Definicion de covariante y de contravariante.

Тензоры ранга 1, 2 и 3 визуализируются с ковариантными и контравариантными компонентами.
se visualizan tensores de rango 1, 2 y 3 con componentes covariantes y contravariantes
definicion de covariante y de contravariante.

Hopping conductivity

The gradual freezing-out of conduction electrons with decreasing temperature eventually leads to a situation in which the main contribution to the electrical conductivity comes from electrons hopping directly between impurities without any excursion to the conduction band. This is called hopping conductivity. Electrons jump from occupied donors to empty ones, and therefore the presence of empty positions on donors is a necessary condition. At low temperatures this condition can be fulfilled only by compensation.
The hopping mechanism of conduction corresponds to a very low mobility, since the electron jumps are associated with a weak overlap of wave-function tails from neighboring donors. Nevertheless, it wins in the competition with band conduction, because only an exponentially small number of free carriers can participate in the latter.

Assumptions in Heisenberg’s 1925 paper

    http://physics.stackexchange.com/questions/18519/assumptions-in-heisenbergs-1925-paper


    http://theorie2.physik.uni-erlangen.de/index.php/Papers_from_the_beginning_of_quantum_mechanics


    My first query is why does he claim the position and period of an electron to be unobservable “in principle”? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically.

Werner Heisenberg obviously disagreed with this assumption of yours and it just happened that his ability to disagree made him a founder of quantum mechanics.

He has spent several years by trying to develop “quantized planetary” models of the helium atom etc. before he understood that this failing project is failing for fundamental reasons. Such a helium with well-defined positions would be described by a chaotic 3-body problem and there would be no way how it could be consistent with the known regular behavior of the helium atom (and other atoms and other coherent systems), including the sharp spectral lines.

So Heisenberg was able to see in 1925 something that you can’t see now: that the electrons can’t be going along any particular trajectories while they’re in the atoms. Instead, what is observed is that they have a totally sharp energy from a possible list, the spectrum – something we can really observe via the photons that atoms emit or absorb. To conclude that electrons can’t be going along particular classical trajectories in the atoms, he didn’t have to wait for measuring apparatuses that would be sufficiently accurate. He was able to make this conclusion out of the available data by “pure thought”, and he was right.

    Secondly, just because a theory dealing with those quantities is inconsistent, or not general enough, why does it imply that we cannot define or measure quantities that that theory deals with? We may be able to measure some quantities perfectly, but still formulate an incorrect theory around them.

Many combinations of options would be possible in a generic hypothetical world and you’re right that the combination of options you mentioned would be logically possible in another world but Heisenberg was talking about our world. He learned his message from special relativity that one shouldn’t talk about things that can’t be operationally defined – such as the simultaneity of events (which is observer-dependent) and tried to maximally apply this positivist mode of reasoning to the world of atoms. His analysis dictated that he may assume that the electron in the atom has a particular energy for a long time but it can’t have a well-defined position or velocity. So he reformulated physics around the notion of the energy which is measurable and found out the first formulation of quantum mechanics in the energy eigenstate basis Heisenberg picture.

    Finally, is there any ad-hoc basis to decide what these “uncertain” quantities are? More specifically, how could Heisenberg pinpoint position of an electron as an uncertain parameter and not any other quantity (like some electric field, etc.)?

You are mixing apples with oranges here. Heisenberg’s paper wasn’t discussing the electromagnetic field. It was discussing the general logical framework underlying physics and the examples he took were those from mechanics – rigid rotator and anharmonic oscillator – that were meant to be later generalized to a theory of atoms in particular just by a new choice of the potential energy formula.

There’s no observable concept of “electric fields” in the description of an atom or anharmonic oscilator at all. Even in classical physics, one deals with functions of positions and momenta. He figured out that not all functions are equally observable: energy (a particular function of positions and momenta) is much more observable and stable.

The underlying logic he has developed was later (soon) applied to other systems in mechanics such as atoms and molecules as well as field theory such as electromagnetism. But the essence isn’t in describing which degrees of freedom are there (they’re kept as close to those in the corresponding classical theory as possible); the essence of quantum mechanics is in the totally new set of postulates and methods to make predictions.

He realized that the right goal wasn’t just to find another classical theory just with some new degrees of freedom, which is the intrinsic, fundamental, and completely flawed assumption of your whole question from the beginning to the end. He realized that the new insights force physicists to formulate a completely new theory – and he (and others) has (have) already used the completely new term “quantum theory” for it – and he just did so, discovering some of the new explicit quantum formulae for nontrivial predictions (beyond the spectrum of the Hydrogen atom that was “explained” by Bohr’s toy model).

You may repeat many times that a complete conceptual revolution in physics (switching from the classical to the quantum) wasn’t needed and one should have only discussed new classical models with new variables (paying no attention to whether or not they may be actually observed) except that Heisenberg knew that it was needed and the months (and a few years) that followed his discovery made his assumption unquestionable.




1

Thanks for the detailed explanation. But I wanted to confirm the following – Heisenberg did not propose the indeterminacy of position/velocity due to some experimental results, rather, just as special relativity challenged the ad-hoc concept of time (which was used as a parameter for evolution of position, momentum and other quantities in classical mech), Heisenberg challenged the absolute determinacy of position/momentum (which were in turn parameters to describe fields, energy, etc.). And so in this sense it was a theoretical analogy to special rel? Is that correct? –  

2

I see… well definitely the helium model failure was a motivation as well. On a side note, instead of just studying quantum mechanics, I additionally intend to focus on such fundamental matters and questions underlying it. In other words I actually want to study the “Physics” of it, rather than just the mathematical framework blindly (excuse me if I’m being obscure), and understand how each aspect of the theory fits into the physical world. Do you have any suggestions as to how to go about it, and whether studying the original pioneering papers would help in this regard? – 

3

I meant that sometimes during the course of going through the mathematical formalism, it is possible to unknowingly ignore the physical interpretation of some steps taken, or the physical meaning behind some results. So as far as possible, I don’t want to ignore any of that. (Also which “good textbooks” are you referring to?) Again, sorry for the trouble. –

4

That’s very good you don’t want to ignore the physical interpretation and indeed, not too many words are being said about it in most cases. However, what is even more important than to appreciate the right physical interpretation of the formulae is to avoid a wrong interpretation of them – such as a classical or “visualizable” interpretation: none of it is ever right in QM. Certain things just don’t have any “easy to imagine” content and the calculated probabilities (and cross sections and allowed eigenvalues etc.) are really everything that is meaningful & “real” from a physical vantage point – 

5

The online reference to link Darrengol could be useful to see the problems H. was addressing at that time. Also a read of Sommerfeld paper, to see the “elipses” of the relativistic atom and how problematics they were, can be illuminating. –


Heisenberg’s paper is deriving its results from an assumption which is stated only obliquely in the paper, and which is central for all the conclusions. This assumption is explained more clearly on Wikipedia.

Heisenberg is dealing with the orbit of an electron in the atom. Let us assume that this orbit is precise, so that the electron has a position on the m-th Bohr orbit as a function of time is Xm(t). The motion is periodic, so you can Fourier transform this motion to get a Fourier series for the electron’s position

X(t)=∑neinωtXmn

The quantity Xmn is the n-th Fourier coefficient of the m-th Bohr orbit. This quantity is associated with the frequency nω where ω=2π/T is the classical orbit (radian) frequency and T is the classical orbital period. Notice that the classical Fourier frequencies are multiples of a least common multiple, which is (2π times) the reciprocal period.

The fundamental reason Heisenberg rejects this description (which is very close to Bohr’s original idea, and developed by Kramers and Heisenberg) is the fact that these integer spaced frequencies nω are not observed in atomic transitions.

the frequencies that you do observe are the quantum frequencies, which are the difference in energy between the n-th Bohr orbit and the m-th Bohr orbit. There is a fundamental mismatch between the classical orbital description with its integer tower of frequencies and the observed electromagnetic wave emission of the atom, which has a completely different non-integerly spaced collection of frequencies.

The quantum frequencies are given by En−Em, the difference in energy of the n-th and m-th orbit, which however do become integer spaced when n and m are both large. In this limit, called the correspondence limit, En−Em=∂E∂J(n−m) where the partial derivative is of the classical energy with respect to the classical action variable J.

So in the correspondence limit, the classical orbit description is valid, because the frequencies you observe in atomic transitions match the frequencies you would deduce by Fourier transforming a sharp classical orbit.

But what about at smaller quantum numbers? Here Heisenberg makes a radical new assumption. He takes the quantities Xnm, which are the n-th Fourier coefficient of the m-th Bohr orbit, and says that they appear in quantum mechanics with the frequency En−Em, not with the frequency 2πnT! This idea is already present in Bohr to some extent, even in 1913 Bohr states that the transition from orbit n to orbit m should correspond to the classical Fourier component of motion somehow, but Bohr does not develop this idea fully, leaving it vague.

