Download The Quantum Hall Effect Michael Richardson

The Quantum Hall Effect
Michael Richardson
In 1985, Klaus von Klitzing was awarded the Nobel Prize for his discovery of the quantized Hall effect. The
quantum Hall effect has provided an amazingly accurate method for calibrating resistance. Some of the
successful explanations of the effect are summarized in the following.
I. INTRODUCTION
The classical Hall effect was discovered in 1879 by
Edwin Hall. He correctly speculated that the charge
carriers in a wire would be deflected to one side of the
wire if a magnetic field were set up perpendicular to
the flow of the current. In other words, the magnetic
field led to a transverse voltage in the wire. This
transverse voltage could be measured experimentally
and was seen to increase as the magnetic field
increased.
It is possible to demonstrate the same effect in a
two dimensional electron gas. The electrons are
trapped at the interface between a metal and a
semiconductor. Then a current is established in one
direction and a magnetic field is applied perpendicular
to the electron gas.
In 1980, von Klitzing did experiments to test the
Hall effect on a two dimensional electron gas at very
low temperature and high magnetic field. He found
that the Hall conductivity  (current divided by the
transverse voltage) had plateaus of constant
conductivity as the magnetic field was varied. More
astonishingly, the values of the conductivity at these
steps were integer multiples of e2/h that were measured
to an accuracy of one part in 10 million.
Ever since, theorists have attempted to explain the
phenomenon. A fairly simple explanation involving
Landau levels can be made for an ideal system. Yet
more complicated explanations are needed to explain
real systems.
II. A SIMPLE APPROACH
If the electron gas is subjected to a strong magnetic
field, then the electron orbitals split into Landau
levels. The number of orbitals in each level is
𝐷 = (2𝜋𝑒𝐵/ℏ𝑐)(𝐿/2𝜋)2 = 𝑒𝐵𝐿2 /ℎ𝑐.
If the voltage is adjusted so that the Fermi level of the
electron gas falls right between two Landau levels,
then the total number of electrons in the electron gas is
given by
𝑁 = 𝑠𝐷
One can rearrange these equations to determine that
𝜎 = 𝑠𝑒 2 /ℎ.
No elastic collisions are possible from one state to
another state in the same Landau level because all the
states in the level are filled. Because the magnetic
field is strong and the temperature is low, ℏ𝜔𝑐 ≫ 𝑘𝐵 𝑇.
Therefore, the electrons are unlikely to absorb the
necessary energy to move to the next Landau level
from a phonon. As a result, the conductivity remains
quantized while the Fermi energy is between the
Landau levels.
III. LAUGHLIN’S EXPLANATION
According to the above explanation, the occurrence
of the quantum Hall effect would not be expected for
systems with partially filled Landau levels. Yet it does
occur in such systems.
Robert Laughlin used the principle of gauge
invariance to explain the quantization of the
conductance. He imagined a 2D electron system bent
into a cylinder so that the current moves in a loop.
The surface of the cylinder experiences a strong
magnetic field that is everywhere normal to the
cylinder. Now consider a magnetic flux through the
cylinder. Any change in this flux will create an
electromotive force that will cause the charges to
move around the loop and, due to the magnetic field,
deflect to one side of the cylinder. If the change in the
magnetic flux is equal to an integral multiple of the
elementary quantum of magnetic flux ℎ𝑐/𝑒, then the
Hamiltonian describing the system will remain
unchanged.
This is because, according to the
Aharonov-Bohm principle, the system is gauge
invariant under flux changes of integral multiples of
ℎ𝑐/𝑒.
To calculate the Hall conductance, one divides the
transverse current by the Hall (transverse) voltage.
𝜎 = 𝐼/𝑉𝐻
The current is found by using the electromagnetic
relationships
𝜕𝑈
𝜕𝜑
= −𝑉𝐻 𝐼 = 𝐼𝑐
,
𝜕𝑡
𝜕𝑡
𝐼=𝑐
𝛿𝑈
𝛿𝜑
The energy 𝛿𝑈 is equal to 𝑁𝑒𝑉𝐻 , where N is the
number of electrons that are moved to the other side of
the cylinder due to the emf from the change in flux.
