
(11)
m∗
In some situations it is more convenient to refer to the resistivity % of a conductor. This is
defined as the inverse of the conductivity % = 1/σ.
σ = ene µ =
In the above discussion it was assumed the all electrons take part in the transport. From a quantum mechanical point of view this is not correct, since here only electrons close to the Fermi
energy are transferred into vacant states close to EF . Electrons in the center of the Fermi sphere
will not find a vacant state they can occupy. However, even in a rigorous quantum mechanical treatment we would obtain the same result for the conductance. This can be understood
by a simple argument [3]. By applying an electric field the Fermi sphere is slightly displaced.
The amount of electrons taking part in the transport can be estimated by: ne vdrif t /vF . These
electrons have a velocity of approximately vF so that we obtain for the current density
vdrif t
vF ,
(12)
j=e n
vF
which is identical to Eq. (9).
4.2
Einstein relation
For an electron gas at zero temperature the sum of drift current jE = σE and diffusion current
jD = eD∇ne must vanish at thermodynamic equilibrium. The quantity D is the diffusion
constant.
jE + jD = σE + eD∇ne = 0 .
(13)
Thermodynamic equilibrium implies that the electrochemical potential µ is constant ∇µ = 0.
In general, the electrochemical potential is defined by the sum of the electrostatic potential
energy −eV and the Fermi energy, or chemical potential, EF :
µ = −eV + EF ,
(14)
where EF is measured from the bottom of the conduction band. By using the definition of µ we
can write the gradient of µ as
∇µ = eE +
1
∇ne .
D(EF )
(15)
Quantum Transport in Semiconductor Nanostructures
V4.11
Here, we made use of dEF /dne = 1/D(EF ). Inserting the expression for ∇ne resulting from
Eq. (13) we can write
1
σ
∇µ = e −
E.
(16)
D(EF ) De
As pointed out above, at thermal equilibrium the electrochemical potential is constant ∇µ = 0.
Since the electric field E is not necessarily zero, it implies for the conductance that
σ = e2 D(EF )D .
(17)
This is the Einstein relation, which connects the conductivity σ to the diffusion constant D. Only
if the conductivity and the diffusion constant are related by the Einstein relation the current will
be zero in equilibrium, as required. Thus for the determination of the conductance one first
calculates the diffusion constant at the Fermi energy. Note, that the Einstein relation in the form
as given above is only valid for a degenerate electron gas.
Let us consider a two-dimensional electron gas. If we take the expression for the conductance
and make use of D2D = n2D /EF and of Eq. (11), we obtain the following relation for the
diffusion constant
1
1
D2D = vF2 τe = vF le .
(18)
2
2
This can be generalized to the dimensionality d = 1, 2 or 3
1
1
Dd = vF2 τe = vF le .
d
d
4.3
(19)
Typical transport parameters of a low-dimensional semiconductor structures
It is instructive to calculate the elastic mean free path for the heterostructures introduced in
Sect. 2.1, in order to get a feeling for the lengths which are obtained for high mobility twodimensional electron gases. 1 The elastic mean free path can be obtained from le = vF τe .
√
The Fermi velocity vF can be calculated from the electron concentration vF = ~ 2πn2D /m∗ ,
whereas the elastic scattering time τe is extracted from the mobility µe = eτe /m∗ . For the values
of µe and n2D of typical AlGaAs/GaAs and InGaAs/InP two-dimensional electron gases (see
Table 2) an elastic mean free path of 8.7 µm is obtained for the AlGaAs/GaAs heterostructure,
while le = 5.8 µm for the InGaAs/InP layer system.
The mobility values observed in semiconductor nanowires, i.e. InAs nanowires, are usually
much lower. The reason for that is, that owing to the large surface to volume ratio, the surface
scattering contribution can be quite large. Typical mobility values for InAs nanowires are in the
order of a few 1000 cm2 /Vs [15].
V4.12
Thomas Schäpers
AlGaAs/GaAs InGaAs/InP
n (cm−2 )
2.8 × 1011
6 × 1011
µe (cm2 /Vs)
106
450 000
m∗ /me
0.067
0.04
le (µm)
8.7
5.8
λF (nm)
47
32
Table 2: Typical parameters of a two-dimenensional electron gas in an AlGaAs/GaAs heterostructure and in an In0.77 Ga0.23 As/InP heterostructure.