Heisenberg then states that if X_{nm} are quantum Fourier coefficients, then it is immediate that their time development should be

Xnm(t)=ei(En−Em)tXnm(0)

Here you can recognize the Heisenberg equation of motion for the matrix elements of X. This is required by the correspondence principle, to match the frequency of classical Fourier coefficients for large orbits. It is also incompatible with the picture of sharp orbits, because the X matrix elements no longer make integer-spaced towers which can be used to reconstruct a periodic classical orbit. Further, the coefficients with opposite frequencies are complex conjugates of each other Xmn=X∗nm, in the classical picture, it would be Xm,n=X∗m,−n.

Part of the difference is a trivial shifting: the classical n=0 point is shifted to n=m in the matrix description, just because the near-diagonal part is the classical motion, not the 0 column. This shifting is expressed by the correspondence rule that Xclm,n=Xm(m+n), where the left hand side is the classical Fourier coefficients, and the right hand side is the quantum matrix elements. But even with this shifting, the conjugation relations are off.

The complex conjugation in QM reflects along the diagonal of the matrix, it doesn’t reflect the horizontal row along a vertical line running down the middle. You can see how the classical limit emerges by looking at large m,m+p in the matrix, The reflection to m+p,m is p units away from the diagonal to the left, while the original position is p units to the right. So when the rows become continuous and the columns stay discrete, the complex conjugation relations reproduce those of classical mechanics on the Fourier coefficients.

But things are not quite right, because the stuff to the left of the midpoint in a given row is not the complex conjugate of the right. This means that if you try to write down the classical orbit as a function of time, you will fail, producing complex quantities which are not periodic, just some nonsense.

It is important to see Heisenberg’s intuition here— he was sure that the quantum Xmn is a complete description of the quantum motion, but it does not include the classical orbits. His conviction is that the orbit was not a part of the description, that it was a redundant classical idea that was no longer useful, and the fact that his description did not allow you to reproduce the orbit was a positive sign, not an incompleteness.

Other stuff in the paper

The next step is to derive the multiplication law. This is explained on Wikipedia, but it is pretty obvious from the classical law for multiplying Fourier series by convolution. The result is matrix multiplication.

Heisenberg then derives the on-diagonal part of the canonical commutation relations from some complicated radiation sum-rules he did with Kramers. The derivation on Wikipedia is more elementary, but uses essentially the same ingredients, without relying on Kramers-Heisenberg sum rules, and without doing ad-hoc tricks like differentiating with respect to n. The derivation of the on-diagonal canonical commutation relation is the main hurdle that makes this paper magical— it is difficult to follow, you need to do it a different way today.

Why uncertainty?

The uncertainty principle, although only explicitly formulated in 1927, is already present in 1925 to a large extent, except not stated in terms of complementary variables.

Heisenberg’s matrices only allow you to reconstruct a fuzzy orbit, it is only a classical periodic orbit to the extent the the frequencies are integer spaced. So for Heisenberg, the quantities which are “uncertain” are not uncertain yet in a statistical sense (that comes later, after Born’s interpretation of the wavefunction), but they are uncertain in the sense that they cannot be reconstructed in a quantum system.

Heisenberg would have said that the momentum is also uncertain, because the momentum fourier series cannot be reconstructed from the matrix elements of P. The energy would be certain, because the energy levels are precise in the description (ignoring back-reaction from the EM field emissions).

This is an artifact of the fact that Heisenberg was working in frequency space, so that the Hamiltonian was diagonal. In this picture, every quantity which does not commute with H would be considered uncertain, because it would necessarily have off-diagonal matrix elements that do not allow you to reconstruct it’s time variation precisely.

This concept of uncertainty is not the same as the 1927 uncertainty, which came after further developments clarified the notion of state. In 1925, Heisenberg wan’t sure how to describe the state, he could only describe the analogs of classical motion in the Bohr orbits.

So the notion of fuzziness of quantity in the 1925 paper should be considered an ill-definedness of the classical quantity as a function of time, not as a statistical statement about the values of observation of that quantity (at least not yet).

Imagenes Raman

By combining a high precision translation stage, sensitive multichannel detectors, large magnification optical  microscope objectives and bright spectrometers high spatial resolution Raman  imaging becomes possible.

One understands by imaging the representation of  Raman spectral parameters over a certain area of which each pixel has an associated  spectrum; from those one can extract the Raman parameters using fitting routines;  the images are defined by three coordinates (x, y, z), where x and y are the surface  spatial coordinates, and z represents the corresponding Raman parameter which the  distribution is imaged.
The parameters of a Raman peak are the frequency  (wavenumber), intensity, polarization, and linewidth. When more than one Raman  band is present, the ratios between the different Raman bands can be also obtained.
Light ongoing onto a solid undergoes several interaction processes with the solid, as  described in previous chapters, among them light scattering, either elastic or inelastic. The Raman effect consists of the coupling of the electromagnetic field of  the incident light with the optical phonons through the induced electric dipole  moment.
The incident photons, characterized by their energy, wavevector, and  polarization, are inelastically scattered by the crystal. The inelastic component  (Raman) of the outgoing light is characterized by its corresponding intensity,  energy, wavevector, and polarization, which are determined by the structure and
nature of the solid.

 The inelastic component (Raman) of the outgoing light is characterized by its corresponding intensity, energy, wavevector, and polarization, which are determined by the structure and nature of the solid.
The relation between the incident and the scattered light is governed by the corresponding Raman scattering selection rules, which are determined by the lattice symmetry and the nature of the crystal [38]. The energy  exchange between the incident light and the solid can only take the values of the phonon energies. When the lattice absorbs the energy necessary to excite a phonon
the Stokes (S) component of the spectrum is observed, whereas the release of a
phonon to the electromagnetic wave results in the anti-Stokes (AS) component of
the Raman spectrum.
Usually, Raman experiments deal with the S component,  because its intensity is several times that of the AS component. The AS-component takes a relevant role when dealing with temperature measurements. We will discuss about temperature later on, because contact-less temperature measurements are very
important for the device reliability analysis; on the other hand, the thermal aspects  are crucial in nanostructures, where the thermal conductivity, and heat dissipation are dramatically affected by the low dimensionality [39]; e.g. Si nanowires  (NWs) immersed in low heat dissipation media present evident signs of overheating in the presence of a laser beam for the acquisition of the Raman spectrum [40–42].

Strain. The bond lengths in strained materials are different from those of the
unstrained ones; therefore, strain shifts the Raman peaks; the sign of that shift
depends of the type of strain (compressive or tensile), Dx being proportional to
the strain [97]. One needs to establish the strain tensor to give accurate strain
estimations, which in the case of l-R experiments is limited by the scattering
geometry. This is a successful application of l-R spectroscopy applied to
semiconductor devices, for which an exhaustive literature is available [30–32,
58, 98–103]. Strain can influence the electronic properties of the semiconductor
structures; for example, the strain can reverse the order of heavy and hole
subbands [104], or may produce changes in the band offsets [105] of the
heterostructures.

Hibridacion sp, sp2 y sp3

Definicion de los enlaces sp

Hibridizacion:es el proceso en el cual dos o mas orbitales de la capa de valencia de un atomo se empalman para formar el mismo numero de orbitales identicos con las mismas energias.
En el caso de la hibridacion sp , hibridacion digonal, un orbital S y un orbital P se empalman para dar como resultado dos orbitales sp identicos. que estan sobre la misma linea (el angulo entre ellos es de 180^o )

Definicion de los enlaces sp2

Hibidizacion Trigonal: Proceso de translapado de un orbital S y dos orbitales P para dar como resultado tres orbitales hibridos identicos se llama sp2 o hibridizacion  trigonal.
Los tres orbitales toman las direcciones de los orbitales P y cada uno de ellos tiene 33.3% de caracter S y 66.6% de caracter P.
El angulo entre ellos es de 120^o y es plano.
Los orbitales sp2 son mas pequenios que los orbitales sp3.

Definicion de los enlaces sp3

Hibridacion tetrahedral:  Proceso de translape de un orbital S y tres orbitales P que da como resultado cuatro orbitales hibridos identicos, llamados sp3 o hibridacion tetrahedral.
Cada orbital hibrido tiene 25% de caracter sigma y 75% de caracter P. Los orbitales hibridos estan orientadas a lo largo de cuatro esquinas de un tetrahedro regular en un angulo de 109^o 28'

Aplicacion al Carbono

Estado pasivo y excitado del carbono.
En estado excitado, un electron del nivel 2s se promueve al nivel 2p (donde solo habia dos electrones) y abre la posibilidad a 4 enlaces.  
 










.

Enlaces sigma y enlaces pi

Como son los enlaces sigma y los enlaces pi? Por que se forman? De la configuracion electronica del atomo, a la configuracion electronica de la molecula, en la cual los electrones ocupan diferentes niveles en los orbitales moleculares (esto explica el tipo de enlace, cuando es covalente, ionico).









