13 Magnetic Quantum Well and Magnetic (Quantum) Dot

Dr. Anchal Srivastava

epgp books

 

 

Contents of this Unit

 

1.  Introduction.

1.1. Magnetic Quantum Well.

1.2. Magnetic Quantum Wire.

1.3. Magnetic Quantum Dot.

2. Integration of fluorescence and magnetic properties of magnetic quantum dots.

3. Size and dimensionality effects.

4. Single-electron tunnelling.

5. Superconductivity.

6. Summary.

 

Learning Outcomes

After studying this module, you shall be able to

  1. Learn about the basic difference in magnetic quantum well, magnetic quantum wire and magnetic quantum dots.
  2. Learn about the integrated properties magnetic quantum dots with fluorescent nano-materials.
  3. Learn about size and dimension effects of magnetic nano-particles as quantum dot and wells.
  4. Learn about what are SETs devices and how they use quantum dots and how Quantum dot coupled to an external circuit through source and a drain leads.
  5. Learn about an over-view how superconductivity is connected to quantum wire and their properties.

 

1. INTRODUCTION

 

As we know that if dimension of a material is continuously reduced from a large or macroscopic size from centimetre to nano-dimension, the properties remain the same at first, then small changes begin to occur, until finally when the size drops below 100 nm, dramatic changes in properties can occur.

 

1.1. MAGNETIC QUANTUM WELL:

 

If a magnetic nanoparticle’s one dimension is reduced to the nano-range while the other two dimensions remain large, then we obtain a structure known as a quantum well. Magnetic properties of materials can be tailored in nanostructures, such as thin films, wires and dots. And the electronic states that are relevant for magnetic phenomena, such as oscillatory coupling, giant magnetoresistance (GMR) and spin-polarized tunnelling. These states are probed by high-resolution photoemission and inverse photoemission.

 

The state of the art in surface preparation and analysis has made it feasible to produce new types of materials that are structured on the nanometer scale. A good example of an engineered solid is the magnetoresistive reading head for data stored on a magnetic hard disk. The design of these heads has evolved rapidly, which has led to an increase in the data density of hard disks faster than Moore’s law.

 

Giant magnetoresistance (GMR) reading heads consist of sandwiches of cobalt, copper, and permalloy (nickel-iron) that change their electrical resistance when exposed to the magnetic field of a stored bit. The discovery of this effect was recognized by the 2007 Nobel Prize in Physics. The sandwich structures are known as spin valves, since they preferentially transmit electrons of one spin orientation. A related phenomenon is oscillatory magnetic coupling, an oscillation in the magnetic orientation of two layers with film thickness.

 

To find out which electrons are producing these effects we have investigated their energy levels by inverse photoemission. Thereby, low energy electrons impinge onto the surface and drop into unoccupied energy levels emitting ultraviolet photons. This study discovered quantized electronic states in magnetic multilayers that are connected to their special properties.

 

Fig: 1 -The figure shows periodic changes in the density of electron states when the thickness of a copper film is changed, just a couple of atomic layers at a time (top panel).

 

A way of understanding the experiment is shown on the right: Electrons in the Cu film are acting like waves and give maximum intensity when an integer number of oscillations fits into the film. These structures can be viewed as the smallest man-made interferometers, right at the atomic limit. They allow us to map out the wave function of electrons.

 

It turns out that the density-of-states maxima are correlated with oscillations of the magnetic coupling in multilayers (bottom panel, from Qiu et al., Phys. Rev. B 46, 8659 (1992)). The magnetoresistance oscillates with the same period. This provides the clue for understanding these magnetic phenomena as the effects of quantized electron levels in nanostructures.

 

By manipulating the interfaces in magnetic multilayers, one can enhance the spin-dependent reflectivity and optimize the sensitivity of a magnetic reading head.

 

1.2. MAGNETIC QUANTUM WIRE:

 

If a magnetic nanoparticle’s two dimensions are so reduced and one remains large, the resulting structure is referred to as a quantum wire.

 

1.3. MAGNETIC QUANTUM DOT:

 

The extreme case of this process of size reduction in which all three dimensions for magnetic nano-particles, reach the low nano-meter range is called a quantum dot. The word quantum is associated with these three types of nanostructures because the changes in properties arise from the quantum-mechanical nature of physics in the domain of the ultra-small.

 

Multimodal contrast agents based on highly luminescent quantum dots (QDs) combined with magnetic nanoparticles (MNPs) or ions form an exciting class of new materials for bio-imaging. With two functionalities integrated in a single nanoparticle, a sensitive contrast agent for two very powerful and highly complementary imaging techniques [fluorescence imaging and magnetic resonance imaging (MRI)] is obtained.