The change in flux is the elementary quantum of
magnetic flux. So
𝐼=𝑐
𝑁𝑒𝑉𝐻
ℎ𝑐/𝑒
Given that the Hall conductance is 𝜎 = 𝐼/𝑉𝐻 , the
above equation yields
𝑒2
𝜎=𝑁
ℎ
Because the system returns to its original state after the
flux change, one assumes that if the system were to
undergo the same flux change again, then the same
number of electrons would be transferred. Yet
quantum mechanically, two systems in the same state
do not have to produce exactly the same results under
the same measurement. Therefore, something more
must ensure that the average number of electrons
transferred is also an integer.
1
2𝜋
𝐾𝑑𝐴 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑆
where the integral is over a surface with no boundaries
and K is the local curvature of the surface.
Shiing-shen Chern generalized this formula to the
geometry of eigenstates that can be parameterized by
variables over a toroidal surface. The “adiabatic
curvature” is found by taking the limit of the Berry
phase mismatch divided by the loop as the loop size
goes to zero.
This formula shows that as long as the adiabatic
curvature is the same for each cycle of increasing
magnetic flux, the same integer will be produced every
time. This information, combined with Laughlin’s
explanation demonstrates why the Hall conductance is
quantized in integer multiples of 𝑒 2 /ℎ.
V. CONCLUSION
The incredible precision with which one may
measure the value of 𝑒 2 /ℎ has allowed for a new
standard for resistance (ℎ/𝑒 2 ). The quantum Hall
effect also allows a precise calculation of the fine
structure constant. The theory behind the effect is still
being researched by many scientists.
IV. TOPOLOGICAL QUANTUM NUMBERS
After the system undergoes the cycle of changing
its flux by ℎ𝑐/𝑒, it returns to the same state. Its wave
function is the same as before, except that the phase of
the wave function may be different. This accumulated
phase is known as the Berry phase. When it is
compared to the accumulation of phase by vectors
undergoing parallel transport on a curved surface, the
Hall conductance can be compared to the curvature of
the surface. In other words, just as the curvature of a
surface is linked to the accumulation of phase in
parallel transport, so is the Hall conductance linked to
the Berry phase that arises under the adiabatic cycle of
the wave function.
Doing so allows a formula from topology to be
used in order to explain the integral quantum Hall
effect. The formula, by Gauss and Charles Bonnet, is
References
C. Kittel, Introduction to Solid State Physics. 498-503
(2005)
J. Avron, D. Osadchy, R. Seiler, Physics Today. 56, 8
(2003)
http://en.wikipedia.org/wiki/Quantum_Hall_effect
http://nobelprize.org/nobel_prizes/physics/
laureates/1985/
Left-handed Materials
Josh Holt
March 5, 2008
1
Theoretical Introduction
In the 1960’s, Victor Veselago investigated the optical repercussions of a hypothetical medium
with simultaneously negative and real electric permittivity ² and magnetic permeability µ. 11
At a flat interface of his conjectured material with “normal” material, the direction of wave
propagation is opposite to the Poynting vector of energy propagation; thus electromagnetic
waves obey the left-hand, instead of the right-hand, rule. Veselago coined such media exhibiting this effect left-handed materials (LHMs).
One must then choose the negative root
q
of the index of refraction given by n = ± ²µ/²0 µ0 (where the naught subscript indicate
respective free-space quantities). Although the ensuing consequences of a negative refractive
index, including negative refraction, reverse Doppler shift and reversal of Cherenkov radiation, are peculiar, they do not violate any fundamental physical laws. For example, negative
refraction allows creation of a so-called a “Veselago lens,” a flat, parallel slab of LHM inside
a right-handed material (RHM) with the condition that both media have the same isotropic
refractive index and the same impedance. Rays from a point source impinging on a Veselago
lens would be refocused to a point on the opposite side of the material.