.

El advenimiento del microcosmos

Название книги: Microcosm The Quantum Revolution In Economics And Technology . Pagina 30

Автор: Gilder, George -

Ключевые слова: Filosofia Ensayo

The “informativeness” of subatomic matter is the key to modem electronics. Because the world of the microcosm consists not of inert and opaque solids but of vibrant, complex, and comprehensible fields, it constitutes a useful arena for modern information technology. It is because there is so much analogical information in the microcosm that the microcosm is a uniquely powerful medium for information.The quantum-regulated movement of electrons across quantum-mapped crystalline paths epitomizes information technology, and in-formation technology epitomizes the quantum era.

In the atom of information, this era acquires its definitive symbol. What was once a blank solid is now revealed in part as information, what was once an inert particle now shines with patterns and probabilities, what was once opaque and concrete is now a transparent tracery of physical laws. Far from plunging reality into clouds, quantum theory makes the universe radically more intelligible. The new science does not estrange human beings from their environment. Since thought is the most distinctly human power, the quantum world is actually more anthropomorphic than the world of Newtonian masses and forces.

A more intelligible universe, penetrable by the human mind, endows people with greater power to create wealth. But it also radically changes the way wealth is created. Throughout previous human history, the creation of wealth depended chiefly upon the extraction, transport, combination, and modification of heavy materials against the resistance of gravity, the constraints of entropy, and the constrictions of time and space. When things are large and approached out-side-in, it is expensive to move and manipulate them. Their costs derive from the weight, rarity, entropy, and resistance of their matter. But small things, virtually devoid of matter, move less like weights than like thoughts. In the microcosm, the costs of fuel and materials decline drastically; the expense devolves from matter to mind. Just as quantum science overthrew Newtonian matter in the explanation of the universe, the quantum economy overthrows Newtonian matter in the creation of new wealth.

El primer microprocesador

El primer microprocesador

The planar process was a logical outgrowth of the diffusion and oxide masking process. Planarization was the creation of physicist Jean Hoerni of newly-formed Fairchild Semiconductor. Hoerni observed the production limitations of conventional 3-dimensional transistor designs (e.g., the "mesa" transistor). Hoerni reasoned that a design based on a "plain" would be superior. Thus, the planar transistor, as the name implies, was flat. Flattening the mesa enabled electrical connections to be made, not laboriously by hand, but by depositing an evaporated metal film on appropriate portions of the semiconductor wafer. Using a lithographic process of a series of etched and plated regions on a thin, flat surface or wafer of silicon, the "chip" was born out of the planar transistor. Like the printing process itself, the planar process allowed for significantly greater rates of production output at even higher yields.

More importantly, the planar process enabled the integration of circuits on a single substrate since electrical connections between circuits could be accomplished internal to the chip. Robert Noyce of Fairchild quickly recognized this. As Gordon Moore recalls:

"When we were patenting this [planar transistor] we recognized it was a significant change, and the patent attorney asked us if we really thought through all the ramifications of it. And we hadn't, so Noyce got a group together to see what they could come up with and right away he saw that this gave us a reason now you could run the metal up over the top without shorting out the junctions, so you could actually connect this one to the next-door neighbor or some other thing."

Fairchild introduced the first planar transistor in 1959 and the first planar IC in 1961. Moore views the 1959 innovation of the planar transistor as the origin of "Moore's Law."

Perhaps more than any other single process innovation, planarization set the industry on its historical exponential pace of progress. As one early industrial technologist noted, "The planar process is the key to the whole of semiconductor work." George Gilder's account in his 1989 treatise, Microcosm, is more eloquent:

"Known as the planar integrated circuit, Fairchild's concept comprised the essential device and process that dominates the industry today. . . Ultimately it moved the industry deep into the microcosm..."

Bob Schaller. The Origin, Nature, and Implications of "MOORE'S LAW" The Benchmark of Progress in Semiconductor Electronics. 1996. http://research.microsoft.com/en-us/um/people/gray/Moore_Law.html

En 1968, Robert Noyce decide abandonar la compañía Fairchild Semiconductor, para poder fundar en 1969, junto a Andrew Grove y Gordon Moore, la compañía Integrated Electronics, conocida como Intel.

En 1970, Intel consiguió almacenar cadenas de ceros y unos, desarrollando la primera memoria RAM.  En 1971, consiguen integrar la memoria RAM en la CPU, con lo que consiguen crear el primer ordenador comercial al alcance del consumidor, el Altair 8800. En 1971, nació el primer microprocesador, denominado 4004, compuesto por 4 chips desarrollados por Ted Hoff y otros 2 chips de memoria. Poco después, Intel comercializó el 8008. En 1981, Intel desarrolló los procesadores de 16 bits 8086 y los de 8 bits 8088. Estos procesadores permitieron a IBM, por primera vez, confeccionar el primer PC. En 1982, Intel desarrolló el 286 capaz de ofrecer compatibilidad con sus predecesores.

En 1985, llegó el 386, un microprocesador de 32 bits. Fue adoptado por Compaq para su computadora personal Compaq Deskpro 386. En 1989 la compañía desarrolló Intel 486 de 1,2 millones de transistores. En 1990, Noyce investigaba acerca de los microchips, hasta que el 3 de junio falleció tras un fallo cardíaco. Después de su muerte, la compañía Intel prosiguió desarrollando los microprocesadores a través de la línea Pentium, consiguiendo que la mayoría de computadoras tengan como cerebro, un Pentium o un Celeron. En el año 2000, Jack Kilby recibe el Premio Nobel de Física, conjuntamente con Robert Noyce, por su trabajo acerca de los microprocesadores

The exemplary technology of this era is the microchip—the computer inscribed on a tiny piece of processed material. More than any other invention, this device epitomizes the overthrow of matter. Consider a parable of the microchip once told by Gordon Moore, chairman of Intel and a founding father of Silicon Valley: “We needed a substrate for our chip. So we looked at the substrate of the earth itself. It was mostly sand. So we used that. “We needed a metal conductor for the wires and switches on the chip. We looked at all the metals in the earth and found alutninum was the most abundant. So we used that. “We needed an insulator and too saw that the silicon in sand mixed with the oxygen in the air to form silicon dioxide—a kind of glass. The perfect insulator to protect the chip. So we used that.” The result was a technology—metal oxide silicon (MOS), made from metal, sand, and air—in which materials costs are less than 1 percent of total expense. Combining millions of components on a single chip, operating in billionths of seconds, these devices transcend most of the previous constraints of matter. The most valuable substance in this, the fundamental product of the era, is the idea for the design.

The overthrow of matter in economics is made possible by the previous overthrow of matter in physics. All the cascading devaluations of matter in the global economy and society originate with the fundamental transfiguration of matter in quantum science. Max Planck, the discoverer of the quantum, offered the key when he asserted that the new science entailed a movement from the “visible and directly controllable to the invisible sphere, from the macrocosm to the microcosm.” The macrocosm may be defined as the visible domain of matter, seen from the outside and ruled by the laws of classical physics. The microcosm is the invisible domain, ruled and revealed by the laws of modem physics.

Microcosm The Quantum Revolution In Economics And Technology .  Pagina 18
Gilder, George

Gilder, the quantum era

According to Gilder, the quantum era is still unfolding in a fourfold transformation of the world—in science, technology, business, politics—and even in philosophy. But all the changes converge in one epochal event: the overthrow of matter.
The change originates in the microcosm of quantum theory itself – which overthrew matter in the physical sciences.

At the foundation of the universe, Isaac Newton’s hard, inert, and indivisible solids gave way to a rich panoply of paradoxical sparks, comprising waves and particles that violate every principle of Newtonian solidity.

At the root of all the cascading changes of modern economic life—devaluing material resources in technology, business, and geopolitics—is this original overthrow of material solidity in the science of matter itself.
The second step in the overthrow of matter came in the use of quantum theory to overcome the material limits of weight, heat, and force in the creation of new machines. The industrial age essentially managed and manipulated matter from the outside, lifting it against gravity, moving it against friction, melting or burning it to change its form. The quantum era manipulates matter from the inside, adapting its inner structure to human purposes.

In the microchip, combining millions of components operating in billionths of seconds in a space the size of the wing of a fly, human beings built a machine that overcame all the conventional limits of mechanical time and space. Made essentially of the silicon in sand—one of the most common substances in earth—microchips find their value not in their substance but in their intellectual content: their design or software.

The third great manifestation of the overthrow of matter is the impact of this technology on the world of business. By overcoming the constraints of material resources, the microchip has devalued most large accumulations of physical capital and made possible the launching of global economic enterprises by one entrepreneur at a workstation.

With the overthrow of the constraints of material scarcity, gravity, and friction, large bureaucracies in government and business lose their power over individual creators and entrepreneurs.
The fourth phase of the overthrow of matter is the collapse of the value of natural resources and territory in determining the distribution of power among nations. The microcosms of science, technology, and enterprise have converged in a global quantum economy that transcends all the usual measures of national power and wealth.

