Magnetic and size dependent properties of nanomaterials

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Magnetic and size dependent properties of nanomaterials

Introduction

It is one of the more often used units for very small lengths i.e. nanometer (nm). One nanometer equals to one thousandth of a micrometer or one millionth of a millimeter or ten angstrom. It is often associated with the field of nanotechnology and the wavelength of light. Nanotechnology can offer potential solution to many problems using emerging nano techniques. Depending upon the interdisciplinary character of nanotechnology, there are many research fields and several potential applications. But irrespective of the field, materials are getting importance. Nanomaterials means, material with one (or) two (or) zero dimensions and with an internal structure which could exhibit novel characteristics compared to the same material without nanoscale features. Any physical substance with structural dimensions between 1-100 nm can be defined as nanomaterials.

2. Luminescence

Luminescence is the term applied to the re-emission of previously absorbed radiation.  In molecular photoluminescence, photons of electromagnetic radiation are absorbed by molecules raising them to some existed state and then, on returning to the ground state, the molecules re-emit the radiation back and this is known as luminescence. Some important luminescences are photoluminescence, thermo luminescence, bioluminescence and cathodoluminescence. In this presentation details of photoluminescence will be discussed.

2.1 Photoluminescence from bulk matter:

If a photon has energy greater than that of the band gap energy of a material, then it can be absorbed by that material (mostly semiconducting type).  The absorbed energy raise an electron from the valence band (leaving behind the holes) up to the conduction band across the forbidden energy gap. In this process of photoexcitation, the electron generally has excess energy and that will be mostly lost before coming to rest at the lowest energy level in the conduction band. From this point, the electron eventually falls back down to the valence band and recombine with the holes. As it falls down, the energy it loses is converted back into a luminescent photon. This energy of the emitted photon is a direct measure of the band gap energy.

            Photoluminescence includes fluorescence and phosphorescence. In fluorescence the energy transitions do not involve a change in electron spin, on the other hand phosphorescence involves a change in electron spin and is therefore much slower than the fluorescence process. Fluorescence emission involves a lifetime of 10^-8 sec (electron life time in the excited state or conduction state) and therefore happens quickly after initial photon absorption. [Some fluorescent nanomaterials have been successfully used as fluorescent labels for a variety of bio-analytical purposes such as detection of DNA, proteins and other biomolecules, cellular labeling and etc.] In phosphorescence type emission the life of time of excited electrons is about 10^-3 sec and therefore happens slowly after initial photon absorption.

2.2 Photoluminescence Excitation and Emission:

 In physics, excitation means elevation in energy level above an arbitrary baseline energy state. The below Fig.1 schematically represent the excitation process. An incoming photon of energy greater than that of band gap supplies energy to the valence band electron and therefore this valence electron is removed from the valence band [Valence band: A representation of bond state, here electrons are bound to individual atoms and therefore they can’t make a free movement  inside the material (it means no current is possible) ]. The as removed electrons are moving to the conduction band [Conduction band: A representation of a free state, here the electrons are bound to the individual atoms of crystal network but belongs to the whole network, so that they can move freely around the material but definitely not out of the material] and spends its time there for a while. After spending an initial time of 10^-8 or 10^-3 Secs, the electrons return back to the valence band. Since energy is conserved; this returning process gives back the photon absorbed and is known as “emission” process.

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2.3 Effect of Size reduction:

When the materials (mostly semiconductors) are in the bulk form then band theory is applicable as discussed above. But when the materials size is reduced to nanoscale level then it is impossible to see such a bands (that is valence and conduction bands). Instead of bands there will be discrete energy levels¹ as shown below Fig.2 These discrete energy levels have the nomenclature as valence and conduction energy levels (well separated lines in the conduction and valence band). Here also there exists a forbidden gap and therefore energy absorption is required for an electron to get excited into the conduction energy levels (take care not conduction bands).

                        

2.4 What makes discreteness?

A simple answer for the conversion of a “band” into “discrete energy level” is a cut in the number of atoms (upon size reduction). We all know that matter is formed by the joining of atomic orbital’s together and this joining, also known as over lapping of energy levels, occurs at the outer most levels known as valence levels of atoms. The over lapping of energy levels may not be so perfect due to the thermal vibration of atoms and therefore there will be a minimum energy level separation between the overlapping energy levels.  When the size of the matter is big then it means more number of atomic orbital’s or energy levels are overlapping. This in turn results a collection of closely packed energy levels also known as bands. The bound states are known as valence band and the free states are known as conduction band. If one can closely observe the bands there also exists discreteness but with a very minimum energy separation. Because of the minimum energy separation, just room temperature energy of about 23 to 25 meV (milli electron volt) is sufficient for the electrons to get promoted from one level to the other and therefore, within in the band, either conduction or valence band, the electrons can make free movement from bottom to top. That is getting energy from room temperature the electrons can move from one level to the other within the band. Once reached top of the band the electrons loses their energy by collision and therefore comes down to the bottom of the band. As this process is random the net current will be zero. However for the electrons to move from valence band to conduction band a minimum energy, equivalent to that of forbidden gap is required. Getting this energy the electrons from valence band can make a movement towards conduction band, leaving behind holes in valence bands. [Within the conduction band the electrons can move freely from bottom to top levels getting energy from room temperature and therefore these motion with mostly will be random and therefore the net current will be zero. However by applying a biasing voltage, we can give directionality to the electrons and therefore a non-zero net current will be resulted. Remember, here also there exists randomness and therefore the electron movement towards the positive end of the potential is slowed down. This slowing down procedure gives raise to heat and that’s why most of the bulk devices and their products produce lot of heat.] These excited electrons in the conduction band, in the absence of external biasing voltage, will recombine with the holes in the valence band and thereby re-emits the radiation.

