Highlights in Topics
"Progress in Nonlinear Nano-Optics"
Recently it was reported that the eBook edition of our book "Progress in Nonlinear Nano-Optics",
(Editors: M. Sakabe, C. Lienau, R. Grunwald, Springer-Verlag, Heidelberg, 2014) was among
the 25% most downloaded eBooks in the relevant Springer Book Collection with 7,435 chapter
downloads in the year 2015. Contact: R. Grunwald
28 April 2016:
Quantum Swing -
a pendulum that moves forward and backwards at the same time
18 February 2016:
Amplification of Sound Waves at Extreme Frequencies
28th September 2015:
Hot means slow: Electron plasma oscillations tuned down with light
Oscillations of an optically heated electron plasma depend sensitively on the plasma temperature. Ultrafast heating and cooling of a plasma in the semiconductor zinc oxide (ZnO) leads to pronounced shifts of plasma frequency, holding a strong potential for novel switching applications in optoelectronics.
A plasma is a special state of matter in which a large number of electrons form are negatively charged cloud of particles which is separated from a positively charged background of ions. Plasma exists in many systems including hot stars, the ionosphere and other ionized gases, as well as solid state materials. The electric forces between electrons and ions allow for generating periodic spatial motions of the electron cloud relative to the ions, so called plasma oscillations or plasmons. Recently, plasmons in metals and semiconductors have received strong interest. They display peculiar optical properties and hold strong potential for applications in high-speed optoelectronics and optical microscopy with sub-wavelength spatial resolution.
A basic and interesting question is: Can one manipulate plasma oscillations with light and, in particular, modify their frequency? This would allow for switching the electric and optical properties for a short period of time, changes most helpful for novel optoelectronic devices. In the current issue of Physical Review Letters [115, 147401 (2015)], a joint research team from the Max-Born-Institute and Humboldt University in Berlin demonstrates a novel concept for ultrafast plasmon switching in the semiconductor ZnO (Movie). In their experiments, the researchers investigated plasma oscillations in a 100 nanometer thick crystalline ZnO layer containing a high density of approximately 1020 free electrons per cubic centimeter. Plasma oscillations are excited by an infrared pulse of 150 fs duration (1 fs = 10-15 s) and the frequency shift of the infrared plasmon absorption band is measured with a second delayed and much weaker probe pulse. The shift of the absorption band allows for extracting the momentary plasma frequency as a function of time (Fig. 1). The experiments give direct evidence of a transient shift of plasma oscillations to lower frequency. The strong frequency reduction by 20% lasts for only 400 fs after which the original plasma frequency is restored. Over the period of the experiment, the electron density remains unchanged.
The physical origin of the frequency reduction lies in the transient heating of the electron plasma by the infrared excitation pulse. The electrons reach a peak temperature of ≈3300 K and populate a very wide range of the conduction band of ZnO (Fig. 2). In this range, the average electron mass is higher than in the initial state and, thus, the plasma frequency is reduced. The hot electrons transfer most of their thermal excess energy to the crystal lattice within some 400 fs. As a result, both the average electron mass and the plasma frequency return to their original values. All experimental observations are in excellent agreement with theoretical calculations.
||Fig. 1: Experimentally observed time-dependent shift of the plasma frequency in a thin ZnO layer. Left: 3D-plot of the absorption change as a function of the probe frequency and time delay between pump and probe pulses. Right: concept of a transient difference spectrum. The cold plasma (blue) shows an absorption peak at the plasma frequency of the cold electron gas. The pump pulse heats the plasma resulting in a red-shift of the plasmon resonance (red). In the time-resolved experiments we measured the so called difference spectrum, i.e., the absorption of the hot plasma minus that of the cold plasma (black).
||Fig. 2: The conduction band of ZnO shows a non-parabolic band structure, i.e., the electron energy as a function of the electron momentum follows a hyperbola rather than a parabola. As a result electrons at the conduction band minimum are quite light (low energy, small mass) compared to the much heavier electrons (large mass) at high energies. A cold plasma (left) contains essentially light electrons whereas a hot plasma (right) contains many heavy electrons at high energies.
||Animation: Right: plasma oscillations in a thin ZnO layer. Negatively charged electrons (blue clouds) oscillate collectively versus the positively charged ions (red dots). Left: Such a plasma oscillation resembles strongly a classical pendulum, i.e., a massive ball hanging on a elastic spring. (i) For negative times t < 0 (time counter upper left), the oscillation frequency is quite high due to the small mass of the electrons. (ii) During the period 0 < t < 100 fs fs the pump pulse heats the electron plasma (lighters below, temperature display upper right) resulting in an elevated mass of the electrons in ZnO (right) or increased mass in the pendulum (left). (iii) For t >100 fs the probe pulse measures the plasma oscillation frequency again, now showing a distinctly slower motion.