 

2.  INTEGRATION OF FLUORESCENCE AND MAGNETIC PROPERTIES OF MAGNETIC QUANTUM DOTS:

 

This is done by describing the developments for four different approaches to integrate the fluorescence and magnetic properties in a single nanoparticle.

 

  1. The first type of particles is created by the growth of hetero-structures in which a QD is either overgrown with a layer of a magnetic material or linked to a (superpara, or ferro) MNP.
  2. The second approach involves doping of paramagnetic ions into QDs.
  3. A third option is to use silica or polymer nanoparticles as a matrix for the incorporation of both QDs and MNPs.
  4. Finally, it is possible to introduce chelating molecules with paramagnetic ions (e.g., Gd-DTPA) into the coordination shell of the QDs. All different approaches have resulted in recent breakthroughs and the demonstration of the capability of bio-imaging using both functionalities.

 

In this semiconductor hetero-structure, lattice mismatch strain between the CdSe quantum dots (QDs) and the ZnCdSe quantum well (QW) produces a three-dimensional confinement potential in the QW itself, effectively confining carriers to small regions in the sample termed strain induced quantum well dots, or SIQWD’s.

Fig:1- Schematic picture of the changes in the conduction and valence bands of the ZnCdSe QW due to the strain fields of the CdSe QDs.

 

In the two structures below, carriers confined to quantum dots interact with a local magnetic environment provided by paramagnetic Mn2+ ions (spin 5/2). In the left, a layer of MnSe is grown sufficiently close to the CdSe QDs so that there is significant overlap of the carrier’s wave function with that of the Mn ions. The right structure is a variant of the SIQWD samples in which the dot-like regions produced by strain are present in a Digital Magnetic Hetero-structure (DMH – a quantum well with incorporated Mn ions).

 

 

Fig:2- The plot above traces the spin splitting of one of the magnetic quantum dot photoluminescence peaks as a magnetic field is applied.

 

The single peak at zero field is replaced with two peaks of opposite circular polarizations. The enhanced Zeeman splitting of the magnetic SIQWD sample is attributed to enhanced local fields resulting from contributions of nearby Mn ions.

 

3. SIZE AND DIMENSIONALITY EFFECTS:

 

We are accustomed to studying electronic systems that exist in three dimensions, and are large or macroscopic in size. In this case the conduction electrons are delocalized and move freely throughout the entire conducting medium such as a copper wire. It is clear that all the wire dimensions are very large compared to the distances between atoms. The situation changes when one or more dimensions of the copper becomes so small that it approaches several times the spacing between the atoms in the lattice. When this occurs, the delocalization is impeded and the electrons experience confinement. For example, consider a flat plate of copper that is 10 cm long, 10 cm wide, and only 3.6 nm thick. This thickness corresponds to the length of only 10 unit cells, which means that 20% of the atoms are in unit cells at the surface of the copper. The conduction electrons would be delocalized in the plane of the plate, but confined in the narrow dimension, a configuration referred to as a quantum well.

 

TABLE NO.1 OF CONDUCTION ELECTRON CONTENT OF SMALLER SIZE (ON LEFT) AND LARGER SIZE (ON RIGHT) QUANTUM STRUCTURES CONTAINING DONOR CONCENTRATION OF 1014 TO 1018 CM-CUBE.

 

 

TABLE NO.2 OF DELOCALIZATION AND CONFINEMENT DIMENSIONALITIES OF QUANTUM NANO-STRUCTURES.

 

The electrons are delocalized and move freely along the wire, but are confined in the transverse directions.

 

Many of the properties of good conductors of electricity are explained by the assumption that the valence electrons of a metal dissociate themselves from their atoms and become delocalized conduction electrons that move freely through the background of positive ions such as sodium ions or silver ions. On the average, they travel a mean free path distance l between collisions. These electrons act like a gas called a Fermi gas in their ability to move with very little hindrance throughout the metal. They have an energy of motion called kinetic energy, E = ½ mV2 = P 2/2m where m is the mass of the electron, v is its speed or velocity, and p = mv is its momentum. This model provides a good explanation of Ohm’s law, whereby the voltage V and current I are proportional to each other through the resistance R, that is, V = IR.

 

In a quantum-mechanical description the component of the electron’s momentum along the x direction px has the value  , where = is Planck’s universal constant of nature, and the quantity Kx is the x component of the wave vector k. Each particular electron has unique Kx, Ky and Kz values, that these values of the various electrons form a lattice in k space, which is called reciprocal space. At the temperature of absolute zero, the electrons of the Fermi gas occupy all the lattice points in reciprocal space out to a distance kF from the origin k = 0, corresponding to a value of the energy called the Fermi energy EF which is given by,

We assume that the sample is a cube of side L so its volume Vin ordinary coordinate space is V = L3. The distance between two adjacent electrons in k space is 2π/L and at the temperature of absolute zero all the conduction electrons are equally spread out inside a sphere of radius kF and of volume 4πkF3/3 in k space.