3a
a)
b)
3
O
φ
2
Q
A
1
Fig. 7: a) Possible trajectories of an electron propagating from point A to end point Q. The trajectory (3a) represents a closed loop. b) Detail of a closed loop with a magnetic flux penetrating
this loop.
4.4
Weak localization
Interference effects of electron waves can even be seen in large samples where the phase coherence length is much smaller than the dimensions of the sample. This effect is called weak
localization. It is observed if the temperature is sufficiently low so that the phase coherence
time τφ is much larger than the elastic scattering time.
In Fig. 7 some typical trajectories of an electron starting at point A and arriving at Q are
sketched. Here, we assumed that the elastic mean free path le is considerably smaller than
the distance between A and Q. An electron undergoes many elastic scattering events on its way.
However, during elastic scattering the electron does not loose its phase memory. Since the
phase coherence length is assumed to be much longer than the distance between A and Q we
can assume that the phase information is kept. After Feynman we can describe each path j by a
complex amplitude given by
Aj = Cj exp(iϕj ) .
(20)
Here, ϕj is the phase shift the electron acquires on its way from A to Q due to propagating a
certain distance. For free electron propagation the phase accumulation along the path j can be
Quantum Transport in Semiconductor Nanostructures
V4.13
calculated from the action Sj by
ϕj =
with the non-relativistic action defined by
Z tQ
Z
Sj =
L(ṙ, r, t)dt =
tA
Sj
~
tQ
tA
m 2
ṙ dt =
2
(21)
Z
path j
1
p dr.
2
(22)
with L(ṙ, r, t) = (m/2)ṙ2 the Lagrangian function of a free propagating electron. tA is the
time when the electron starts at A and tQ the time when it arrives at Q. However, the electron
not only acquires a phase shift during free propagation but also a well defined phase shifts by
the elastic scattering events, so that the total phase accumulated along the path is the sum of
both contributions. The total probability PAQ for an electron to be transported from A to Q is
determined by the square of the total amplitude
2
X
iϕj 2
(23)
Cj e .
PAQ = |Aj | = j
In systems with a large number of possible paths usually the phases ϕj are randomly distributed.
Therefore, the wave nature should have no effect on the electron transport due to averaging.
The fact, that nevertheless an increase of the resistance is observed compared to the classical
transport, is a result of closed loops [see e.g. Fig. 7, trajectory 3a)]. Along these loops, an
electron can propagate in two opposite orientations with the corresponding complex amplitudes
A1,2 = C1,2 exp(iϕ1,2 ). The current contribution of the current returning to the starting point of
the loop (O) is given by
POO = |A1 + A2 |2 = |C1 |2 + |C2 |2 + 2Re(C1∗ e−iϕ1 C2 eiϕ2 ) .
(24)
Since for time reversed pathes C1 = C2 and ϕ1 = ϕ2 , we obtain
|A1 + A2 |2 = 4|C1 |2 .
(25)
For a classical non-phase coherent transport regime the probability would simply be |C1 |2 +
|C2 |2 , which is a factor of two smaller than for the phase coherent transport. A larger probability
to return to the origin implies that the current through the sample is reduced. The carriers are
localized in the loop. The localization does not dependent on the size of the loop as long as
its length is smaller than the phase coherence length. It is important to notice that constructive
interference occurs for all possible closed loops in the conductor and thus not averaged out. As
a result, the total resistance is thus increased compared to the classical case.
Below we will briefly sketch, how the correction of the conductance due to the weak localization
can be obtained quantitatively [4]. For the weak localization effect we are only interested in
those processes where the electrons return to its starting point. From the diffusion equation of
a two-dimensional system one obtains for the return probability 1/(4πDt). For the total return
probability we have to takes care that the phase of the electrons is preserved, which gives us a
pre-factor exp(−t/τφ ). For a return of an electron it is necessary that it is at least once elastically
V4.14
Thomas Schäpers
scattered, thus we have to include a pre-factor [1 − exp(−t/τe )]. If the phase coherence length
is much smaller than the width of the wire so that the sample is effectively two-dimensional the
correction to the conductance is can thus be expressed as
Z ∞
1
2~
dt
δσloc = − σ0
1 − e−t/τe e−t/τφ
me
4πDt
0
2
e
τφ
= − 2 ln 1 +
.