Original Publication: Physical Review Letters 115, 147401
Ultrafast Nonlinear Response of Bulk Plasmons in Highly Doped ZnO Layers
Tobias Tyborski, Sascha Kalusniak, Sergey Sadofev, Fritz Henneberger, Michael Woerner, and Thomas Elsaesser
Dr. Michael Woerner Tel. 030 6392 1470
Prof. Dr. Thomas Elsaesser Tel. 030 6392 1400
AC/DC for terahertz waves - rectification with picosecond clock rates
09th April 2014
Researchers at the Max-Born-Institute in Berlin, Germany discover an ultrafast rectifier for terahertz radiation. In the unit cells of a lithium niobate crystal alternating currents (AC) with a frequency 1000 times higher than that of modern computer systems are transformed into a direct current (DC), thereby generating simultaneously a series of overtones of the terahertz radiation.
When the guitarist Angus Young of the Australian hard rock band AC/DC touches the strings of his electric guitar, a strongly distorted sound rings out from the loudspeaker. The origin of the electronically generated overtones is the rectifying effect in the electronic tubes of the guitar amplifier. In the simplest case an (A)lternating (C)urrent generates a (D)irect (C)urrent, an effect which finds its application in telecommunications at much higher radio or mobile phone frequencies. From a physics point of view the highly interesting question arises: up to which cut-off frequencies can one generate directed currents (DC) and which microscopic mechanisms underlie them?
For the generation of a direct current out of alternating currents the material used must feature a preferred direction. This condition is fulfilled by ferroelectric crystals, in which the spatial separation of positively and negatively charged ions is connected to a static electric polarization. Most ferroelectrics are electric insulators, i.e., low electric fields cannot cause any detectable electric currents in the material. A drastic change of this behavior is observed if one applies for a short period an extremely high electric field in the range of several 100.000 volts per centimeter. At such field strengths, bound electrons, the so called valence electrons can be freed for a short period by means of the quantum mechanical tunneling process leading in turn to a current through the crystal.
Now, researchers at the Max-Born-Institute in Berlin, Germany investigated the properties of such a current for the first time and report their results in the current issue of the journal Physical Review Letters 112.146602 (2014)). Using ultrashort, intense terahertz pulses (1 Terahertz = 1012 Hz, period of a field oscillation 1 picosecond=10-12 seconds) they applied an AC field to a thin lithium niobate (LiNbO3) crystal which causes an electric current in the material. The properties of this current were studied in detail by measuring and analyzing the electric field radiated by the accelerated electrons. Besides an oscillating current with the frequency of the applied terahertz field (2 THz) and several overtones of the latter, the researchers observed the signature of a directed current (DC) along the c-axis the preferred direction of the ferroelectric LiNbO3 crystal.
The rectified current along the ferroelectric c-axis has its origin in the interplay of quantum mechanical tunneling of electrons between the valence and several conduction bands of the LiNbO3 crystal and the deceleration of electrons by friction processes. The tunneling process generates free electrons which in absence of friction would spatially oscillate in time with the applied terahertz field. The friction destroys this oscillatory motion, a mechanism called decoherence. Due to the asymmetry of the tunneling barrier along the ferroelectric c-axis decoherence results in a spatially asymmetric transport, i.e., the tunneling barrier lets pass more electrons from right to left than from left to right. This mechanism is operative within each unit cell of the crystal, i.e., on a sub-nanometer length scale, and causes the rectification of the terahertz field. The effect can be exploited at even higher frequencies, offering novel interesting applications in high frequency electronics.
