Quantum Theory



Nothing is Real:
A Brief Introduction to Quantum Theory and Schrödinger's Cat


Quantum mechanics provides the fundamental underpinning of all modern science. The equations describe the behaviour of very small objects, generally speaking the size of atoms or smaller, and they provide the only understanding of the world of the very small. Without quantum mechanics, chemistry would still be in the Dark Ages,there would be no science of molecular biology, no understanding of DNA, no genetic engineering. Scientists couldn't build lasers, or nuclear power stations, and they couldn't explain why the sun stays hot.

Until the end of the nineteenth century, classical physics appeared to be sufficient to explain all physical phenomena. The universe was conceived as containing matter which consisted of particles, obeying Newton's laws of motion, and radiation (waves) following Maxwell's equations of electromagnetism. However, during the late nineteenth century and the first quarter of the twentieth, experimental evidence accumulated which required new concepts to be developed which were radically different from those of classical physics. These new concepts were quantum theory.

Quantum theory makes some very strange predictions. In fact the world of quantum mechanics is so strange that Albert Einstein found it incomprehensible, and refused to accept all of the implications of the theory developed by Erwin Schrödinger and his colleagues. Einstein, and many other scientists, found it more comfortable to believe that the equations of quantum mechanics simply represent some sort of mathematical fudge, which just happens to give a reasonable working guide to the behaviour of atomic and subatomic particles but that conceals some deeper truth that corresponds more closely to our everyday sense of reality. For what quantum theory says is that nothing is real and that we cannot say anything about what things are doing when we are not looking at them. Schrödinger's mythical cat was invoked to make the difference between the quantum work and the everyday world clear. In the world of quantum theory, the laws of physics familiar from the everyday world no longer work, instead, events are governed by probabilities.

The Paradox of Schrödinger's Cat

It is true to say that a radioactive atom might decay (emitting an electron or other such sub-atomic particle) or it might not, making it possible to set up an experiment in which there is a fifty:fifty chance that one of the atoms in a lump of radioactiive material will decay in a certain time and that a detector will register the decay if it does happen.

Schrödinger was just as upset as Einstein about the implications of quantum theory and so tried to show the absurdity of those implications by imagining such an experiment and in 1935 Schrödinger published an essay describing the conceptual problems in quantum theory. A brief paragraph in this essay describes the cat paradox.1

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The Psi function for the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

In other words, such an experiment can be set up in a closed container, which contains a live cat and a phial of poison arranged that if the radioactive decay does occur then the phial is broken and the cat dies. In the everyday world, there is a fifty:fifty chance that the cat dies, and without looking inside the box we can say, quite happily, that the cat is either dead or alive. However, according to quantum theory, neither of the two possibilities open to the radioactive material, and therefore to the cat, has any reality unless it is observed. The atomic decay has neither happened, nor not happened. The cat has neither been killed nor not killed. Until we look inside the box to see what has happened.

There are two broad ways of interpreting the paradox of Schrodinger's cat. The conventional, Copenhagen, interpretation looks at these possibilities from a different perspective, and says, in effect, that both wave functions are equally unreal, and that only one of them crystallises as reality when we look inside the box. Hugh Everett provides us with an alternative interpretation which accepts the quantum equations at face value and says that both cats are real. There is a live cat, and there is a dead cat; but they are located in different worlds. It is not that the radioactive atom inside the box did or did not decay, but that it did both. Faced with a decision, the whole world, or universe, split into two versions of itself, identical in all respects except that in one the cat is dead, and in the other, the cat is alive. More information on the Copenhagen Interpretation can be found at the Stanford Encyclopedia of Philosophy as can more information on the Many Worlds Interpretation.

Theorists who accept the pure version of quantum theory say that the cat exists in some indeterminate state, neither dead nor alive, until an observer looks into the box to see how things are getting on. However, this particular experiment should not be taken too far as there are no pure 'alive' or 'dead' cat states. A cat is a complex thermodynamical system which at finite temperature is describable (in principle anyway) by a density matrix with no coherence, and hence no interference, between 'alive' and 'dead' states.