(26)
2π ~
τe
The localization vanishes, if the inelastic scattering time τφ is smaller than τe , since then the
logarithmic factor tends towards zero. The ratio of the weak localization to the Drude conductivity δσloc /σ0 is of the order of 1/kF le . For a quasi one-dimensional structure of width W with
lφ W the diffusion is effectively reduced to one dimension, so that the return probability can
now be expressed by W −1 (4πDt)−1/2 The correction of the conductance for this case is given
by [4]
"
−1/2 #
e2 lφ
τφ
δσloc = − 2
1− 1+
.
(27)
π ~W
τe
A comparison of the one- and two-dimensional case reveals that the correction to the resistivity
is much larger for the one-dimensional case.
If the sample is penetrated by a magnetic field the phase accumulation is modified since the
Lagrangian function L of an electron with charge e is now given by 2
m
(28)
L(ṙ, r, t) = ṙ2 − [e v · A]
2
The phase accumulation along a closed loop with area S can thus be expressed by
I
ie
2πΦ
C1 → C1 exp −
Adl = C1 exp i
,
~
Φ0
(29)
where, A is the vector potential and Φ is the magnetic flux penetration the loop. For a propagation in the opposite direction one obtains
2πΦ
C2 → C2 exp −i
(30)
Φ0
The phase difference between both trajectories is therefore
∆ϕ = 4π
Φ
2S
= 2 .
Φ0
lm
(31)
Thus by the presence of a vector potential the localization is partially lifted. In a diffusive
conductor usually many loops of different sizes are found. For a particular magnetic field the
localization is lifted to a different extent depending of the size of the loops. On average the
degree of localization decreases with increasing magnetic field, which results in a continuous
decrease of the resistance. As an example, the increased resistance at zero magnetic field due
to weak localization of an AlGaAs/GaAs two-dimensional electron gas is shown in Fig. 8.
2
e = −1.602 × 10−19 C
Quantum Transport in Semiconductor Nanostructures
V4.15
Fig. 8: Weak localization in a two-dimensional electron gas in an AlGaAs/GaAs heterostructure. (Measurement by M. Hagedorn, FZ Jülich)
4.5
Weak antilocalization
Up to now the effect of the spin on the electron interference was neglected. This approach
is valid as long as the spin orientation is conserved. In many materials the spin changes its
orientation while the electron propagates along the closed loops. This can be either be due to
spin dependent scattering or due to spin precession originating from the presence of spin-orbit
coupling.
Let us assume |si i is the initial spin state, generally being a superposition of the spin up and
spin down state. In principle there are two mechanisms how the spin orientation can be changed.
First, the potential profile of the scattering centers can lead to spin-orbit coupling. This results
in a spin rotation, while the electron is scattered at the impurities (Elliot-Yafet-mechanism) [see
Fig. 9 (a)]. The second mechanism is the so-called Dyakonov-Perel mechanism. Here, the spin
precesses while the electron propagates between the scattering centres, as illustrated in Fig. 9
(b). The origin of the spin precession can either be the lack of inversion symmetry, i.e. in
zincblende crystals (Dresselhaus effect) [16], or an asymmetric potential shape of the quantum
well forming a two-dimensional electron gas (Rashba effect) [17].
Regardless of the underlying mechanism, if an electron propagates along a closed loop, its spin
orientation is changed. The modification of the spin orientation can be expressed by a spin
rotation matrix U [18]. For the propagation along the loop in clockwise (cw) direction the final
state |scw i can be expressed by
|scw i = U |si i ,
(32)
with U the corresponding rotation matrix. For the propagation along the loop in counterclock-
V4.16
Thomas Schäpers
Fig. 9: Typical closed trajectory in clockwise and counterclockwise directions with spin scattering at the impurities. The initial spin state |si is transformed to the final spin states |scw i or
|sccw i. The spin orientation is preserved during the propagation between the scattering centers.
(b): Situation where the spin precesses during the propagation between the scattering centers.
wise directions (ccw) the final spin state is given by
|sccw i = U −1 |si i .
(33)
Here, we made use of the fact that the rotation matrix of the counterclockwise propagation is the
inverse of U . For the interference between the clockwise and counterclockwise electron waves
not only the spatial component is relevant but also the interference of the spin component:
hsccw |scw i = hU −1 s|U si
= hs|U † U si
= hs|U 2 si .