Interest in LHMs irrupted after the work by Pendry which argued that the Veselago
lens is a “perfect lens” in the sense that it gives a perfect image of the point source. 5 His
statement is based upon the observation that the evanescent waves of a form exp(iky y − κx)
that usually decay in the near-field region are amplified by the LHM. Pendry claimed that the
amplified evanescent waves restore a perfect image in both the near-field and far-field regions,
but he did not present a solution in coordinate space. That the cherished diffraction limit
could be violated did not set well with many physicists. A real-space solution was offered by
Ziolkowski and Heyman, 12 revealing that Pendry’s solution diverges exponentially at each
point of a 3D domain near the focus, just where the fields of the evenescent waves increase due
to amplification. Simultaneous attacks against so-called “superlensing” followed from many
groups. 2,8,3 Eventually the debate settled until it was generally agreed that superlensing can
be found in the near-field regime while resolution in the far-field remains on the order of
wavelength. 6 Even so, implications of negative refraction and the possibility of beating the
diffraction limit could not be ignored. The theoretical controversy amplified the need to
reveal new experimental evidence of negative refraction.
1
Left-handed Materials
2
Solid State II
Experimental Evidence
Since large magnetic response, in general, and a negative permeability at optical frequencies, in particular, do not occur in natural materials, metamaterials have been contrived to
facilitate ² < 0 and µ < 0 simultaneously.
The earliest probe for negative refraction used a two-dimensional array of split-ring resonators (which couple to and alter the local magnetic field) to produce negative µ over a
particular frequency region and wire elements to produce negative ², showing microwave
transmission at a positive from normal angle using a teflon prism and transmission at a negative angle using a manufactured LHM metamaterial prism. 10 An immediate debate questioning the veracity of the results arose 1 but, ultimately, the effects of negative refraction
were repeatably verified.
Permeability, permittivity, and refractive index are bulk, effective medium properties.
Although metamaterial consist of discrete scattering elements, it may be approximated as an
effective medium for wavelengths that are larger than the unit cell size. This approximation
is analogous to the effect of the periodic Bloch potential on band electrons in condensed
matter. The advantage in these metamaterials, of course, is that they can be scaled to any
particular wavelength of light, whereas the dispersion relation of any solid-state electronic
device is effectively pinned to a fixed lattice spacing. Photonic crystals and structures have
naturally followed to implement negative refraction within optical frequencies. It was shown
that a dielectric photonic crystal made of non-magnetic materials can behave as a LHM with
negative ² and µ if it has a negative group velocity in the vicinity of the Γ-point of the second
Brillouin zone both theoretically 7 and experimentally. 4,9 However, the exsistance of surface
plasmons in photonic crystals make interpretation of experiments and simulations difficult.
3
Conclusions
Physical understanding since Veselago’s initial theory has bloomed into applications probably beyond what he originally imagined (see, for example, Ref. 12). Applications founded on
negative refraction include beam steerers, modulators, band-pass filters, and lenses permitting subwavelength point source focusing. The primary players which have stakes in LHM
technology include the telecom industry (which relies on the ultrafast switching of the optical
1.5 µm wavelength for high-throughput communication), data storage and imaging technology (who are continually trying to work around the diffraction limit). The non-intuitive
effects of LHMs have driven research in metamaterial and led to their much improved understanding.
References
[1] N. Garcia and M. Nieto-Vesperinas. Is there an experimental verif ication of a negative
index of refraction yet? Opt. Lett., 27:885–887, 2002.
Left-handed Materials
Solid State II
[2] N. Garcia and M. Nieto-Vesperinas. Left-handed materials do not make a perfect lens.
Phys. Rev. Lett., 88:207403, 2002.
[3] F. D. M. Haldane. Electromagnetic surface modes at interfaces with negative refractive
index make a ”not-quite-perfect” lens. cond-mat/0206420, 2002.
[4] P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar. Negative
refraction and left-handed electromagnetism in microwave photonic crystals. Phys. Rev.
Lett., 92:127401, 2004.
[5] J. B. Pendry. Negative refraction make a perfect lens. Phys. Rev. Lett., 85:3966–3969,
2000.
[6] V. A. Podolskiv and E. E. Narimanov. Near-sighted superlens. Opt. Lett., 30:75–77,
2005.
[7] A. L. Pokrovsky and A. L. Efros. Sign of refractive index and group velocity in lefthanded media. Solid State Comm., 124:283–287, 2002.