Basically, nothing is real unless it is observed.2



Exchange Bias, Fact or Fiction?:
A brief introduction to exchange bias


Magnetic recording systems, such as the hard disk, constitute the main form of data storage and retrieval in present-day computer and data processing systems. Yet, despite some fundamental changes to the components, today’s hard disk has barely changed in composition since 1973 and the Winchester hard disk. To record information, data is written and stored as magnetisation patterns on the magnetic recording medium. The data is written using a write head and retrieved using a read head. It is in read head technology that some of the most fundamental changes have been made. Early heads were primitive affairs being electromagnetic elements with wire wrapped around a laminated iron core. In the 1970s Thin Film Inductive (TFI) technology was introduced, but this did not overcome the problems of the previous heads. First, they had to switch constantly between the reading and writing functions, slowing down the performance. Second, it became impossible to increase the sensitivity of the thin film head without reducing its ability to write data.

Then in 1991, IBM introduced Magnetoresistive head technology and this paved the way forward for future developments. The magnetoresistive head features a separate read element, which is a thin strip of magnetic material that changes its resistance in the vicinity of a magnetic field. As the disk passes under the head, changes in magnetic orientation of the bits on its surface create resistance changes in the magnetoresistive strip. Although this made the development of larger capacity hard drives possible, it had its own limitations. Once aerial densities (the quantity of data that can fit into a particular space, usually measured in Gigabits per square inch) reached above three Gigabits per square inch, the changes in resistance became much smaller and rendered the magnetoresistive head ineffective.

By 1994, IBM had come up with a technology that used a phenomenon called the Giant MagnetoResistive (GMR) effect. This is the tendency of materials comprising thin, alternating layers of metallic elements to exaggerate small changes in resistance. Most magnetoresistive sensor devices used, such as magnetic random access memories (MRAM) or in read head technology, have an arrangement of an antiferromagnetic (AF) layer, a “pinned” or exchange-biased ferromagnetic layer (F1), and an “unpinned” or free ferromagnetic layer (F2). There is also a spacer layer between the pinned and unpinned ferromagnetic layers, which may be a nonmagnetic conductive layer, as in giant magnetoresistive spin valves, or an insulator as in magnetoresistive tunnel junctions (MTJs). The coupling between the AF layer and the pinned layer F1 is called exchange anisotropy (or bias) and ensures that the magnetic field required to switch the pinned layer is different (larger) than that for the free layer. Thus, F1 and F2 may have either parallel or antiparallel magnetic configurations, which are needed to obtain the magnetoresistive sensing effect.3

IBM then applied this research to the magnetoresistive read element in a hard disk head, by separating the two magnetic layers with a spacer element, and using a strong antiferromagnetic layer to 'pin' the orientation of one of the magnetic layers in place. IBM found that when the weak magnetic field from a bit on the hard disk passed underneath this structure (which IBM called a spin valve), the magnetic orientation of the unpinned magnetic layer rotated relative to that of the pinned layer. This generated a significant change in electrical resistance due to the giant magnetoresistive effect. The result was a far more sensitive read head capable of dealing with much higher aerial densities.

Magnetic tunnel junction devices can be used as cells for storing one bit of data. This is done is a similar manner to the spin valves, i.e. by placing the magnetic tunnel-barrier at the intersection of two orthogonal lines. A two dimensional array of these cells can be set up to produce a complete MRAM, in this case the individual cells are at intersections of word and bit-lines providing the necessary local magnetic fields.

Recent research has shown that it is possible to manipulate the spins in a semiconductor and has given confidence that spin-ensembles in semiconductors have a long enough lifespan so that they can be manipulated by voltages, allowing them to migrate in a coherent fashion. Current thinking is that when the spin-injection process (i.e. injecting spin-polarised electrons from a contact into the semiconductor heterostructure) is efficient, many different device concepts can be envisaged that could lead to novel computing approaches such as Quantum Computing.4 A basic Quantum Computer has already been achieved by an IBM research team in which five fluorine atoms interacted to become five quantum bits (Qubits). Radio frequency pulses were then used to alter atomic spins in order to 'program' the computer. The computer tackled a classic permutation problem and came up with the answer in a single step, a process that would have taken a conventional computer several steps.5