El transistor (bardeen y los colaboradores en el premio nobel)

El transistor (bardeen y los colaboradores en el premio nobel)

En el centro de todo esta el electron.

Most interested people understand much of what electrons do. But very few have any clear idea of what an electron actually is, or its implications for the concept of matter and its overthrow in the world economy.

From the telephone to the human brain, from the television set to the computer, information mostly flows in the form of electrons. This function of electrons has quantum roots. As in Planck’s black body radiation, electrons do not respond to applied energy in a continuous, proportional, or linear way. They are non-linear; they have quantum thresholds and resonances. These quantum functions shape their electrical properties. In order to move through a solid, electrons must be freed from their atoms, jumping from one energy state to a free state across measurable energy “band gaps” in strict accordance with quantum rules. These rules give electrons identifiable and controllable features that can be used to convey information.

With controlled pulses of electrons down wires, computers could be interconnected around the world. With controlled flows of electrons in and out of tiny capacitors, computer memories could be constantly read, written, and restored.

Crossing decisively into the microcosm, Heisenberg declared that the waves which Bohr had examined in recreating the atom were not conventional waves at all. Designated “probability amplitudes,” they were waves or fields that defined the statistical likelihood of finding an electron at any particular location. This was a climactic step in the overthrow of materialism in physics. With the electron itself depicted as a wave and the wave depicted as a probability field, the specific particle in this theory had disappeared into a cloud. With it disappeared the last shreds of Newtonian logic and mechanistic solidity.

As Bohr put it, quantum theory required “a final renunciation of the classical idea of causality and a radical revision of our attitude toward the problem of physical reality.”

Microcosm The Quantum Revolution In Economics And Technology .  Pagina 25

Los encargados de aplicar en forma práctica la teoría cuántica, fueron los inventores del transistor en 1948, lo cual significó un hito en la historia del desarrollo de las tecnologias de información.

John Bardeen (1908-1991), William B. Shockley (1910-1989), and Walter H. Brattain (1902-1987)

Bardeen en su Nobel Lecture de 1947 establece que en la raiz de toda la investigacion que condujo al desarrollo del primer transistor estuvo la Wilson’s quantum mechanical theory, based on the energy band model, and describing conduction in terms of excess electrons and holes. It is fundamental to all subsequent developments. The theory shows how the concentration of carriers depends on the temperature and on impurities.

JO H N BA R D E E N Semiconductor research leading to the point contact transistor

Nobel Lecture, December 11, 1956

Del mecanicismo causistico a lo probabilistico

Del mecanicismo causistico a lo probabilistico

Today most sophisticated people imagine that they have transcended Newton and have come to terms with the findings of modern science. But they have not. As an intellectual faith, materialist logic still prevails.

We still believe that the solid world we see and feel—governed by determinate chains of cause and effect, rooted in Newtonian masses and forces—is real and in some sense definitive. The atom may not be ultimate, but they assume some other particle is, perhaps the quark.

At the foundations of the physical world, so it is supposed, are physical solids—”building blocks”—that resemble in some way the solids we see. They link together in causal chains of mechanical logic like a set of cogs and levers. These solids are deemed to comprise all matter, from atoms and billiard balls to bricks and the human brain.

Announced in 1913 and proved for the single electron of the hydrogen atom, the Bohr model was the first great vindication of quantum theory. One test of scientific advance is whether it extends the realms of human understanding and control.

The established physics could not explain the effectiveness of chemistry, let alone extend it to atoms. Unlike a solar system, atoms do not exist in majestic isolation. Ceaselessly in movement, they endlessly jiggle together in what is called Brownian motion. We even step on them. In a world of Newtonian continuities, electron orbits would vary continually as atoms collided with one another. Constantly knocked loose in these collisions, electrons in a conductor should flow far more copiously and respond to heat more massively than experiments showed.

Reunifying chemistry and physics in the microcosm, the new model of the atom explained the apparent solidity of the physical world. Establishing a gap, called a band gap, between an electron in its ground state and an electron excited to a higher energy level, the new physics showed why the constant collisions of atoms do not cause the atomic structure to collapse. A small collision will not affect an atom. An electron will not respond to any small disturbance. It will react only if it receives its necessary quantum of energy, defined by its resonant frequency times Planck’s constant.

Microcosm The Quantum Revolution In Economics And Technology .  Pagina 21

Heisenberg y los observables

Heisenberg y los observables

Generalmente es reconocido el hecho de que el papel publicado en julio de 1925 por Werner Heisenberg es el que di fin a la “Teora Cuntica Vieja”, y que con la exposicin de su Mecnica Matricial se di entrada a la Mecnica Cuntica tal como se conoce y se practica en la actualidad.

Heisenber mismo, en su famoso articulo fundacional de la nueva mecánica cuántica, resaltaria el hecho de que “it is well known that the formal rules which are used in quantum theory for calculating observable quantities such as the energy of the hydrogen atom may be seriously criticzised on the grounds that they contain, as basic element, relationships between quantities that are apparently unobservable in principle, e.g., position and period of revolution of electron... Experience however shows that only the hydrogen atom and its Stark effect are amenable to treatment by these formal rules of quantum teory”.

Heisenberg también critica el principio de correspondencia: “It has become the practice to characterize this failure of the quantum-theoretical rules as a deviation from classical mechanics, since the rules themselves were essentially derived from classical mechanics.”

Y en concordancia con Husserl:

On doit s'accommoder du fait que ce n'est qu' travers le processus de connaissance lui-mme que se dcide ce qu'on doit entendre par "connaissance". [...] Toute formulation dans le langage est toujours, non seulement une saisie de la ralit, mais aussi une manire de la mettre en forme et de l'idaliser [...] La connaissance n'est sans doute en dernire instance rien d'autre que l'agencement non pas l'agencement de quelque chose qui serait dj disponible en tant qu'objet de notre conscience ou de notre perception, mais plutt l'agencement de quelque chose qui ne devient un veritable contenu de conscience ou un processus perceptif qu' travers cet agencement meme (Whm, 363-364)

"Philosophie. Le manuscrit de 1942"

Heisenberg, W. Philosophie. Le manuscrit de 1942. Introduction et traduction par C. Chevalley (490 p.). Editions du Seuil, 1998. Premire dition en allemand : Ordnung der Wirklichkeit, 1989. Seconde dition francaise : Arla, 2003, 2010 (173 p.)

da el paso fundamental que destraba el callejon sin salida a que habia llegado la mecanica cuantica.

“In this situation it seems sensible to discard all hope of observing hitherto unobservable quantites, such as the position and period of electron, and to concede that the partial agreement of the quantum rules with experience is more or less fortuitous. Instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur.”

Dando a luz la mecánica matricial. Y abriendo una nueva dimension para la fisica.

Quantum-theoretical re-interpretation of kinematic and mechanical relations. W.Heisenberg

As hitherto defined, quantum mechanics enables the radiation

emitted by the atom, the energy values of the stationary states, and other

parameters characteristic for the stationary states to be treated. The theory

hence complies with the experimental data contained in atomic spectra. In all those cases, however, where a visual description is required of a transient event, e.g. When interpreting Wilson photographs, the formalism of the theory does not seem to allow an adequate representation of the experimental state of affairs. At this point Schrödinger’s wave mechanics, menawhile developed on the basis of the de Broglie’s theses, came to the assistance of quantum mechanics.

Tal como Heisenberg lo considera, el cambio de la mecánica clásica a la mecánica cuántica fue así:

In classical physics the aim of research was to investigate objective processes occurring in space and time, and to discover the laws governing their progress from the initial conditions. In classical physics a problem was considered solved when a particular phenomenon had been proved to occur objectively in space and time, and it have been shown to obey the general rules of classical physics as formulated by differential equations.

The manner in which the knowledge of each process had been acquired, what observations may possibly have led to its experimental determination, was completely immaterial, and it was also immaterial for the consequences of the classical theory, which possible observations were to verify the predicitions of the theory. In the quantum theory, however, the situation is completely different. The very fact that the formalism of quantum mechanics cannot be interpreted as visual description of a phenomenom occurring in space and time shows that quantum mechanics is in no way concerned with the objective determination of space - time phenomena. On the contrary, the formalism of quantum mechanics should be used in such a way that the probability for the outcome of a further experiment may be concluded from the determination of an experimental situation in an atomic system, providing that the system is subject to no perturbations other than those necessitated by performing the two experiments.

Nobel Lecture 1939 Heisenberg.

La catástrofe del átomo de Bohr.

La catastrofe del atomo de Bohr.

Bohr cuantizó las orbitas planetarias enunciando el principio, más fundamental de que el ímpetu angular del sistema es un múltiplo entero de la constante de Planck (entre 2p); pero, a pesar de su gran éxito en los átomos hidrogenoides, finalmente se debió concluir que esta cuantización es incorrecta.