[8] A. L. Pokrovsky and A. L. Efros. Diffraction theory and focusing of light by a slab of
left-handed material. Physica B, 338:333–337, 2003.
[9] Vladimir M. Shalaev, Wenshan Cai, Uday K. Chettiar, Hsiao-Kuan Yuan, Andrey K.
Sarychev, Vladimir P . Drachev, , and Alexander V. Kildishev. Negative index of
refraction in optical metamaterials. Opt. Lett., 30:3356–3358, 2005.
[10] R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a negative index
of refraction. Science, 292:77–79, 2001.
[11] V. G. Veselago. Properties of materials having simultaneously negative values of the
dielectric(²) and magnetic (µ) susceptibilities. Sov. Phys.-Solid State, 85:3966–3969,
1967.
[12] R. W. Ziolkowski and E. Heyman. Wave propagation in media having negative permittivity and permeability. Phys. Rev. E, 64:056625, 2001.
1
Graphene: Carbon, Now in Stunning 2D
Jon Paul Johnson, University of Utah, Physics 5520
If thin is in, graphene has a bright future as
the thinnest stable material known. If you
were to remove a single atomic layer from
graphite, one of the more traditionally-built
forms of carbon, you would have graphene—
a sheet of carbon atoms in a honeycomb
array with the thickness of a single carbon
atom. The discovery of stable graphene
sheets in 2004 came as a surprise1, since it
was assumed that it was energetically
unfavorable for carbon to be in this
configuration and not rolled up in a carbon
nanotube, a fullerene, or some other more
stable three-dimensional folded shape. Now
this intriguing material with its predicted
bizarre electrical properties is accessible,
which has attracted the attention of electrical
engineers looking for a replacement for
silicon transistor channels as the limits of that
material are expected to be reached within a
few decades. Graphene also provides an
unlikely sandbox to study quantum effects
because its electronic structure yields charge
carriers that are massless fermions.
among the forest of thicker graphite flakes.
The thin flakes have a slightly different
appearance from the substrate, and the
contrast is maximized for certain SiO2
thicknesses.
Much of the interest in graphene comes from
the electrical dispersion relation of electron
waves in a honeycomb lattice, which at the
Fermi level (also known as the neutrality
point in graphene) is gapless and linear. See
Figure 1. This had been predicted long before
stable graphene was discovered2,3. Sp2hybridized bonds connect the carbon atoms
in the lattice, with the last electron per carbon
almost completely delocalized. Since effective
mass depends on the curvature of the
dispersion relation, its linearity at the Fermi
level implies charge carriers with zero
effective mass, and these carriers are
quasiparticles that behave as relativistic
particles with a speed that is lower than the
normal speed of light by about a factor of
300.
This
enhances
some
quantum
electrodynamic effects, which depend on the
Graphene was first discovered by a group at
the University of Manchester which, in an
effort to study thin sheets of graphite,
presumably got more than they bargained for
when they discovered the presence of
individual flakes of graphene in their
samples. Sample preparation was (and is)
rather more hunter-gatherer than agricultural
in nature: “micromechanical cleavage” of a
piece of graphite involves either peeling
layers off of it with adhesive tape or drawing
with it on a solid SiO2 surface. Then a search
is performed with an optical microscope to
find the few thin graphene flakes scattered
Figure 1. The dispersion relation for graphene.
1
2
electrons and holes used to carry charge in
today’s devices, but first there must be
advances made in epitaxial growth of this
intriguing material6. Without a way to create
graphene layers compatible with the parallel
processing
techniques
used
by
the
semiconductor industry, graphene is likely to
remain a condensed matter research tool—
though a very interesting and, as of now,
unique one.
inverse of the speed of the particles involved,
to the point that they are measurable even at
room temperature. Specifically, the quantum
Hall effect has been observed in graphene at
room temperature.
The electron transport properties graphene
make it a candidate as a conductive channel
in electronic devices. Graphene is conductive
without doping, and its carriers exhibit very
high mobility and even ballistic transport on
small enough scales at room temperature.