 

The number of conduction electrons with a particular energy depends on the value of the energy and also on the dimensionality of the space. This is because in one dimension the size of the Fermi region containing electrons has the length 2kF in two dimensions it has the area of the Fermi circle πkF2, and in three dimensions it has the volume of the Fermi sphere 4πkF3/3.

 

This means that the number of electrons with an energy E within the narrow range of energy dE = E2 – E1 is proportional to the density of states at that value of energy. The resulting formulas for D(E) for the various dimensions are shown. We see that the density of states decreases with increasing energy for one dimension, is constant for two dimensions, and increases with increasing energy for three dimensions. Thus, the density of states has quite a different behaviour for the three cases. These equations and plots of the density of states are very important in determining the electrical, thermal, and other properties.

 

Fig:3- Fermi-Dirac Distribution f(E) , Indicationg equal density in k space, vs temperature ( Handbook of physics, wiley, 1998, p138)

 

Fig:4 – Density of states D(E) =dN(E)/dE plotted as a function of the energy E for conduction electrons delocalized in one (Q-wire), two (Q-well), and three (bulk) dimensions.

 

The electrons that are present fill up the energy levels starting from the bottom, until all available electrons are in place. No matter how shallow the well, there is always at least one bound state E.

 

Fig:5 – Number of electrons N(E) ( left Side ) and density of states D(E) (right side) plotted against the energy for four quantum structure in the square well-fermi gas approximations.

 

The electrons confined to the potential well move back and forth along the direction x, and the probability of finding an electron at a particular value of x is given by the square of the wave function for the particular level n where the electron is located. There are even and odd wave function that alternate for the levels in the one-dimensional square well, and for the infinite square well we have the expressions

Another characteristic of a particular energy state E, is the number of electrons that can occupy it, and this depends on the number of different combinations of quantum numbers that correspond to this state. we see that the one-dimensional square well has only one allowed value of the quantum number n for each energy state. An electron also has a spin quantum number m, which can take on two values, ms = +1/2 and ms = -1/2 for spin states up and down, respectively.

 

According to the Pauli exclusion principle of quantum mechanics, no two electrons can have the same set of quantum numbers, so each square well energy state E, can be occupied by two electrons, one with spin up, and one with spin down. The number of combinations of quantum numbers corresponding to each spin state is called its degeneracy, and so the degeneracy of all the one-dimensional square well energy levels is 2.

 

This means that many electronic and other properties of metals and semiconductors change dramatically when the dimensionality changes. Some nanostructures of technological interest exhibit both potential well confinement and Fermi gas delocalization, confinement in one or two dimensions, and delocalization in two or one dimensions, so it will be instructive to show how these two strikingly different behaviours coexist.

 

In a three-dimensional Fermi sphere, the energy varies from E = 0 at the origin to E = E, at the Fermi surface, and similarly for the one and two-dimensions. When there is confinement in one or two directions, the conduction electrons will distribute themselves among the corresponding potential well levels that lie below the Fermi level along confinement coordinate directions, in accordance with their respective degeneracies d, and for each case the electrons will delocalize in the remaining dimensions by populating Fermi gas levels in the delocalization direction of the reciprocal lattice.

 

These considerations can be used to predict properties of nanostructures, and one can also identify types of from their properties.

 

Excitons are a common occurrence in semiconductors. When an atom at a lattice site loses an electron, the atom acquires a positive charge that is called a hole. If the hole remains localized at the lattice site, and the detached negative electron remains in its neighbourhood, it will be attracted to the positively charged hole through the Coulomb interaction, and can become bound to form a hydrogen-type atom. Technically speaking, this is called a Mott Wannier type of exciton. The Coulomb force of attraction between two charges Q,= – e and Qa = +e separated by a distance r is given by where e is the electronic charge, k is a universal constant, and is the dielectric constant of the medium.

 

4. SINGLE-ELECTRON TUNNELLING:

 

We have been discussing quantum wells in isolation. To make them useful, they need coupling to their surroundings, to each other, or to electrodes that can add or subtract electrons them. This can easily be done in SET.

Fig:6 – Quantum dot coupled to an external circuit through source and a drain leads.

 

This shows an isolated quantum dot or island coupled through tunnelling to two leads, a source lead that supplies electrons, and a drain lead that removes electrons for use in the external circuit. The applied voltage causes direct current I to flow, with electrons tunnelling into and out of the quantum dot. In accordance with Ohm’s law V = IR the current flow I through the circuit of the applied source-drain voltage divided by the resistance R, and the main contribution to the value of R arises from the process of electron tunnelling from source to quantum dot, and from quantum dot to drain. This device, as described, as a voltage-controlled or field-effect-controlled transistor, commonly referred to as an FET. For large or macroscopic dimensions, the current flow is continuous, and the discreteness of the individual electrons passing through the device manifests itself by the presence of current fluctuations or shot noise.