En general, el modelo de Bohr:

1. No explica de donde surge la relación mvr = nh/2p ; con lo cual sus resultados son asombrosamente congruentes con los hechos experimentales.
   2. Solo es aplicable para el átomo de hidrógeno, para átomos más complejos sus ecuaciones resultan insatisfactorias
   3. Cuando el espectro del átomo de sodio se examina con un espectroscopio de alta resolución, la línea original se descompone en dos, lo que no se explica en su teoría.
   4. La teoría de Bohr no explica por que cuando los espectros de emisión atómica se observan en presencia de un campo magnético surge una multiplicidad de líneas espectrales (efecto Zeeman).
   5. No explica porque algunas líneas espectrales son mas brillantes que otras.
   6. Existía una incoherencia lógica en su teoría , pues al lado de los principios fundamentales de la Física Clásica y el electromagnetismo, se introdujeron postulados nuevos (momento angular y condición de frecuencia) que entraban en contradicción con los principios de los cuales partía.

Las reglas de la cuantización (momento angular y condición de frecuencia) se añadierón a la Física Clásica sin ninguna liga lógica. Bohr mismo hizo un examen crítico de su teoría a fin de mostrar a los jóvenes físicos la necesidad de buscar los principios de la teoría de los fenómenos atómicos

Con el advenimiento de la mecánica ondulatoria de De Broglie y la ecuación de Schrodinger los estados dinámicos quedaron limitados automáticamente a la serie postulada por Bohr y sólo se abandona el concepto clásico del electrón "planetario".

QG-Mendeleiev, Genaro carmona.

Modelo atómico de Niels Bohr

http://wwwgcmendeleiev.blogspot.ru/2009/02/modelo-atomico-de-niels-bohr.html

Pozos Cuanticos. Particula en un pozo de potencial, infinito, finito.

La energia de una particula atrapada en una caja de paredes energeticas de altura infinita, esta cuantizada.
Asi pasa cuando se atrapa una particula en una caja de paredes finitas, solo que la longitud de onda es mayor que la distancia entre las paredes, por el efecto tunel.

En estos videos se explica, y se ven ejemplos de esto. No es perdida de tiempo verlos.

Chapter 3 Metal-semiconductor interfaces and ballistic electron emission microscopy

This chapter is divided into two parts. In the first part the metal-semiconductor (M-S) inter- faces are discussed - viz. the formation of the Schottky barrier (SB), followed by the models to determine the Schottky barrier height (SBH), possible barrier lowering mechanisms like image force lowering, lowering due tunneling and due to electrostatic screening. In the sec- ond part we discuss the basic concepts of hot electron transport, as used in ballistic electron emission microscopy (BEEM). The various modes of operation in BEEM are presented. This is then followed by discussions of the various possible scattering mechanisms for hot elec- trons. The most commonly used model to determine the local Schottky barrier height, called the Bell-Kaiser model is discussed.

3.1    Transport at metal-semiconductor interfaces

We discuss the different transport mechanisms that occur at biased  and  non biased Schottky  interfaces between a metal and a semiconductor. First we discuss common transport models such as thermionic emission and  tunneling across such interfaces [1], [2].  We explain  the  relevance of incorporating tunneling mechanisms to ex- plain the observed current-voltage (I-V) characteristics in our devices  (as presented in Chapter 5). Further, we also discuss in details  the hot electron  transport at simi- lar Schottky  interfaces using the technique of ballistic electron  emission microscopy (BEEM). We explain  the  different contributions of hot  electron  scattering in met- als, semiconductors and  their  interfaces to hot electron  transport. We also discuss the factors  that  influence the hot electron  attenuation length  in metals.   We finally discuss the Bell-Kaiser  model  that  is commonly used  to extract  the local Schottky barrier height  at metal-semiconductor (M-S) interfaces [3].

3.1.1   Schottky barrier formation

When  an n-type semiconductor is brought in contact  with  a metal,  electrons will flow from  the semiconductor to the metal  if the Fermi  level of the semiconductor (SC) is higher than  that of the metal.  Such flow of electrons causes  the Fermi levels of the metal  and  the semiconductor to align.  The electrons moving from the semi- conductor to the metal  leave depleted donors in a region  close to the interface that create  an electric field in the semiconductor. This field causes  band  bending in the semiconductor close to the interface,  leading to the formation of a Schottky  barrier as shown in Fig.  3.1.  Such a barrier is a rectifying barrier for electronic  transport across  the metal  semiconductor interface.  

Figure 3.1: Energy  band  diagram of formation of a metal-semiconductor (n-type)  (M-S) con- tact.  (a) before contact,  (b) after the contact; the formation of a Schottky  junction  for the case where φm > φS . The M-S interface shown in (b) is at equilibrium.

Figure 3.2:  The metal  and  the  semiconductor are  shown in contact  at the  top.   (a) shows the charge  density, (b) electric field and  (c) electrostatic potential in the semiconductor as a function of the distance from the interface (x) into the semiconductor.

In Fig.  3.1 (a) the conduction band,  va- lence band  and Fermi level of the semiconductor are given by EC , EV  and EF S . φm is the work  function of the metal,  which  corresponds to the energy difference between  the vacuum level and the Fermi level of the metal.  χ is the electron  affinity of the semiconductor, which  is measured from  the bottom of the conduction band  to the vacuum level.  The obtained Schottky  barrier allows  electrons to flow from the semiconductor to the metal, but blocks it in the opposite direction, which  makes it a rectifying junction. In this thesis,  the two most important parameters that are to be considered are the depletion layer width (W ) and the Schottky  barrier height  (φB ).

3.1.2   Depletion layer

As mentioned above, when  a metal is brought in contact with a n-type semiconduc- tor, electrons flow from the semiconductor to the metal.  This leaves a region, close to 

the interface,  depleted of mobile  electrons. This region  is called the depletion layer. The depletion width (W ) in a Schottky  junction  can be determined analytically us- ing Poisson’s  equation. The depletion layer  width depends on the semiconductor permittivity ( s ), donor concentration (ND ), built-in  potential (Vbi ) and applied bias (V) [1], [2], following the equation:

(3.1)

Because  of the static  charge  in the depletion layer  an electric  field is present.  The strength of this field depends on the charge carrier density (ND ), the depletion width (W ), the permittivity of the semiconductor ( s ) and  the distance from the interface (x). This dependence is given by [1], [2]:

(3.2)

The electric field is the largest  at the interface,  i.e. for x = 0

(3.3)

The presence of an internal electric  field  across  the  M-S interface results in a po- tential  difference between the metal  and  the semiconductor bulk called the contact potential (V ) which  is given as:

(3.4)

3.1.3   Schottky barrier height

From  Fig.  3.1 (b) it is seen  that  the  Schottky  barrier height  depends on the  work function of the metal (φm ) and the electron  affinity of the semiconductor (χ) as:

φB = (φm − χ)                                        (3.5)

This relation is called the Schottky-Mott relation. This model of determining a Schot- tky  barrier is based  on a few assumptions: (a) The surface  dipole  contribution to φm and  χ do not change  when  the metal  and  semiconductor are brought together. (b) There  are no localized states  present on the surface  of the semiconductor, and it forms  a perfect  contact  with  the metal.   In more  complex  approximations deter- mining the Schottky  barrier height, the influence of image potential, tunneling, and electrostatic screening should be taken  into  account.  These  three  mechanisms are discussed as follows:

Figure 3.3: (a) Left: field caused by an electron  close to the metal-semiconductor interface and surface  charges.   Right:  field caused by two opposite charges on either  side of the interface. (b) Representation of a Schottky  barrier showing the  image  force effect which  lowers  and pulls  the SBH maximum inside  the semiconductor, indicated by the shaded blue region.

1. Image force lowering.

2. Lowering due to tunneling.

3. Lowering due to electrostatic screening.

Schottky barrier lowering by image charge potential

When  an electron  approaches the metal-semiconductor interface,  it attracts surface charges of opposite sign in the metal.  These surface charges in the metal film exactly balance  the field generated by the electron  in the semiconductor, so that it does not penetrate into the metal  as shown in the left side of Fig. 3.3 (a). The field produced by these  surface  charges and  the electron  in the semiconductor is the same  as the field generated by an electron  in the semiconductor and another particle  of opposite charge  in the metal  as shown in the right  side of Fig.  3.3 (a).  This other  particle  is called the image charge.  This image charge in the metal film creates an image charge

Figure 3.4: Field and thermionic-field emission under forward bias. EF M and EF S represent the Fermi levels of the metal and the semiconductor respectively, V is the applied voltage  and Em is the energy where the contribution of thermionic-field emission has its maximum.

potential close to the barrier. This field is the highest at the barrier, because  there the electron  is very close to its image  charge.  At a large distance from the interface,  the electron  hardly feels the attraction of its image  charge  anymore and  the attractive force goes to zero. The potential energy caused by this image charge as a function of distance from the interface is schematically depicted in Fig. 3.3 (b). When the image potential energy is added to the original potential in the depletion layer, we find the barrier shape  that accounts for the image force. This resulting barrier height  is lower by an amount ∆EI given by [1], [2]:

(3.6)

In addition to lowering of the Schottky  barrier, the image charge potential also pulls the potential maximum into the semiconductor as shown in Fig. 3.3 (b) over a dis- tance of ∆z given as follows [1], [2]:

(3.7)

Because  the  maximum potential lies inside  the  semiconductor, the  electrons first travel  a short  distance through the semiconductor before  they  reach  the top of the barrier.