In May 2001, IBM reported that it was the first to mass-produce hard disks using 'antiferromagnetically-coupled media' (AFC). These disks use a special coating on their platters, consisting of two magnetic layers and, sandwiched between them, a microscopic layer of ruthenium. The result is a platter that works like that in a conventional hard drive, but proves far more resistant to the superparamagnetic effect. (As arial density increases, teh magnetic regions that contain the data get smaller and much closer together. Eventually a point is reached where the bits are so small that random atomic level vibrations can cause them to spontaneously flip their magnetic orientation, effectively erasing the recorded data.) All this has led, in recent years, to a growing area of research focusing on the study of surface magnetism in general and of magnetic properties of materials composed of alternating magnetic and non-magnetic layers in particular. In these systems, the interplay between electron transport properties and magnetic behaviour results in a variety of fascinating phenomena. In particular, the GMR effect mentioned above. Several interesting phenomena, such as the oscillatory nature of exchange coupling and saturation magnetoresistance, as a function of nonmagnetic spacer layer thickness, are found to be associated with magnetic multilayers exhibiting GMR. The GMR effect in a superlattice configuration relies on antiferromagnetic coupling between adjacent magnetic layers mediated by intervening nonmagnetic spacers. Although a GMR effect as large as 80 % has been reported in superlattices based on the Co-Cu system, the switching fields required to overcome antiferromagnetic coupling in superlattice structures are large. From an application standpoint, a combination of high GMR and low switching field is required.6

It is clear from the previous discussion that exchange coupling between ferromagnetic and antiferromagnetic systems is an extremely important effect, the fundamental electronic basis of which is an essential pre-requisite for an interpretation of existing experimental studies and the prediction of new systems.

Exchange bias occurs in multilayer systems formed by thin films of ferromagnetic, antiferromagnetic and non-magnetic materials and is the unidirectional exchange coupling between the antiferromagnet and ferromagnet that appears when a sandwich of these is grown or heated in a magnetic field.

Meiklejohn and Bean7 discovered the phenomena of exchange bias over forty years ago. They found that fine particles of partially oxidised Co exhibit magnetisation curves with an unusual displacement along the field axis, as though there were a “bias” field (HEB) in addition to the applied field. Thus a characteristic feature of exchange bias is the shift of the centre of the magnetic hysteresis loop from its normal position at H = 0 to HE ? 0, caused by a unidirectional rather than uniaxial anisotropy. A sin θ torque curve7 (θ is the torque angle), rotational hysteresis in fields greater than 2 K/Ms (K: magnetocrystalline anisotropy, Ms: saturation magnetization of the ferromagnetic layer)7, and enhanced coercivity which is much higher than the coercivity of a corresponding ferromagnetic single film may be observed in the presence of exchange bias.8 They also showed that there was a close connection between anomalous high field rotational hysteresis and exchange bias, although these do not generally co-exist.7 This macroscopic exchange bias effect has been the subject of extensive studies due to its technological relevance and lack of scientific understanding and although extensive experimental and theoretical studies have revealed the mechanism of interlayer coupling in the films, the magnetic interaction in these bilayers is still unclear.

The simplest model7 assumes that the ferromagnetic/antiferromagnetic interface occurs at the ideal uncompensated (i.e. all spins aligned) plane of the antiferromagnet. However, this predicts fields of orders of magnitudes above those observed and fails to explain why exchange bias occurs with a fully compensated interface plane (i.e. no net magnetic moment). Other models have since been proposed, for example Mauri et al9 showed that the formation of a domain wall parallel to the interface lowers the energy required to reverse the magnetisation. However, exchange coupling was simply assumed, which led to questions regarding the origin of the coupling. The fact that this coupling happens at fully compensated interfaces was also not addressed. It was suggested by Malozemoff10 that exchange bias was due to random exchange fields resulting from interface roughness. Malozemoff also pointed out that that the existence of antiferromagnetic domain walls lying perpendicular to the interface would have the effect of subdividing the antiferromagnetic single crystal into multiple domains, resulting in non-vanishing moments over a macroscopic area of the sample, giving a finite value for the exchange bias. Koon11 proposed another model, which shows that even in the case of perfectly smooth interfaces, spin flopping in the antiferromagnetic layers may lead to finite values of the exchange bias.

From the above, it can be seen that although various models have been proposed to deal with different aspects of this phenomenon, they all fail to fully address the problem of exchange bias. It has generally been accepted that the net moment appearing on an uncompensated antiferromagnetic surface that is ferromagnetically or antiferromagnetically coupled with the ferromagnetic moment is responsible for the exchange bias,7,9 and that the nearest-neighbour interactions are ferromagnetic, of the overlap type, while the next-nearest-neighbour couplings and antiferromagnetic, of the superexchange type.12 However, this means that the net exchange coupling energy between the ferromagnetic film and the compensated antiferromagnetic surface is independent of the relative angle between the spins. Matsuyama et al13 have concluded, therefore, that the antiferromagnetic film does not cause the exchange bias in the bilayer.