The carrier velocity (~106 m/s) and the fact
that an applied electric field in either
direction creates charge carriers could make
graphene useful for ultra fast-switching field
effect transistors.
1. A. K. Geim & K. S. Novoselov, The rise of
graphene. Nature Materials 6, 183-191
(2007).
2. P. R. Wallace, The band theory of
graphite, Phys. Rev. 71, 622-634 (1947).
One drawback for the use of graphene in a
more traditional transistor design is actually
the lack of a gap at the neutrality point,
which creates charge carriers for conduction
regardless of whether or not an electric field
is applied to the sheet. One way round this is
to artificially create a semiconducting gap by
introducing quantum confinement effects on
a sheet of graphene. Limiting one of the two
remaining spatial dimensions by creating
ribbons of graphene4 could open a gap that
would make graphene suitable as an
alternative to silicon as the channel material
in field effect transistors. Another way a
semiconducting gap could be manufactured
is by using the lattice mismatch between
graphene and the crystal used as a seed to
grow graphene epitaxially (for example, 5).
3. J. C. Slonczewski & P. R. Weiss, Band
structure of graphite. Phys. Rev. 109, 272279 (1958).
4. L. Brey & H. A. Fertig, Electronic states of
graphene nanoribbons. Phys. Rev. B 73,
235411 (2006).
5. S.Y. Zhou et al., Substrate-induced band
gap opening in epitaxial graphene. Nature
Materials 6, 770-775 (2007).
6. A. K. Geim & A. H. MacDonald,
Graphene: Exploring carbon flatland.
Physics Today 60, 35-41 (2007).
The discovery of graphene nicely rounds out
the collection of available forms of atomically
thin carbon, taking its place with 0D
fullerenes and 1D nanotubes. Graphene has
the potential to make waves in electronic
device design by swapping massless
wavelike quasiparticles for the massive
2
March 5, 2008
Living Green With Thermoelectricity
Ben Mangum
Department of Physics, University of Utah
Thermoelectric devices capable of converting heat directly into electricity and thermoelectric
coolers, which do not contain any moving parts or ozone depleting gases may become a major player
in reducing carbon footprints. While thermoelectric devices have traditionally been plagued by low
efficiencies (ZT < 1), recent technological advances allowed for the fabrication of new nanostructured
materials with much higher figures of merit (ZT ∼ 3). Such advances are demonstrated in materials
that maintain an high electric conductivity and low thermal conductivity by minimizing phononic
contributions. By introducing inclusions at various length scales, phonon contributions can be
minimized. The development of high-efficiency thermoelectric devices will play an increasingly
larger role in society as environmental costs of current technologies are considered more carefully.
I.
(a)
INTRODUCTION
Heat Source
For almost two centuries now, scientists have possessed
both the understanding and the ability to make generators capable of converting heat directly to electricity and refrigerators with no moving parts to wear out.
Thomas Seebeck discovered in 1823 that two dissimilar metals joined together with a temperature gradient
across the junction was capable of deflecting a compass
needle. With later clarification from Ørsted, this discovery meant that such a simple device was capable of generating ‘thermoelectricity’ [1]. Working independently
in 1834 Jean Peltier discovered that applying a voltage
across such a junction was capable of establishing a temperature gradient across the junction [4]. Lord Kelvin
(William Thompson) described the heating or cooling of
a current-carrying conductor with a temperature gradient in 1851, showing that as some metals exhibit cooling while others produce heat as current flows from high
to low potential with an applied temperature gradient.
These reversible processes (the Seebeck effect, the Peltier
effect, and the Thompson effect) are collectively known
as the thermoelectric effect [1].
By 1838 Heinrich Lenz had demonstrated the use of a
thermoelectric (TE) device; by placing a drop of water
between a bismuth wire and an antimony wire and applying a current, the water would freeze and then melt
by reversing the direction of the current [3]. In the 1950’s
a renewed interest in the thermoelectric effect arose as a
consequence of the discovery that doped semiconductors
were much exhibited a large thermoelectric effect, thus
much of the early semiconductor research was in pursuit
of thermoelectric refrigerators [3]. Despite the fact that
these thermoelectric technologies are now centuries old,
there has been very little, or at least very slow, progress
in putting TE devices to practical use. Currently, due
to low efficiencies TE devices are used for niche markets such as laser diode cooling, and power generation for
space bound devices like the Voyager deep space probe
[2]. However, with a new arsenal of tools and techniques,
coupled with the green awakening sweeping the globe,
thermoelectric technologies are ripe for more investigation and implementation into everyday living.