C. Somma, K. Reimann, C. Flytzanis, T. Elsaesser, und M. Woerner:
High-Field Terahertz Bulk Photovoltaic Effect in Lithium Niobate
Physical Review Letters 112.146602 (2014)
Dr. Martin Hempel receives the Adlershof Dissertation Prize 2013
14th February 2014
On February 13, the Adlershof Dissertation Award was granted to Dr. Martin Hempel of the Max-Born-Institute (MBI). In competition with two other nominees and according to the jury's unanimous opinion, he presented best his dissertation work "Defect Mechanism in Diode Lasers at High Optical Output Power: The Catastrophic Optical Damage".
As a student, Martin Hempel was fascinated by research on novel semiconductor diode lasers. As a PhD student at MBI, he investigated the ultimate performance limits of these universal devices. His research was focused on the so-called Catastrophic Optical Damage (COD), a defect mechanism that affects semiconductor diode lasers at very high optical intensities. A volume of ∼1 µm3 of the laser crystal is heated up to ∼1600°C by re-absorbed laser light. This temperature jump takes place in just 1 ns. Subsequently, the COD defect expands in the laser cavity with a velocity of about 90 km/h. The entire diode laser is degraded after a few µs. The final COD defect volume is approx. 1000 times larger than the initial COD starting point. Hempel was able to resolve the spatio-temporal dynamics of the COD process by a combination of methods. His results provide new insight into the physical mechanisms behind the COD. Based on this knowledge, it is possible to identify the root causes for device failures in an efficient and fast manner. In close cooperation with manufacturers of diode lasers this makes it possible to increase output power and reliability of the devices.
About the prize
Each year since 2002, IGAFA, Humboldt University and WISTA-MANAGEMENT GmbH jointly award the Adlershof Dissertation Prize to young scientists for PhD research which was conducted at one of the research institutions in Adlershof and got an excellent grade of the scientific quality of their work. Three selected nominees present their dissertation work in form of short talks to a general audience. The jury then makes the decision which of the finalists not only conducts excellent research but also explains complex topics enthusiastically and in simple terms.
|Figure 1 (click to enlarge)
COD defect pattern in the active layer of a diode laser (grey shaded area). The thermal signature of the COD defect front was mapped temporally resolved by a thermocamera. This was done by monitoring the side of the laser cavity (along z axis) and the front facet (along x) in parallel. The center of gravity of these signals, seen form two sides of the device, gives the position of the defect front (black dots). Thermography is demonstrated to be a tool to resolve the motion of the COD defect front temporally and spatially at once. In the upper part, thermal images are shown, record at different times while looking on the front facet. The corresponding positions in the x-z-diagram are marked. The constant signal amplitude indicates a constant temperature during the entire COD defect growth. Evaluating the temporal and spatial data gives a velocity of ∼90 km/h for the movement of the defect front along the laser cavity.
|Figure 2 (click to enlarge)
||Electron microscopy image of a device damaged by COD. The front facet is indicated by the dotted line on the left side. The material extrusion is clearly visible there. The degradation of the active laser layer starts at the front facet and is expanded up to the right edge of the image. A crystallographic interpretation of the changes of the material composition in the damaged volume indicates the presence of a temperature around 1600°C
|Figure 3 (click to enlarge))
Dr. Martin Hempel, +49 (0) 30-6392 1453
Generation of terahertz radiation by ionising two-color femtosecond pulses in gases
23rd January 2012
Scientists of the Max-Born- Institute have elucidated theoretically in cooperation with external partners the basic mechanism of the emission of Terahertz radiation in gases, provided an experimental evidence for this and opened new possibilities for the control of the Terahertz spectral parameters. Terahertz radiation (1 terahertz=1THz=1012Hz=1012cycles per second) is light with extremely large wavelength of about 0.3 mm. A frequency of 1 THz is about 50 times higher than the working frequency of cell phones. Nowadays THz radiation finds broad applications, such as for wireless data transfer or for the analysis of materials. Also the full-body scanners at airports use THz radiation for the observations of objects. In research THz pulses of extremely short duration are used for the investigation of basic properties of solids and liquids, e.g. charge transport and electric resistance. These investigations require the generation of short THz flashes which can be realized by the ionisation of gases with ultrashort light pulses. Among the various THz sources, employing two-color femtosecond pulses in gases provides striking performance in terms of high peak fields (up to the MV/cm