(34)
The final expression was obtained by making use of the fact that U is unitary: U −1 = U † , with
U † the adjoint (complex conjugated and transposed) matrix of U . In a diffusive conductor there
are very many of these closed loops. In each loop the spin precession is different. This the spin
contribution to the interference is different, no construction interference can be expected. It can
be shown that by averaging over all possible precession angles, a negative contribution remains
[18]. Thus, the quantum correction to the conductivity is opposite to the case when the spin
orientation is preserved. Instead of an increase of the resistance a decrease of the resistance
occurs [19]. This is the reason why this effect is called weak antilocalization. In fact, as can be
deduced from Eq. (34) weak localization and thus constructive interference is recovered if the
spin orientation is conserved in case that U is the unit matrix 1.
Weak antilocalization measurements of a set of InGaAs/InP wires are shown in Fig. 10. In
contrast to the weak localization effect an enhanced conductivity is found at zero field. If a
magnetic field is applied the weak antilocalization effect is gradually suppressed.
4.6
Universal conductance fluctuations
In transport measurements performed on small samples often irregular fluctuations in the resistance are observed at low temperatures. The fluctuations are reproducible if the measurements
Quantum Transport in Semiconductor Nanostructures
V4.17
Fig. 10: Magneto-conductivity measured on a set of 160 InGaAs/InP wires at various temperatures. The spin-orbit coupling, present in this type of quantum well, results in weak antilocalization, an enhanced conductivity at zero magnetic field. The wires had a geometrical width of
1.2 µm
are repeated for the same sample [20, 21]. However, if different samples with the same geometry and fabricated from the same material are compared it is found that a different fluctuation
pattern belongs to each sample. This is the reason, why the fluctuation pattern of a sample is
sometimes called fingerprint. Conductance fluctuations are observed in metal structures as well
as in semiconductor samples [22, 23, 24].
The reason, why each sample shows a different fluctuation pattern can be found if one realizes
that in the very small structures only few impurities are present. In other words: The sample
possesses only that few impurities, that the particular impurity configuration governs the transport properties. If there is only a limited number of scattering centers it is not allowed anymore
to apply an ensemble average for the theoretical description, since this does not consider the particular spatial distribution of the scattering centers. As an example the conductance fluctuations
of a single InN nanowire are shown in Fig. 11 [25].
The fluctuations cover an interval of approximately ±e2 /h. As proved by a detailed theoretical
analysis the fluctuation amplitude of e2 /h is universal [26, 27]. For a qualitatively explanation
of the physical origin of the conductance fluctuations one can refer to the above discussed electron interference effects. As illustrated in Fig. 12 the electron can propagate along a certain
number of paths in order to cross the wire. The total transmission probability results from the
squared amplitude of all possible trajectories. Among these trajectories one can find a limited
number of paths which meet again after a certain distance. If we apply a magnetic field, these
paths are penetrated by a magnetic flux Φ. The superposition of the electron waves of two
paths which cross twice leads to a flux periodic variation of the transmission probability due
to the Aharonov–Bohm effect [28], if the magnetic field is varied. Usually, in a sample differ-
V4.18
Thomas Schäpers
Fig. 11: Conductance fluctuations of an InN Nanowire as a function of magnetic field measured
at 0.8 K. The nanowire had a diameter of 67 nm and a length of 410 nm. (Measurement by Ch.
Blömers, FZ Jülich, [25])
B=0
f1
f2
f3
reservoir
W
reservoir
L
disordered region
Fig. 12: Electron trajectories in a quantum wire. If a magnetic field is applied, loops are
penetrated by a magnetic flux Φ1 , Φ2 ... .
ent crossed trajectories with different encircled areas can be found. This results in a different
Aharonov–Bohm period for each area. By superposition of the different quasi Aharonov–Bohm
rings an irregular pattern in the conductance is produced [21]. It is important that only a limited
number of trajectories exists, so that an effective average of the oscillations is prevented. Beside a variation of the magnetic field it is also possible to observe conductance fluctuations if
the applied voltage is increased. In this case the Fermi wave length of the electrons is changed.
A detailed theoretical description of conductance fluctuations which is based on the particular
scattering center configuration was provided by Al’tshuler and Aronov [29] and by Lee and
Stone [27]. By using their theoretical approach it was possible to calculate the average oscillation amplitude of the sample specific conductance fluctuations. An important essence of the
theoretical model is that a variation of the magnetic field or the Fermi energy induces the same
kind of fluctuations as an ensemble average (ergodic hypothesis). This allows to use an averaging over the scattering center configurations for the calculation. The conductance fluctuations
can be determined by using the correlation function F , which has the following form
F (∆E, ∆B) = hg(E, B)g(E + ∆E, B + ∆B)i − hg(E, B)i2 .