N
(b)
Cooled Surface
N
P
Cool Side
P
Dissipated Heat
FIG. 1: Schematic of thermoelectric (TE) devices. Panel (a)
shows a TE generator; as heat flows from a heat source to a
heat sink, heat carriers move along the temperature gradient.
Utilizing both n-doped and p-doped semiconductors for the
junctions the heat carriers are also charge carriers and establish a potential difference across the device capable of driving
a load. This is an example of a Seebeck generator. Panel
(b) shows a Peltier cooler, and is essentially the converse of
the effect just described; an applied voltage will preferentially
move heat carriers to one side of the device establishing a
temperature gradient.
II.
THEORY
A schematic of both a TE cooler and a TE generator are shown in Fig. 1. When n-doped and p-doped
semiconductors are joined to both a heat source and a
heat sink by metal contacts, thermal carriers (electrons
or holes) move from hot to cold. As seen in Fig. 1(a)
this motion of thermal, but also charge, carriers creates
a potential difference and thus a current in the material
is created as indicated by the arrows in the figure. In
like manner a Peltier (TE) cooler is shown in Fig. 1(b).
An applied voltage drives the motion of charge carriers
as indicated by the arrows, while this net flow of carriers from one side of the device to the other establishes a
thermal gradient.
The efficiency of any device is thermodynamically
limited to a maximum efficiency (Carnot efficiency) of
Tc /(Th − Tc ) for refrigerators. Modern compressor based
2
refrigeration systems can reach 30-90% of Carnot efficiency, thus if TE devices could reach 30% Carnot efficiency, they could compete with modern compressor
based refrigeration technologies. The efficiency of thermoelectric devices is often characterized by a dimensionless figure of merit ZT, where Z has units of ◦ C −1 and T
is the average temperature. Z is a function of the thermal
conductivity (κ), the electrical coductivity (σ), and the
Seebeck coefficient (S):
Z=
S2σ
κ
(1)
The Seebeck coefficient (S) is simply a measure of a
material’s ability to establish a voltage difference in response to a given temperature difference: S = ∆V /∆T .
An examination of metals and degenerate semiconductors yields [2]:
S=
8π 2 kb2 ∗ π 2/3
m T
3e~2
3n
(2)
where n is the carrier concentration, and m∗ is the effective mass of the carrier [2]. It must also be borne in mind
that both the electrical conductivity and the thermal conductivity are also functions of carrier concentration; σ
also being a function of the electric charge (e) and mobility (µ), σ = neµ. The thermal conductivity has contributions from the motion of electrons and holes, but is
also influenced by phonons in the lattice, thus κ = κe +κl ,
with κe = neµLT , where L is the Lorentz factor. Noting
that σ and κ are directly proportional to the carrier concentration n, but that S ∝ n−2/3 the optimum carrier
concentration can then be found. This sensitive balance
of carrier concentration effects, indicate then that metals, although having high electrical conductivity make
poor thermoelectric materials due to a low Seebeck coefficient and high thermal conductivity. On the other
hand, insulators, while having large Seebeck coefficients
and small electrical contributions to thermal conductivity, have low electrical conductivities also making them
poor TE materials [3]. The optimum balance between
Seebeck coefficient and conductivities is typically struck
with highly doped semiconductors: n ∼ 1019 cm−3 [2].
III.
RECENT ADVANCEMENTS
Optimizing the concentration of charge carriers in a
material is now really the task at hand for determin-
[1] http://en.wikipedia.org/wiki/Thermoelectric effect
[2] G.J. Snyder, and E.S. Toberer, “Complex thermoelectric
materials,” Nature Mater. 7, 105–114 (2008).