Koon11 reported simulations indicating that the FM spins exchange coupled with the spins of the compensated antiferromagnetic plane tend to be directed in the perpendicular direction to the antiferromagnetic easy axis and that the antiferromagnetic spins cant towards the ferromagnetic spin, this experiment was conducted using a Fe3O4/CoO system. Matsuyama et al13 found that the exchange interaction at the interface forces the Fe spin-polarisation distribution to follow the NiO spin configuration even at the compensated NiO (001) surface. By taking into account the spin canting of the NiO, they calculated the spin configurations of Fe/NiO (001), which showed that the Fe and NiO spins at the interface are almost perpendicularly coupled.

Since Matsuyama et al13 could not observe the NiO spin configuration directly with their spin scanning electron microscope; they estimated the Fe and NiO spin configurations using a simple model similar that introduced by Mauri.9 They used three assumptions on which to base their model:

  1. The Fe spins at the interface are in the same direction,
  2. The anisotropy at the NiO surface is the same as that in the bulk, and
  3. Each Fe spin in exchange coupled with its corresponding NiO spin and the lattice mismatch is not considered.

The exchange interaction at the interface makes the NiO-sublattice spins in the topmost layer rotate at certain angles, which are generally different from each other. Thus, the NiO domain wall created at the interface is not an ordinary wall with coherent rotation of each sublattice spin. The rotation angles of the pair of NiO sublattice spins can be expressed by the linear combination of the coherent rotation angle and the canting angle of the spin pair. Allowing the domain wall energy to be approximated by Ew = Er + Ec (Er: the wall energy due to the coherent rotation and Ec: the wall energy due to the canting). It was found that canting the NiO spins towards the Fe spin decreases the exchange energy and, consequently, reduces the total magnetic energy instead of increasing the wall energy.

Stöhr et al14 showed that the NiO spins at the (001) surface are perpendicular to the surface plane using x-ray magnetic linear dichroism, which is inconsistent with the results obtained by Matsuyama et al13 for the NiO spin configuration at the Fe/NiO interface. Matsuyama et al. concluded that this was because the Fe atoms in Fe/NiO were attached to the Ni atoms at the interface. They also concluded that near-perpendicular coupling between the Fe spin and the NiO sublattice spins is realised at the surface and similar to that reported by Koon, except that the NiO spins are not parallel to the interface.

Another recent experiment, conducted by Ohldag et al15 used the antiferromagnet NiO coupled with a thin layer of either Fe or Co in order to determine experimentally the relative alignment of the antiferromagnetic and ferromagnetic spin systems. They showed that the magnetic spins near the surface of a NiO (001) single crystal and those in a thin film of Co or Fe deposited on top align perfectly parallel to each other. This is in complete contrast to the conclusions of Matsuyama et al13 for the same system. The Ohldag15 results reveal a reorientation of the antiferromagnetic spins near the NiO (001) surface upon deposition of the ferromagnetic film. More importantly, they provide clear experimental evidence that symmetry breaking at surfaces and interfaces will in general lead to spin reorientation effects in antiferromagnets. They also found that for the cleaved NiO (001) surface only a subset of the bulk antiferromagnetic domains with formations of novel {110} domain walls were observed.

After either Co or Fe had been deposited, it was found that Co and Fe exhibited the same behaviour after deposition on the surface in the thickness range 0.5 – 2.0 nm, which corresponds to 1.5 – 6 layers of Fe, the antiferromagnetic spins reorient and align, domain-by-domain, parallel to the Fe spin direction, which is in plane. The antiferromagnetic NiO (001) surface resembles a NiO (001) wall parallel to the surface with fourfold domain symmetry about the surface normal. The Fe layer itself assembles into antiferromagnetic domains, indicating a strong uniaxial anisotropy of the Fe parallel to the antiferromagnetic axis. The results clearly demonstrate the sensitivity of the antiferromagnetic spin orientation to surface and interface effects.

The differing results obtained by Ohldag15 and Matsuyama13 demonstrate that any realistic model for the magnetic exchange coupling at ferromagnetic-antiferromagnetic interfaces has to be based on the actual spin structure near the interface which may significantly deviate from that expected from the bulk.

Another group (Miltényi et al.) used Monte Carlo simulations involving a simple model of a ferromagnetic layer on a diluted antiferromagnet to show exchange bias and to explain qualitatively its dilution and temperature dependence. They found that by diluting the antiferromagnet volume domains were formed, which cause and control exchange bias.16 The non-magnetic defects in the volume part of the antiferromagnet led to the formation and stabilization of the antiferromagnetic domains, which are responsible for the exchange bias. It was found by Fraune et al17 that a significant reduction of the exchange bias field was observed in NiO/Ni nanowires compared to un-patterned films. This was found to be consistent with the model proposed by Miltényi et al16 they further concluded that the probable origin of the decrease in the exchange bias filed was due the onset of a transition from the multi-domain to single-domain state in the NiO.