Schottky barrier lowering by tunneling

The  second  mechanism that  can  cause  lowering of the  effective  Schottky  barrier height  is tunneling, either direct or thermally assisted. Under forward bias, in heav- ily doped semiconductors at low temperatures, electrons can tunnel directly from the Fermi level of the semiconductor, through the Schottky  barrier, to the metal.  For reverse bias, tunneling from the metal  to the semiconductor can happen under the same circumstances. The current that arises from these electrons is called field emis- sion as shown in Fig.  3.4. When  electrons have  a certain  thermal energy, they  can also tunnel through the barrier with  thermal assistance. Since the barrier is thinner at higher energies, electrons with  higher energies have  higher tunneling probabil- ity.  On the other  hand, the number of electrons with  higher energies are few.  This implies  that  the electrons with  a certain  amount of energy have  maximum contri- bution to thermionic-field emission (denoted by Em ), as shown in Fig.  3.4.  When the  Schottky  barrier is approximated as a triangular potential barrier, the  tunnel- ing probability (P) for an electron  having an energy ∆E less than  the height  of the barrier is given by following equation [2]:

(3.8)

Here ∆E is the energy of the electron  below the top of the barrier and Vbi is the built- in potential. E00  is a parameter which plays an important role in tunneling theory.  It is the diffusion potential of a Schottky  barrier such that the transmission probability for an electron  whose  energy coincides with  the bottom of the conduction band  at the edge of the depletion region  is equal  to e−1  [2], and is given by:

(3.9)

Here ~ is the Planck constant, Nd  the donor concentration and   s the permittivity of the semiconductor.

From Eqn.  3.8 it can be deduced that  an E00 value  of 0 leads  to a tunnel proba- bility of zero and  a higher value  leads  to a higher tunnel probability. Also, a lower

∆E value,  which  means  a higher electron  energy, leads  to a higher tunnel proba- bility.  When  ∆E becomes  zero,  i.e.  the electron  has an energy equal  to that  of the Schottky  barrier, the tunnel probability goes to 1, which  we would expect, since the electron  has enough energy to overcome the barrier. Because the tunneling electrons can cross the barrier at an energy lower  than  the maximum of the Schottky  barrier, direct  and  thermally assisted tunneling lower  the effective  Schottky  barrier height as shown in Fig. 2.8. The amount of Schottky  barrier height  lowering due  to these

Figure 3.5: Schottky  barrier lowering due to electrostatic screening. φ and φef f represent the original and effective Schottky  barrier height  and ∆EES is the Schottky  barrier lowering due

to electrostatic screening. Adapted from [4].

effects is given by [12]:

(3.10)

Schottky barrier lowering due to electrostatic  screening

The third mechanism that can cause Schottky barrier lowering is electrostatic screen- ing. In ideal Schottky  theory,  the potential distribution in the metal is assumed to be constant. However, this condition may be violated in the metal close to the interface with  the semiconductor, when  the magnitude of the free charge  carriers  induced at the  surface  of the  metal  becomes  large  [5].  This is the  case at an interface with  a large permittivity semiconductor, such as Nb:SrTiO3 . This large permittivity causes a voltage  drop on the  metal  side  of the  junction  due  to the  conservation of elec- tric displacement, as shown in Fig 3.5 [6]. The barrier potential corresponds to the energy that  is needed to excite an electron  from  the bulk  of the metal  to the semi- conductor. From  Fig.  3.5 we can see that  this barrier is lowered by the amount of the voltage  drop in the metal.  For zero applied bias, this value  can be calculated as:

(3.11)

Figure 3.6: Transport processes in a forward-biased Schottky  junction. (a) Thermionic emis- sion over the barrier, (b) Thermally assisted tunneling through the barrier and (c) Direct tun- neling from the bottom of the conduction band.

In contrast to barrier lowering by image  force and  tunneling, the barrier lowering due to electrostatic screening is proportional to the square root of the semiconductor permittivity. Thus for higher values  of the relative  permittivity the Schottky  barrier is reduced by a larger  amount.

3.2    Electronic  transport across a Schottky barrier

For macroscopic characterization of Schottky  junctions, current-voltage measure- ments  (I-V measurements) are  most  commonly used.    In  such  measurements, a varying voltage  is applied across  the  interface and  the  current through the  inter- face is measured as shown in Fig.  3.6. In this circuit,  electrons cross the barrier at the interface between the semiconductor and  the metal.  Following are the various mechanisms by which  electronic  transport across the barrier can take place (Fig 3.6) [1], [2].

1. Thermionic emission over the top of the barrier.

2. Thermally assisted tunneling through the barrier.

3. Direct tunneling through the barrier.

3.2.1   Thermionic  emission

The electrical  transport across  an ideal  Schottky  barrier is described by thermionic emission [1]. By subtracting the current which  flows from the metal to the semicon- ductor JM →S   from the current flowing  from the semiconductor to the metal  JS→M the following expression for the total current I is obtained [1], [2]:

(3.12)

where q is the charge  of the electron,  kB  the Boltzman constant, T the temperature, A* the Richardson constant, φB the barrier height and n the ideality factor (unity  for purely thermionic emission dominated transport), and A∗ is given by:

(3.13)

where me ∗ is the effective mass  of the electron  in the semiconductor. The value  of A∗, the Richardson constant, used in this thesis for Nb doped SrTiO3  semiconductor is 156 Acm−2 K−2 [8]. When temperature is kept constant during a measurement, the only  variables are the  ideality factor  and  Schottky  barrier height  for zero  applied voltage,  so these  parameters can be determined by fitting  the  experimental data. However in practice, resistances appear in the  semiconductor and  the  rest  of the circuit  as well,  and  contribute to the  series  resistance causing the  current-voltage characteristics to deviate from thermionic emission theory at high voltages. Because of the series resistances in the circuit, the applied voltage  does not drop completely at the Schottky  barrier, but  also drops partially in the rest of the circuit  and  in the semiconductor. To reckon for the voltage  loss due to these resistances we can adjust Eqn. 3.12. The voltage  drop over the interface is then  given by V  minus I R instead of V , where R is the  total  resistance of all elements in the  circuit.   This results in following equation for current:

(3.14)

3.2.2   Electron transport by tunneling

Thermally assisted  tunneling and direct tunneling

Since electrons have a higher thermal energy at higher temperatures, thermal emis- sion (represented by (a) in Fig. 3.6) is more dominant in that case. For lower temper- atures, electrons lose their thermal energy and direct  tunneling becomes  dominant. In between these  two  regimes lies the thermally assisted tunneling regime.   While at very  low temperatures electrons tunnel directly through the barrier (direct  tun- neling,  represented by (c) in Fig.  3.6), at intermediate temperatures electrons first get thermally excited and then tunnel at a higher energy corresponding to a thinner part  of the barrier (thermally assisted tunneling, represented by (b) in Fig.  3.6). In the direct tunneling regime  the current is given by [7]:

(3.15)

and

(3.17)

where, E00 is a tunneling parameter (also called as characteristic energy). E00 (T

= 0 K) is 1, (Eqn. 3.9.) hence E0 (T = 0 K) equals  E00 . It also implies  that for this case

Eqn. 3.16 approaches to Eqn. 3.15 at very low temperatures.

3.3    Ballistic electron emission microscopy (BEEM)

Introduction

The technique of ballistic  electron  emission microscopy (BEEM) was developed by Kaiser and Bell in late 1980’s [3]. BEEM is a non destructive technique and is based on a scanning tunneling microscope (STM) [10]. It is a modified form of STM, with an additional contact  at the bottom of the semiconducting substrate, which  can col- lect the electrons traveling through the metal overlayer and across a Schottky  inter- face. Here, the STM tip is used  to inject a distribution of hot electrons into the metal overlayer to be investigated. The hot electrons travel  through the metal  overlayer and  get scattered. A fraction  of these  electrons are able to cross the Schottky  inter- face when  they have the necessary energy and momentum to do so.