range) as well as broad bandwidth (that can exceed 100 THz). Although already observed in the year 2000 and after that studied and applied in many papers, the physical mechanism behind the THz radiation is still controversy discussed in the literature. Initially, this process has been explained by rectification via third-order nonlinearity and later by the plasma current generated by the two-color laser field. Our theoretical study  and experiments at the MBI  related with this study show that THz generation in gases is intrinsically connected to the optically-induced stepwise increase of the plasma density near the maximum amplitudes of the pump fields due to tunneling ionisation leading to the associated emission of a discrete set of attosecond-scale, ultra-broadband bursts. The spectrum is therefore determined by the interference of contributions arising from different ionisation events, showing a remarkable analogy to the linear diffraction theory of gratings. Comprehensive (3+1)-dimensional numerical simulations confirmed this model which offers simple explanations for recent experimental observations and opens new avenues for the governing of THz parameters and THz pulse shapes based on temporal control of the ionization events (such as by the frequency offset of the pump pulses) . The implementation of experiments at the MBI enabled to test this new understanding of THz emission directly by experimental observations . The measured strong broadening of the THz spectra with increasing gas pressure has shown, in good agreement with (3+1)-dimensional numerical simulations, a sensitive dependence on the gas pressure enabling important insight into the basic mechanism of THz emission and the prominent role of propagation effects of the pump pulses. Plasma-induced blueshifts of the driving pulses play a key role in the broadening of the THz spectra with increasing pressure, and also deliver an experimental evidence of the above described mechanism in which THz emission is associated with the step-wise modulation of the tunnelling ionisation current.
Fig.: The mechanism of THz generation. The two-color field E(t) (red) generates free electrons with a step-wise modulation of the electron density (green) near the field amplitude maximum via tunnelling ionisation. This leads to a current (blue pointed) which acts as a source of THz emission. Inset: scheme of the experimental setup.
 I. Babushkin, S. Skupin, A. Husakou, C. Köhler, E. Cabrera-Grenado, L. Berge and J. Herrmann “Tailoring terahertz radiation by controlling tunnel ionisation events in gases “, New Journal of Physics 13, 123029 (2011)
 I. Babushkin, W. Kuehn, C. Köhler, S. Skupin, L. Berge, K. Reiman, M. Woerner, J. Herrmann and T. Elsaesser“, Ultrafast spatiotemporal dynamics of terahertz radiation by ionising two-color femtosecond pulses in gases “,Phys. Rev. Lett. 105, 053903 (2010)
Dr. J. Herrmann, Tel. 030 63921278
The onset of electrical resistance
16th December 2011
Researchers at the Max-Born-Institute, Berlin,
Germany, observed the extremely fast onset of electrical resistance
in a semiconductor by following electron motions in real-time.
When you first learned about electric currents,
you may have asked how the electrons in a solid material move from
the negative to the positive terminal. In principle, they could
move ballistically or ‘fly’ through the solid, without
being affected by the atoms or other charges of the material. But
this actually never happens under normal conditions because the
electrons interact with the vibrating atoms or with impurities.
These collisions typically occur within an extremely short time,
usually about 100 femtoseconds (10-13 seconds, or a tenth of a trillionth
of a second). So the electron motion along the material, rather
than being like running down an empty street, is more like trying
to walk through a very dense crowd. Typically, electrons move only
with a speed of 1m per hour, they are slower than snails.
Though the electrons collide with something very
frequently in the material, these collisions do take a finite time
to occur. Just like if you are walking through a crowd, sometimes
there are small empty spaces where you can walk a little faster
for a short distance. If it were possible to follow the electrons
on an extremely fast (femtosecond) time scale, then you would expect
to see that when the battery is first turned on, for a very short
time, the electrons really do fly unperturbed through the material
before they bump into anything. This is exactly what scientists
at the Max-Born-Institute in Berlin recently did in a semiconductor
material and report in the current issue of the journal Physical
Review Letters [volume 107, 256602 (2011)]. Extremely short bursts
of terahertz light (1 terahertz = 1012 Hz, 1 trillion
oscillations per second) were used instead of the battery (light
has an electric field, just like a battery) to accelerate optically
generated free electrons in a piece of gallium arsenide. The accelerated
electrons generate another electric field, which, if measured with
femtosecond time resolution, indicates exactly what they are doing.