The introduction of probes sensitive to local surface observations, (for example the mean magnetisation in the surface region) such as low-energy electron diffraction (LEED) and spin-polarised LEED, spin-polarised secondary-electron spectroscopy, and spin-polarised photoemission spectroscopy, and more recently metastable helium atom scattering, have opened up the possibility of investigating surface magnetic critical behaviour experimentally.18 Another possibility in studying antiferromagnetic surfaces arises in the form of x-ray magnetic linear dichroism (XMLD) spectroscopy carried out by means of surface sensitive electron yield detection, which measures the expectation value of the square of the magnetic moment.19 Stöhr et al14 used this technique to successfully image surfaces with antiferromagnetic contrast, thus proving its worth. Nonlinear optics have also been used to probe antiferromagnetically ordered surfaces and have proved to be very sensitive.


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References and more information

  1. E. Schrödinger, "Die gegenwartige Situation in der Quantenmechanik," Naturwissenschaftern. 23 : pp. 807-812; 823-823, 844-849. (1935). English translation: John D. Trimmer, Proceedings of the American Philosophical Society, 124, 323-38 (1980), Reprinted in Quantum Theory and Measurement, p 152 (1983).
  2. Two very good books on quantum theory, both by John Gribbin, are called; "In Search Of Schrödinger's Cat" (ISBN 0 552 12555 5) published by Black Swann, and "Schrödinger's Kittens and the search for reality" (ISBN 1 85799 402 7) published by Phoenix.
  3. S. Dubourg, J. F. Bobo, B. Warot, E. Snoeck and J. C. Ousset, Phys. Rev. B, 64, 54416, (2001).
  4. R. Compañó (Ed.), Technology Roadmap for Nanoelectronics, EC IST, (2000).
  5. S. Andrews, P.C.W., 24, 230, 238 & 250, (Dec. 2001).
  6. H. D. Chopra, D. X. Yang, P. J. Chen, D. C. Parks and W. F. Egelhoff, Jr., Phys. Rev. B, 61, 9642, (2000).
  7. W. P. Meiklejohn and C. P. Bean, Phys. Rev., 102, 1413 (1956);
    W. P. Meiklejohn and C. P. Bean, Phys. Rev., 105, 904, (1957).
  8. D. X. Yang, H. D. Chopra, P. J. Chen, H. J. Brown, L. J. Swartzendruber and W. F. Egelhoff Jr., J. Appl. Phys., 87, 4942, (2000).
  9. C. Mauri, H. C. Siegmann, P. S. Bagus, and E. Kay, J. Appl. Phys., 62, 3047, (1987).
  10. A. P. Malozemoff, Phys. Rev. B, 35, 3679 (1987);
    A. P. Malozemoff, J. Appl. Phys., 63, 3874 (1988);
    A. P. Malozemoff, Phys. Rev. B, 37, 7673 (1988).
  11. N. C. Koon, Phys. Rev. Lett., 78, 4865, (1997).
  12. P. W. Anderson, Phys. Rev., 79, 350, (1950).
  13. H. Matsuyama, C. Haginoya and K. Koike, Phys. Rev. Lett., 85, 646, (2001).
  14. J. Stöhr, A. Scholl, T. J. Regan, S. Anders, J. Lüning, M. R. Scheinfein, H. A. Padmore and R. L. White, Phys. Rev. Lett., 83, 1862, (1999).
  15. H. Ohldag, A. Scholl, F. Nolting, S. Anders, F. U. Hillebrecht and J. Stör, Phys. Rev. Lett., 86, 2878, (2001).
  16. P. Miltényi, M. Gierlings, J. Keller, B. Beschoten, G. Güntherodt, U. Nowark and K. D. Usadel, Phys. Rev. Lett., 84, 4224, (2000).
  17. M. Fraune, U. Rüdiger, G. Güntherodt, S. Cardoso and P. Freitas, Appl. Phys. Lett., 77, 3815, (2000).
  18. M. Marynowski, W. Franzen, M. El-Batanouny and V. Staemmler, Phys. Rev. B, 60, 6053, (1999).
  19. A. Dähn, W. Hübner and K. H. Brennemann, Phys. Rev. Lett., 77, 3929, (1996).