BEEM has been used  for studying hot electron  transport in thin films and multi- layers using  conventional semiconductors like Si, GaAs [11], [12], [13], [14]. Energy and spatial dependence of carrier  transport, at the nanoscale and across buried lay- ers  and  interfaces using  current perpendicular to the  plane  of the  device  can  be

Figure 3.7: Schematic  of a BEEM setup  with  its circuit  diagram. A STM tip injects hot elec- trons  in a metal  overlayer via tunneling through the vacuum barrier. The electrons travel  to the  M-S interface.   The white  arrows represent the  spatial distribution of injected  hot  elec- trons.   The electrons with  proper momentum and  enough energy reach  the semiconductor. Due to the momentum criteria,  electrons outside the acceptance cone (in purple) are reflected back from the M-S interface.

investigated using  BEEM. The basic schematic of BEEM with  its circuit  diagram is shown in Fig.  3.6. A negative bias, VT , is applied to the tip to inject electrons into the metal film, as tunnel current, IT . The electrons travel through the film, across the interface and  are collected  in the semiconductor as a BEEM current, IB . The BEEM current constitutes a fraction  of electrons which  have  the  proper energy and  mo- mentum to overcome the Schottky  barrier height. Such an energy and  momentum filter is represented by an acceptance cone at the Schottky  interface as shown in Fig.

3.8. The energy schematic of the BEEM is shown in Fig. 3.9.

Hot electrons and their scattering mechanisms

When  the injected  electrons have  an energy a few tenths of an electron  volt above the Fermi  level of the system they  are referred to as "hot" electrons. By applying a bias of a few eV to the STM tip with  respect  to the Fermi level of the metal  layer we inject a distribution of electrons into the metal  overlayer. As kB T is 25 meV at room temperature, a similar  analysis yields an equivalent temperature of ≈ 12000 K for 1 eV [14]. Such an analogy leads to the term "hot" electrons when  the energy of the injected  electron  is few eV above  the Fermi level of the metallic  film. Scattering

Figure 3.8: Energy  schematics of the  BEEM technique.  It shows  the  hot  electron  distribu- tion  injected  into  the metal  overlayer.  Subsequently, a fraction  of them  get collected  in the conduction band  of the semiconductor.

mechanisms for hot  electrons and  for electrons at the  Fermi  level  are very  differ- ent.  Hot  electrons injected  at an energy eVT  can scatter  into all unoccupied states between eVT  and  EF , according to Fermi’s golden rule.  Such an electron-electron (e-e) scattering for hot electrons results in inelastic  scattering (loss of energy)  and  is a dominant scattering mechanism. In contrast, at the Fermi  level elastic  or quasi- elastic scatterings are the dominant scattering mechanism. When  the hot electrons reach the interface without being scattered inelastically or elastically,  they are called "ballistic" electrons.

The hot electron  transport in BEEM can be divided in different steps:

1. Injection of the hot electrons from the tip into the metal base.

2. Transport of hot electrons through the metal base.

3. Transmission of hot electrons across the metal-semiconductor interface.

4. Collection  at the conduction band  of the semiconductor.

Charge carriers  are  injected  from  the  tip  by tunneling into  unoccupied states of the  thin  metal  base.   This results in momentum and  energy distribution of the injected carriers  at the metal surface.  After injection, the hot electrons travel through the  metal  film and  are  scattered by cold  electrons (lying  close  to Fermi  level)  by inelastic  scattering.  However, an  energy independent elastic  [15] or quasi-elastic scattering by with  either  defects,  grain boundaries, phonons, magnons etc. can also occur. When the electrons reach the interface and satisfy the energy and momentum criteria  at the interface,  they  can be transmitted through and  enter  the conduction band  of the semiconductor and constitute the BEEM current. Due to the local nature of injecting  electrons and the requirement of lateral  momentum conservation at the Schottky  interface,  this technique results in a very high spatial resolution [16] .

3.4    BEEM Theory

In order  to extract  the Schottky  barrier height  from spectroscopy measurements, a theoretical model  is needed to fit the  data.   The first theoretical description deal- ing with the transport of hot charge  carriers  through a metal-semiconductor system in a BEEM setup  was  proposed by L.D. Bell and  W.J. Kaiser  [3].  For all the work presented in this thesis,  electrons are the charge  carriers  responsible for the BEEM current, due to the use of n-type semiconducting substrates (Nb doped SrTiO3 ).

3.4.1   Tunnel  injection  of non-equilibrium charge carriers

The  applied potential between the  tip  and  the  metal  base,  called  the  tip  voltage VT , will determine the energy of the injected  electrons. Tunneling across the poten- tial barrier between the tip and  the metal  will always result  in a distribution of the energy and momentum of the electrons. In common BEEM theory [3], the tunnel in- jection of non-equilibrium electrons from the tip into the base is assumed to behave according to the planar tunneling theory [17]. Although it has been shown that it is not always valid to use planar tunneling theory,  the voltage  spectroscopy measure- ments  with  BEEM are found to agree well with  planar tunneling based  theory [18]. At tip voltages close to the threshold this  results in a sharply peaked distribution of the injected  electrons perpendicular to the M-S interface.   Therefore the injected electrons will have little momentum parallel to the metal base (kk     k⊥).

3.4.2   Transport across the metal base

Due to scattering the spatial and energetic distribution of the electrons will broaden when  traversing the metal  base.  When  assuming a free electron  like behavior the attenuation of the electrons can be described by a single parameter called the atten- uation length, λ(E), which  in principle is energy dependent.  The attenuation can

Figure 3.9: Four different scattering mechanisms in a forward biased  BEEM, where the solid spheres represent electrons and hollow  spheres represent holes. (1) is a purely ballistic trans- port (red), (2) is inelastic scattering of hot electrons in the metal overlayer (pink) can also lead to secondary electrons, (3) elastic scattering in the metal  over layer and (4) impact  ionization where an electron-hole pair is created (green).

then be described by an exponentially decaying function depending on the injection angle θ away  from the surface  and metal film thickness d:

(3.18)

Since the electrons are injected  with  almost  zero parallel momentum kk = 0 we can assume cos(θ) ≈ 1, simplifying the equation.

3.4.3   Scattering mechanisms

All of the different scattering processes which  are relevant in this thesis occur in the metal base and interface.  In Fig. 3.9, the most prominent scattering mechanisms are depicted, which  are:

Ballistic  charge carriers   Ballistic transport is the unscattered propagation of elec- trons  through the  metal  base.   These  electrons do  not  lose  energy or  undergo a change  in momentum and  form  an  important contribution to the  BEEM current.

If the electrons travel  ballistically through the metal  base they  might  have  enough energy, depending on VT , to surmount the Schottky  barrier at the M-S interface.

Inelastic  scattering    If the electrons are scattered inelastically their  energy will be reduced. The processes dominating this form of scattering, at the energies relevant for this  thesis  is electron-electron (e-e) scattering [3] and  will  typically result  in a reduction of half the electron  energy. At low tip voltage  this effectively  means  that any  inelastically scattered electron  will  not  have  enough energy to surmount the Schottky  barrier. However at higher tip voltages, at least twice that of the Schottky barrier, the  collision  might  result  in a secondary electron  with  enough energy to surmount the Schottky  barrier, while the primary electron still has energy above the Schottky  barrier, thereby increasing the BEEM current. Although phonon scattering can also result  in energy loss, they  are not  taken  into  account since the change  in energy is negligibly small  (in the order  of kB T ) in comparison with  e-e scattering. Although plasmon excitations also cause  inelastic  scattering they  are not  relevant since the electron  energies relevant for this thesis are too low for plasmon excitations to occur.

Elastic  scattering    This  form  of scattering will  change  the  momentum but  con- serves  the  total,  kinetic,  energy of the  electrons.  Therefore any  elastic  scattering will result  in a broadening of the distribution of angular momentum. Since trans- mission across  the M-S interface depends sensitively on the momentum, as shown in section 3.4.4, elastic scattering will also have an effect on the BEEM current. Grain boundaries, defects and any inhomogeneities in general are the main elastic scatter- ing sites.

Impact ionization   When an electron  with high enough energy enters  the semicon- ductor, it could  transfer a part  of this energy to an electron  in the valence  band.   If enough energy is transferred, it could  excite the  electron  to the  conduction band creating an electron-hole pair.  This electron  could then contribute to the BEEM cur- rent.  However, for this to occur  the impacting electron  should have  an excess en- ergy  nearly  more  than  twice  the  semiconductor band  gap.   Since all experiments performed in this thesis are below this limit, impact  ionization is absent.

Transport across the metal base

Scattering causes broadening of the spatial and energetic distribution of the injected electrons while  traveling through the metal  overlayer. Scattering in the metal  over- layer  can be quantified by hot  electron  attenuation length  When  assuming a free electron  like behavior the attenuation of the electrons can be described by a single parameter called  the attenuation length, λ(E), which  is usually an energy depen- dent  parameter (Eqn. 3.18).