The researchers saw that the electrons travelled unperturbed in
the direction of the electric field when the battery was first turned
on. About 300 femtoseconds later, their velocity slowed down due
In the attached movie, we show a cartoon of what
is happening in the gallium arsenide crystal. Electrons (blue balls)
and holes (red balls) show random thermal motion before the terahertz
pulse hits the sample. The electric field (green arrow) accelerates
electrons and holes in opposite directions. After onset of scattering
this motion is slowed down and results in a heated electron-hole
gas, i.e., in faster thermal motion.
The present experiments allowed the researchers to determine which
type of collision is mainly responsible for the velocity loss. Interestingly,
they found that the main collision partners were not atomic vibrations
but positively charged particles called holes. A hole is just a
missing electron in the valence band of the semiconductor, which
can itself be viewed as a positively charged particle with a mass
6 times higher than the electron. Optical excitation of the semiconductor
generates both free electrons and holes which the terahertz bursts,
our battery, move in opposite directions. Because the holes have
such a large mass, they do not move very fast, but they do get in
the way of the electrons, making them slower.
Such a direct understanding of electric friction will be useful
in the future for designing more efficient and faster electronics,
and perhaps for finding new tricks to reduce electrical resistance.
Two former PhD students of the Max-Born-Institute receive
the Carl Ramsauer Prize 2011
14th November 2011
Christian Eickhoff and Dr. Wilhelm Kühn are among this year's
winners of the Carl Ramsauer Prize of the Physikalische Gesellschaft
zu Berlin (PGzB), an award for outstanding PhD work.
Christian Eickhoff (third from left in the picture) received his
doctor degree at the Freie Universität Berlin and Wilhelm Kühn
(second from left in the picture) is awarded for his doctoral thesis
at Humboldt University. They did their research work at the Max-Born-Institut
für Nichtlineare Optik und Kurzzeitspektroskopie (MBI). The
prizes will be awarded on November 17th, 2011, at the Physics Department
of FU Berlin.
Dr. Christian Eickhoff's dissertation was devoted
to “Time-resolved two-photon photoemission at the Si(001)-surface:
Hot electron dynamics and two-dimensional Fano resonance.”
He investigated the (100) surface of silicon, the basic material
of the present semiconductor technology. The ongoing miniaturization
and increasing efficiency of devices require a basic understanding
of the electronic and and dynamic properties of carriers at the
Si surface. Eickhoff designed a complex experiment consisting of
a laser system which provides widely tunable photon energies, and
a ultrahigh vacuum setup including a two-dimensional detector for
photoelectrons. He observed the dynamics of optically excited electrons
in surface states and in the conduction band of silicon with a time
resolution of several tens of femtoseconds (1 fs = 1015
sec). He also investigated the influence of elastic and inelastic
scattering processes on energy relaxation and surface recombination.
He showed how optical excitation of carriers leads to complex two-dimensional
interference phenomena which are accounted for by Fano resonances
in the initial and intermediate state. His work explained why the
elevated temperature of hot electrons in the conduction band is
maintained over an unexpectedly long period of many picoseconds
(1 ps = 1012 sec). The extraction of hot electrons may
find application for increasing the efficiency of third-generation
Dr. Wilhelm Kühn has developed a novel method
of nonlinear spectroscopy in the THz frequency domain (1 THz = 1012
Hz) and applied it in basic solid state research. The nonlinear
interaction of light and matter is measured in two independent dimensions
in time to derive two-dimensional (2D) spectra in the frequency
domain. Such spectra give insight into the couplings between different
excitations of the system under study and their time evolution.
THz oscillations are extremely slow compared to visible light, an
oscillation period lasts approximately 250 fs. Focusing the THz
beam allows for generating high electric fields (ca. 300 kV/cm)
and applying them for the acceleration of charge carriers in solids.
With this novel method, Kühn studied the transport properties
of electrons in the semiconductor gallium arsenide (GaAs). He observed
that strongly accelerated electrons move without appreciable friction
('ballistic motion'). At very high kinetic energies, however, they
slow down and are accelerated into the opposite direction, due to
their negative effective mass. In this way, the resulting circular
motion of electrons, the so-called Bloch oscillations, was observed
for the first time in a bulk crystal.
In a second experiment, Kühn investigated a semiconductor model
system displaying an extremely strong coupling of electrons and
the crystal lattice. He was able to show the formation of a new
particle, the polaron which consists of an electron and a cloud
of phonons. His results which were confirmed by theoretical calculations,
also demonstrate how and into which channels electron energy is
transferred to the crystal lattice.