 

For an FET-type nanostructure the dimensions of the quantum dot are in the low nano-meter range, and the attached electrodes can have cross sections comparable in size. For disk and spherical shaped dots of radius r the capacitance is given by

Where is the dimensionless dielectric constant of the semiconducting material that forms the dot, and is the dielectric constant of free space.

 

An example of single-electron tunnelling is provided by a line of ligand-stabilized Au55 nanoparticles. The cluster of 55 gold atoms is encased in an insulating coating called a ligand shell that is adjustable in thickness, and has a typical value of 0.7nm.

 

5. SUPERCONDUCTIVITY:

 

Superconductors exhibit some properties that are analogous to those of quantum dots, quantum wires, and quantum wells.

 

Superconductivity is present in a material when pairs of electrons condense into a bound state called a Cooper pair with dimensions of the order of the coherence length. Thus, Cooper pairs, the basic charge carriers of the supercurrent, can be looked on as nanoparticles.

 

Vortices may be looked on as the magnetic analog of quantum wires in the sense that they confine one quantum unit of magnetic flux in the transverse direction, but set no limit longitudinally. The transverse dimensions of their core are in the nano-meter range, but their length is ordinarily macroscopic.

 

6. SUMMARY:

 

  1. If a magnetic nanoparticle’s one dimension is reduced to the nano-range while the other two dimensions remain large, then we obtain a structure known as a quantum well.
  2. If a magnetic nanoparticle’s two dimensions are so reduced and one remains large, the resulting structure is referred to as a quantum wire.
  3. The extreme case of this process of size reduction in which all three dimensions for magnetic nano-particles, reach the low nano-meter range is called a quantum dot.
  4. This is done by describing the developments for four different approaches to integrate the fluorescence and magnetic properties in a single nanoparticle.
  5.    The first type of particles is created by the growth of hetero-structures in which a QD is either overgrown with a layer of a magnetic material or linked to a (superpara, or ferro) MNP.
  6. The second approach involves doping of paramagnetic ions into QDs.
  7. A third option is to use silica or polymer nanoparticles as a matrix for the incorporation of both QDs and MNPs.
  8. Finally, it is possible to introduce chelating molecules with paramagnetic ions (e.g., Gd-DTPA) into the coordination shell of the QDs. All different approaches have resulted in recent breakthroughs and the demonstration of the capability of bio-imaging using both functionalities.
  9. In this semiconductor hetero-structure, lattice mismatch strain between the CdSe quantum dots (QDs) and the ZnCdSe quantum well (QW) produces a three-dimensional confinement potential in the QW itself, effectively confining carriers to small regions in the sample termed strain induced quantum well dots, or SIQWD’s.
  10. The conduction electrons would be delocalized in the plane of the plate, but confined in the narrow dimension, a configuration referred to as a quantum well.
  11. In a quantum-mechanical description the component of the electron’s momentum along the x direction px has the value    , where  = is Planck’s universal constant of nature, and the quantity Kx is the x component of the wave vector k. Each particular electron has unique Kx, Ky and Kz values, that these values of the various electrons form a lattice in k space, which is called reciprocal space. At the temperature of absolute zero, the electrons of the Fermi gas occupy all the lattice points in reciprocal space out to a distance kF from the origin k = 0, corresponding to a value of the energy called the Fermi energy EF which is given by,
  12. According to the Pauli exclusion principle of quantum mechanics, no two electrons can have the same set of quantum numbers, so each square well energy state E, can be occupied by two electrons, one with spin up, and one with spin down.
  13. We have been discussing quantum wells in isolation. To make them useful, they need coupling to their surroundings, to each other, or to electrodes that can add or subtract electrons them. This can easily be done in SET. Quantum dot coupled to an external circuit through source and a drain leads
  14. For an FET-type nanostructure the dimensions of the quantum dot are in the low nano-meter range, and the attached electrodes can have cross sections comparable in size.
you can view video on Magnetic Quantum Welland Magnetic (Quantum) Dot

Suggested Readings

  1. Charles Poole, The Introduction of Nanotechnology.
  2. W.I. Fletcher – An Engineering approach to Digital Design, PHI, 3Edition 3, Hand book of nanotechnology by B. Bhushan.
  3. .Chapter:  –  Nanostructured  Materials  for  Engineering  Applications,  pp  23-39,  Nanomagnetic
  4. Materials. Authors: – Silvana Da Dalt, Priscila Chaves Panta, Juliano Cantarelli Toniolo.
  5. Nanomagnetic Materials: A review, Authors: T.K.S.P. Gupta and S.S. Verma. Technical Proceedings of the 2003 Nanotechnology Conference and Trade Show, Volume 3, Nano Composites Chapter 6.