[3] G. Mahan, B. Sales, J. Sharp, “Thermoelectric Materials:
New Approaches to an Old Problem,” Phys. Today, March
ing the efficiency of a material for potential use as a
TE device, however, to make use of quantum mechanical predictions, firstly the electronic band structure of
the material must be known [4]. The inability to predict
crystal structures is the major bottleneck in predicting
which new materials might be suitable for TE devices.
Recent progress in x-ray diffraction techniques, the ability to carefully grow new materials through the use of
nano-wires and thin film deposition, along with modern
advances in computing technologies all increase the likelihood of finding new and better TE materials [4].
Theoretical predictions suggest that the TE efficiencies can be improved by quantum confinement of the
electron charge carriers. In a quantum confined structure, high confinement and low dimensionality leads to
narrow electronic energy bands. This equates to high
effective masses and thus large Seebeck coefficients [2].
Being able to create a ‘phonon glass’ while maintaining
an ‘electron crystal’ is seen as another key to creating
high ZT materials. The thermal conductivity contribution due to phonons can be reduced by increased phonon
scattering accomplished by a variety of methods such as
scattering phonons within the unit cell by creating point
defects or alloying, as well as scattering phonons at interfaces through the use of multiphase composites with
nanometer scale structuring [2]. To date, the most efficient TE materials are made from superlattice nanowires
with a ZT of 2.5 - 3 [5].
IV.
CONCLUSIONS
Despite relatively low efficiencies of even modern thermoelectric devices, great improvements have been made
in recent years. Several years ago the most efficient devices had a figure of merit only as high as ZT ∼ 1,
whereas today nanostructured materials are rated as high
as ZT ∼ 3. It is predicted that the heretofore elusive barrier of ZT > 4 will soon be broken, thus allowing thermoelectric coolers to compete directly with current compressor based refrigeration technologies. The optimization of carrier concentrations is certainly vital to creating
high ZT devices, but the next major advances will likely
come from finding ways to maintain an ’electronic crystal’
while simultaneously being a ’phonon glass.’ In an era
where the environmental cost of any technology or device
is become ever more scrutinized, thermoelectric power
generation is also poised to move out of its niche market
as ever more efficient TE materials are made available.
1997 pp. 42–47
[4] F.J. DiSalvo, “Thermoelectric Cooling and Power Generation,” Science, 285, 703–706 (1999).
[5] K. Walter, https://www.llnl.gov/str/May07/Williamson.html
The Quantum Cascade Laser
February 27, 2008
Nick Borys
Introduction
The quantum cascade laser (QCL) is a novel semiconductor laser device which allows for
the creation of laser sources in a broad wavelength range of approximately 1µm up
through 30µm (near-infrared to terahertz radiation). This wavelength range is
exceptionally important in chemical detection due to the majority of vibrational and
rotational energy levels of molecules within this energy range. Consequently, QCLs have
direct applications in environmental analysis, trace detection, and defense and security
technology. Further, QCLs are the first reliable and low-cost laser source in this
wavelength ranges since material stability and processing complications make the regime
hard to access with conventional semiconductor diode lasers [1].
energy
small DOS
Although the QCL is based off of semiconductor materials, it is fundamentally different
from conventional diode lasers. A diode laser relies upon the recombination of an
electron in the conduction band and a hole in the valance band within the semiconductorbased active region. Consequently, the wavelength of a diode laser is defined and limited
by the material dependent bandgap energy. On the other hand, a QCL is based on a
layering of semiconductor materials such that layers of neighboring energy wells and
barriers are created (Figure 1). The physical thickness of the wells and barriers define the
energy levels of the laser. So, rather than relying on the material properties alone, the
wavelength of the QCL can be tuned simply by modifying the thickness of the layers in
the active region consequently opening up the broad wavelength regime of the QCL [1].
distance
g
large DOS
n=3
g
n=3
n=2
n=1
n=2
Injector
Active Region
n=1
Figure 1 Energy diagram of a QCL
Technical Description of the Quantum Cascade Laser
The QCL was first demonstrated in 1994 by a group at Bell Labs that consisted of
Federico Capasso, Claire Gmachi, Deborah Sivco, Alfred Cho, Jerome Faist, Carol
Sirtori, and Albert Hutchinson [2]. As mentioned above, it is principally different than a
semiconductor diode laser in that the QCL operates with electrons cascading through a
series of potential wells that make up the conduction band. Since the QCL only involves
conduction electrons, and not valance band holes, it is referred to as a unipolar laser [1].