Electrons and Lattice Vibrations—A Strong Team in
the Nano World
4th August 2011
Semiconductor electronics generates, controls, and amplifies electrical
current in devices like the transistor. The carriers of the electric
current are mobile electrons, which move with high velocities through
the crystal lattice of the semiconductor. Doing this, they lose
part of their kinetic energy by causing atoms in the lattice to
vibrate. In semiconductors like gallium arsenide the positively
and negatively charged ions of the crystal lattice vibrate with
an extremely short period of 100 fs (1 fs = 10-15 s = 1 billionth
part of one millionth of a second). In the microcosm of electrons
and ions such vibrations are quantized. This means that the vibrational
energy can only be an integer multiple of a vibrational quantum,
also known as a phonon. When an electron interacts with the crystal
lattice (the so called electron-phonon interaction), energy is transferred
from the electron to the lattice in the form of such vibrational
Berlin researchers report in the latest issue of
the scientific journal Physical Review Letters that the strength
of the electron-phonon interaction depends sensitively on the electron
size, i.e., on the spatial extent of its charge cloud. Experiments
in the time range of the lattice vibration show that reducing the
electron size leads to an increase of the interaction by up to a
factor of 50. This results in a strong coupling of the movements
of electrons and ions. Electron and phonon together form a new quasi
particle, a polaron.
To visualize this phenomenon, the researchers used
a nanostructure made from gallium arsenide and gallium aluminum
arsenide, in which the energies of the movements of electrons and
ions were tuned to each other. The coupling of both movements was
shown by a new optical technique. Several ultrashort light pulses
in the infrared excite the system under study. The subsequent emission
of light by the moving charge carriers is measured in real time.
In this way two-dimensional nonlinear spectra (see Fig.) are generated,
which allow the detailed investigation of coupled transitions and
the determination of the electron-phonon coupling strength. From
the coupling strength one finds the size of the electron cloud,
which is just 3-4 nanometers (1 nanometer = 10-9 m = 1 billionth
of one meter). Furthermore, this new method shows for the first
time the importance of electron-phonon coupling for optical spectra
of semiconductors. This is of interest for the development of optoelectronic
devices with custom-tailored optical and electric properties.
Negative Mass and High Speed: How Electrons Go Their Own
12th April 2010
Wilhelm Kuehn beim Einjustieren des ExperimentsIsaac
Newton found in the 17th century that a force causes a body to accelerate.
The inertial mass of the body is the ratio between force and acceleration,
thus, given the same force, a light body is accelerated more strongly
than a heavy body. A body's mass is positive, meaning that the acceleration
is in the same direction as the force. Charged elementary particles
as the free electron, which has a mass of only 10-30 = 0. ...(29
zeros !)...1 kilograms, can be accelerated in electric fields to
extremely high speeds. If the electric field is small, the motion
of electrons in crystals is governed by the same laws. In this regime,
the mass of a crystal electron is only a small part of the mass
of a free electron.
Wilhelm Kuehn, Michael Woerner, Klaus Reimann and Thomas Elsaesser
from the MBI have now demonstrated that crystal electrons in extremely
high electric fields exhibit a completely different behavior. Their
mass even becomes negative. They report in the latest issue of Physical
Review Letters 104, 146602 (2010) that the electron is accelerated
within the extremely short time of 100 femtoseconds = 0.000 000
000 000 1 seconds to a speed of 4 million kilometers per hour. Afterwards
the electron comes to a stop and even moves backward. This means
that the acceleration is in opposite direction to the force, which
can only be explained by a negative inertial mass of the electron.
In the experiments, electrons in the semiconductor
crystal gallium arsenide are accelerated by an extremely short electrical
pulse with a field strength of 30 million Volts per meter. At the
same time the speed of the electrons is measured with high precision
as a function of time. The duration of the electric pulse is only
300 femtoseconds. This extremely short duration is essential as
otherwise the crystal could be damaged.
DThe new results agree with calculations of the
Nobel Prize winner Felix Bloch undertook more than 80 years ago.
They open up a new regime of charge transport with new possibilities
for future microelectronics devices. The observed frequencies are
in the terahertz range (1 THz = 1000 GHz = 1012 Hz),
about 1000 times higher than the clock rate of the newest PCs.