3.4.4   Transmission across the M-S interface

The transmission across  the  barrier is dependent on the  energy and  the  momen- tum  of the incoming electrons. Assuming the electrons satisfy  the 2-d free electron model,  their energy would be given as:

(3.19)

where m is the  rest  mass  of the  electron  and  k⊥ and  kk are the  momentum of the electron  perpendicular and parallel to the M-S interface,  respectively. Here, kk is assumed to have both Kx and ky  components. The energy of the electron  just at the maximum of the Schottky  barrier height  can now be expressed as:

(3.20)

where m∗ is the effective mass of the electron  inside  the semiconductor and  the subscript of kS   denotes the  momentum in the  semiconductor.  If we now  assume conservation of parallel momentum we can obtain  an analytical expression for the maximum allowed parallel moment kkS .  This argument would only be fully con- vincing  for a fully  epitaxial system without defects,  any  deviations from  such  a system would break  the symmetry and  therefore conservation of parallel momen- tum  could  be lost to a certain  degree. Despite this, experimental evidence for non- epitaxial Au/Si systems showing momentum conservation has been observed [19]. We can equate Eqns. 3.20 and 3.21 which  will give us an expression for Ek:

(3.21)

The maximum amount of parallel energy Ekmax would be obtained if the elec tron would have exactly zero perpendicular energy left after crossing the Schottky barrier i.e. E⊥S = 0. This shows that due to the effective mass there is a restriction on the amount of parallel momentum:

(3.22)

If the electron has more parallel momentum than Ekmax it cannot be transmitted across the M-S interface and will bounce back into the metal base. This effect is much  like the total refraction of light at an interface of two media  having different refractive indices.  Since we have found the maximum parallel momentum that elec- trons can have and we know that the maximum total energy is the Fermi energy, the minimum perpendicular energy Emin is equal  to:

(3.23)

We can now express this momentum requirement in the form of an acceptance cone at the M-S interface:

(3.24)

3.4.5   BEEM transport models

In order  to extract  the Schottky  barrier height  from  spectroscopy measurements a theoretical model  is used.  The first theoretical description dealing with the transport of hot-charge carriers  through a metal-semiconductor system in a BEEM setup  was proposed by L.D. Bell and  W.J. Kaiser [3]. The tunnel current between tip and  top metal based  on planar tunneling theory can be written as:

(3.25)

T (E⊥) is the tunnel probability for an electron  to tunnel through the vacuum barrier over  the  transverse and  parallel (to the  interface) energies, E⊥  and  Ek.   A is the constant related to effective tunneling area, f (E) is the Fermi distribution function, and VT  is the applied tip voltage.

According to the widely used  Bell-Kaiser (BK) model  [3], BEEM transmission is the fraction  of the ballistically transmitted tunnel current:

(3.26)

where R is an attenuation factor  due  to scattering in the  metal  base  and  the  M-S interface.  According to the BK model,  R is considered to be energy independent but it can also be weakly dependent on energy (other  parameters are:  

and      )

For VT  just above φB , close to threshold, above Eqns. 3.26 and 3.27 predict:

(3.27)

Figure 3.10: Typical spectra for direct (left) and the reverse (right) BEEM spectroscopy.

Such  a quadratic onset  considers classical  transmission across  the  M-S interface with parabolic conduction band minimum in the semiconductor. Considering quan- tum  mechanical transmission across the M-S interface another model  was given by Ludeke-Prietsch (LP model)  according to which  IB ∝IT (VT  − φB )2.5  [12].  It was found that near the threshold regime,  no significant difference between the BK and LP models can  be resolved beyond experimental error.   For the  Schottky  barrier extraction in our experimental measurement we have  considered the BK model  in- stead of the LP model  and we have seen a better match of SBH with the macroscopic I − V measurements.

3.4.6   BEEM spectroscopy

Direct and Reverse BEEM spectroscopy

In direct  BEEM spectroscopy, which  is one of the most  commonly used  modes in BEEM, a negative bias, VT , is applied to the tip with  respect  to the metal  forming a Schottky  barrier contact  with  a n-type semiconductor. When  the sample-tip bias is below  the Schottky  barrier height, no BEEM transmission is observed. However, IB  increases after a certain  onset  that corresponds to the local SBH. A typical  direct BEEM spectra thus  can be obtained by recording the BEEM current with  respect  to the sample-tip bias at a fixed tip position (shown in fig.  3.8 ). Usually, to improve

Figure 3.11: Energy  schematic of the reverse BEEM spectroscopy.

the signal to noise ratio, several  BEEM spectra are recorded to obtain  a single aver- aged spectrum. The final BEEM spectrum provides valuable information on energy dependence of hot electron  transport in the metal  film as well as the M-S interface. The onset  of the  BEEM spectra determines the  local Schottky  barrier height  with high  accuracy ( of ±0.02 eV) whereas the spectral shape  carries  information about scattering in the metal film, across the M-S interface and in the semiconductor.

In Reverse  BEEM spectroscopy, a positive bias is applied to the  STM tip  with respect  to the metal layer and a distribution of electrons is extracted from the metal overlayer (grown on a n-type SC) to the STM tip.  Reverse  BEEM is based  on the collection  of only  secondary electrons in the  conduction band  of the semiconduc- tor. These secondary electrons are produced by electron-electron scattering which is similar  to Auger  like scattering.

In the case of R-BEEM [20], hot holes  injected  by the tip (corresponding to hot electron  extraction from  the  metal  overlayer) lose energy by creating a secondary electron-hole (e-h) pair.   The energy of the  injected  holes  is transferred to the  ex- cited  electrons up  to a maximum of EF,m + eVT .  If the  secondary electrons have enough energy and  momentum to surmount the  barrier, they  can be collected  as collector current with the same sign as direct BEEM. Considering free electrons and zero temperature, R-BEEM transmission can be written as:

(3.28)

where EF,m  is the base Fermi energy, P (E, E⊥)  is the probability of creation of ex- cited electrons from the injected  hot holes.  The excited  electrons are then  collected above φB with proper momentum. Near threshold, the above expression of R-BEEM transmission can be simplified as:

(3.29)

3.4.7   Hot electron attenuation  length

With increasing metal  base thickness the BEEM current is attenuated. The total at- tenuation length, λ, is related to the inelastic  attenuation length, λI , and  the elastic attenuation length, λe , as described by Matthiessen’s rule:

(3.30)

λe   corresponds to scattering due  to different factors  viz.  defects,  grain  bound- aries, phonons, magnons, polarons etc. From equation 3.18 it is clear that the trans- mission is exponentially dependent on the thickness of the metal  base.  Therefore, by varying the  metal  base  layer  thickness and  measuring the  BEEM transmission at a particular energy, a plot can be obtained of the transmission versus metal  base thickness and energy. The slope of the plot (semi-log)  gives the electron  attenuation length  at a particular energy. The energy dependence of the attenuation length  can now be obtained by repeating this process  at different energies. Although there  are possibilities of extracting the two different attenuation lengths λi and λe  from λ, this is generally not straight forward.

3.4.8   BEEM sample requirements

In BEEM, the measured signals  are often very low (tenths  of pA) and  it is thus  im- portant to reduce noise in the system to measure such tiny currents. The most  im- portant source  of noise in the BEEM signal  arises from the feedback resistors of the operational amplifier (op-amp) circuits,  which  amplify the BEEM current, and from the Schottky  interface.  The feedback resistors of the op-amp circuits  (BEEM current is monitored by a two-stage-op-amp) add noise to the BEEM channel. Although this is hard  to avoid,  the other  noise source  related to the sample can be reduced.  The

Figure 3.12: As the electrons pass through the metal over layer they are inelastically scattered resulting in an exponential decay of the transmission.

voltage  fluctuation across  a resistor at finite temperature is known as Johnson noise

and is given by

(3.31)

where ∆ V is the root mean  square of the voltage  fluctuations, kB  is the Boltzmann constant, T the measurement temperature, B the measurement bandwidth and R the value  of the resistor.

The current passing through a Schottky diode,  described by thermionic emission theory is given  by Eqn.  3.14. In BEEM we are interested in the zero bias resistance of the diode  which  is given as

(3.33)

Thus, by increasing the zero bias resistance R0  of the diode,  its contribution to noise can be decreased. To make sure the diode  is not dominating the noise, its resistance should be higher than the resistance of the op-amp. For common M-S interfaces such as Au/Si the SBH is ≈ 0.8 V, the junction  resistance is of the order  of 1 GΩ at room temperature for a diode  area with  a diameter of 150 µm and thus  is high enough to make sure the sample is not dominating the noise. This is thus an important sample requirement for BEEM studies [11].

3.5    Conclusions

In this  chapter we  discussed the  different transport mechanisms across  a metal- semiconductor (M-S) interfaces along  with  the  different models that  are  essential to the determination of the Schottky  barrier height. We also discussed the different modes of ballistic  electron  emission microscopy that  are used  in this thesis  and  the commonly used  model  to interpret hot  electron  transmission in metal  layers  and across their interfaces with a semiconductor.

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