The QCL consists of two fundamental regions (Figure 1): the injector and the active
region. Electrons are injected from the injector ground state (dashed line labeled “g” in
Figure 1) into the n=3 state of the active region. The injected electrons then radiatively
relax into the n=2 state and quickly, further relax into the n=1 state. From the n=1 state,
the electrons can then tunnel into another injector region and leave the active region, thus
completing the fundamental light emission process of the QCL [1, 2].
The ground-state of the injector region combined with the three energy levels in the
active region essentially comprises a four-level laser system. Electrons can efficiently
tunnel through the injector region into the n=3 state of the active layer. The scattering
rate of n=3 state to the n=2 state is quite large (several picoseconds). However, the
scattering rate from the n=2 state to the n=1 state is an order of magnitude smaller (~0.3
ps). This arrangement of scattering rates accomplished by engineering the well and
barrier widths such that there is minimal wavefunction overlap of the n=3 and n=2 states
and maximal overlap between the n=2 and n=1 states. The net consequence of the
scattering rates is that the n=3 state remains populated by electrons from the injector
ground state while the n=2 state is quickly depopulated resulting in population inversion
between the two states – a requirement for laser emission [1, 2].
Additionally, the injector region is engineered so that it promotes population inversion
between the two states in the active region. The wells and barriers of the injector region
are designed such that the ground-state energy is very close to the n=3 energy level of the
active region. However, this equality is only for electrons being injected. Notice, in
Figure 1, the n=3 energy level of the active region falls in an energy range where the
injector has a large density of states. On the right-hand side, however, the n=3 energy
level falls in an energy range of the injector region that has a low density of states. This
engineered configuration helps prohibit electron tunneling into the subsequent injector
region directly from the n=3 state thus making it more likely the injected electrons will
radiatively relax into the n=2 state [1, 2].
As illustrated in Figure 1, a typical QCL consists of several active regions separated by
injectors, referred to as stages. As of 1999, a typical QCL has a 25-75 stages.
Consequently, one single electron in a QCL produces up to N photons for a device that
has N stages as compared to a diode laser where an injected electron will only produce a
single photon (i.e., once an electron combines with a hole in a diode laser, it will not
produce any additional photons). This, combined with the ability to support larger device
currents, leads to QCLs outperforming diode lasers by up to factors of 1000 [1].
Finally, the full QCL device has the numerous semiconductor layers comprising the
stages clad with semiconducting material of a lower refractive index so that the radiation
is guided through the device. In more complicated commercial laser setups this cladding
is also grated such that single mode can be significantly amplified over another [1, 2].
Summary
Since its invention 1994, the QCL has quickly found a considerable amount of practical
applications. A primary reason for this is the large wavelength range QCLs can be
engineered for. This wavelength range, combined with the relative ease of production
such that QCLs can be produced in large volumes, opens up laser spectroscopy solutions
for a large amount of industries that cannot otherwise afford expensive laser systems [3].
As an example, QCLs are poised to provide a significantly better solution for the large
gas-sensing market [4].
Furthermore, the QCL is also finding use for theoretical studies. In 2007, a paper was
published in Nature that documented using QCLs to directly probe the phase of laser
emission. This method is claimed to open up the ability to probe the dynamics of the
photons as they traverse the laser cavity and should find possible use in studying laser
dynamics [5]. So in addition to a significant impact in several industries, the QCL has
what appears to be a promising future in scientific fields as well.
References
[1]
F Capasso, C Gmachl, D L Sivco, A Y Cho, Physics World, 27-33 (June 1999).
[2]
J Faist, F Capasso, D L Sivco, C Sirtori, A L Hutchinson, A Cho, Science 264,
553-556 (1994).
[3]
I Howieson, Laser Focus World 41, (2005).
[4]
E Normand, Laser Focus World 43, (2007).
[5]
D Citrin, Nature 449, 669-670 (2007).