(35)
The averaging h..i has to be understood as an averaging over the impurity configurations. The
conductance G is normalized to g = G/(e2 /h). The correlation function F is a measure how
strong the conductance at an energy E and at a magnetic field B is correlated to the corresponding value at (B + ∆B) and at (E + ∆E). The magnitude of the conductance fluctuations, the
Quantum Transport in Semiconductor Nanostructures
mean squared deviation, is given by the value of F (0, 0)
F (0, 0) = hg(E, B) − hg(E, B)ii2 = g(E, B)2 − hg(E, B)i2 .
V4.19
(36)
As long as the phase coherence is preserved the value of F (0, 0) is independent of the length
and almost independent of the particular shape of the sample. The value of F (0, 0) is in the
order of one under these conditions. Consequently the conductance fluctuations ∆G possess a
universal amplitude of
e2 p
e2
∆G =
(F (0, 0) ≈
.
(37)
h
h
The universal magnitude of the conductance fluctuations is found for example in the measurement shown in Fig. 11.
The reduction of the correlation function F (∆E, ∆B) can be explained by comparing the energy to the correlation energy or Thouless energy ET h and the applied magnetic field to the
correlation field Bc . The correlation energy has a relative large value which is not expected if
one would average of the large number of uncorrelated trajectories. The magnitude of the correlation energy can be estimated by using the uncertainty relation. The time an electron requires
to cross a sample of length L is given by
t=
L2
,
D
(38)
where D is the diffusion constant. The resulting energy uncertainty is given by:
ET h ≈ ~/t =
~D
.
L2
(39)
The correlation energy or Thouless energy ET h is thus a measure of the width of the energy
levels in a sample of length L. Only if the energy is changed by ∆E which is comparable to
ET h , the next energy level is reached and the conductance is changed. The correlation field Bc
is defined by the value where one additional flux quantum Φ0 = h/e penetrates the sample of
length L and width W
h 1
Bc ≈
.
(40)
e LW
If the temperature is increased the Fermi distribution is smeared out. If the temperature is still
sufficiently low so that the width of the Fermi distribution is smaller than ET h , the maximum
fluctuation amplitude is observed. This situation is shown in Fig. 13, where the width of the
energy levels is compared to the width smearing of the Fermi distribution. For the case that the
smearing of the Fermi distribution exceeds ET h , a number of approximately N = (kB T )/ET h
segments contribute.
Since these
√ N segments are uncorrelated the fluctuation amplitude de√
creases with 1/ N , thus 1/ T . This behavior was experimentally confirmed, as shown in
Fig. 14 [25]. At low temperature the fluctuation amplitude is almost constant.
If the tempera√
ture is increased above a critical value a continuous decrease close to 1/ T is observed.
Similarly to the increase of temperature the conductance fluctuations decrease if the length L
of the wire exceeds the phase coherence length lϕ . In this situation one can cut the wire in
V4.20
Thomas Schäpers
E
k BT
E Th
Fig. 13: Comparison of the energetic width ET h with the smearing of the Fermi distribution
function. If the temperature is increased (dashed line) a larger number of channels contribute
to the conductance.
N = L/lϕ phase coherent pieces connected in series. (see Fig. 15). Each of these segments
produces resistance fluctuations ∆Rs , so that the total resistance fluctuations are given by
∆R =
√
N Rs .
(41)
By using the total resistance R = N · Rs , where Rs is the resistance of a single segment, the
total conductance fluctuations can be calculated
√
∆R 2e2 N
2e2 −3/2
=
N
.
(42)
∆G = − 2 =
R
h N2
h
If N is substituted by the ratio between total length and phase coherence length we obtain
2e2
∆G =
h
3/2
lφ
.
L
(43)
It is important to notice that no exponential decrease of the fluctuations with length is expected.
In contrast only a relatively weak decrease of ∆G with increasing length is prediced. This was
indeed experimentally confirmed [30].
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Quantum Transport in Semiconductor Nanostructures
V4.21
Fig. 14: Decrease of the conductance fluctuations with temperature for InN nanowires [25].
inelastic
scattering event
W
...
reservoir
le
lf
reservoir
L
Fig. 15: Wire with length L lϕ. The conductance fluctuations are determined by cutting the
wire into N = lϕ /L coherent pieces.
V4.22
Thomas Schäpers
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