            When the size of the matter is reduced, obviously the number of atomic levels overlapping to form a band of states is reduced. If this size reduction process continues up to nanoscale level then there will be fewer atoms and hence fewer overlapping of energy levels. In such a situation, the band (of states) appears with discrete states. Remember, now also we have valence states, conduction states (not bands) and forbidden gap. So we need to supply the energy externally to excite the electrons from the valence states into conduction states. One point, one has to remember is that upon size reduction the forbidden gap width is increased and therefore the resulting recombination (in the absence applied biasing voltage) gives a high energy photon.  Therefore by closely studying the emission wavelength one can indirectly obtain the size of the nanoparticle. However, the relation between the size and the emission wavelength is unique to given material of particular composition and crystal structure and therefore one has to take care in co-relating the emission wavelength to the size. For example factors such electron and hole effective masses etc has to be taken into account for the calculation size of the nanoparticle from the wavelength.²

3. Conclusion

            In conclusion, one can measure the emission wavelength using spectrofluorimeter and therefore can obtain the size of the nanoparticle indirectly. Many scientists are using this luminescence technique to find the nanoparticle size.² In particular this emission mechanism will be very much useful to semiconcondcting nanoparticles only. In the case of metals, since there is no forbidden gap, the emission process will be different. 

            The below Fig.3 gives the size dependent emission color obtained from CdSe quantum nanoparticles.

                   

Magnetic and Optical Properties of Nanoparticles

1.0       Diamagnetism, paramagnetism and ferromagnetism

There are three categories of magnetism that we need to consider: diamagnetism, paramagnetism and ferromagnetism. Diamagnetism is a fundamental property of all atoms (molecules), and the magnetization is very small and opposed to the applied magnetic field direction. Many materials exhibit paramagnetism, where a magnetization develops parallel to the applied magnetic field as the field is increased from zero, but again the strength of the magnetization is small. In the language of the physicist, ferromagnetism is the property of those materials which are intrinsically magnetically ordered and which develop spontaneous magnetization without the need to apply a field. The ordering mechanism is the quantum mechanical exchange interaction. It is this final category of magnetic material that will concern us in this chapter. A variation on ferromagnetism is ferrimagnetism, where different atoms possess different moment strengths but there is still an ordered state below a certain critical temperature. We will now define some key terms. The magnetic induction B has the units of tesla (T). The magnetic field strength H can be defined by

        -----------          (1)

where m₀ = 4p x 10^-7 Hm^-1 is the permeability of free space. This gives the horizontal component of the earth’s magnetic field strength as approximately 16 Am^-1 in London. The flux j = BA can be defined, where A is a cross-sectional area. Flux has the units of weber (W).

If a ferromagnetic material is now placed in a field H, Equation (1) becomes

 -------------- (2)

where M is the magnetization of the sample (the magnetic dipole moment per unit volume). We define c as the susceptibility of the magnetic material, and m_r = 1 + c as the relative permeability of the material (both m and c are dimensionless). Table 1 summarizes this classification scheme, and gives values for typical susceptibilities in each category.

Table 1 Classification of magnetic materials by susceptibility

Diamagnet (c < 0)	Paramagnet (c > 0)	Ferromagnet (c >> 0)
Cu: - 0.11 x 10-5	Al: 0.82 x 10-5
Au: - 0.19 x 10-5	Ca: 1.40 x 10-5	Fe:>102
Pb: - 0.18 x 10-5	Ta: 1.10 x 10-5

1.1       Hysteresis Loops

Most of the magnetic properties of a material can be derived from its hysteresis loop. Figure 1 schematically illustrates a magnetization vs field (M–H) hysteresis loop. When the external magnetic field is sufficiently large, all the spins within a magnetic material align with the applied magnetic field. In this state, the magnetization of the material achieves its maximum value, and this value is called the saturation magnetization, Ms. When the external magnetic field becomes weaker, the spins in the material cease to be aligned with the external magnetic field, so the total magnetization of the material decreases. For a ferromagnetic material, when the external magnetic field decreases to zero, the material still has a residual magnetic moment, and the value of the magnetization at zero field is called the remanent magnetization, M_r. The remanence ratio is defined as the ratio of the remanent magnetization to the saturation magnetization, M_r/M_s, which varies from 0 to 1. To bring the material back to zero magnetization, a magnetic field in the negative direction should be applied, and the magnitude of the field is called the coercive field, H_c. The re-orientation and growth of spontaneously magnetized domains within a magnetic material depends on both microstructural features such as vacancies, impurities or grain boundaries, and intrinsic features such as magnetocrystalline anisotropy as well as the shape and size of the particle. In most cases, the hysteresis loop of a magnetic material should be experimentally measured using, a vibrating sample magnetometer (VSM) or superconducting quantum interference device (SQUID) magnetometer, and it is not possible to predict a priori what the hysteresis loop will look like.

           

Materials with different magnetic properties have different shapes of hysteresis loops. Figure 2 shows a schematic diagram of a blood vessel into which some magnetic nanoparticles have been injected, and the magnetic properties of both the injected particles and the ambient biomolecules in the blood stream. Generally speaking, the blood vessel and the biomaterials in the blood vessel are either diamagnetic or paramagnetic, while the injected magnetic particles are either ferromagnetic or superparamagnetic, depending on their sizes.

Broadly, all materials can be regarded as magnetic materials because all the materials respond to magnetic fields to some extent. However, they are usually classified based on their volumetric magnetic susceptibility, χ, describing the relationship between the magnetic field H and the magnetization M induced in a material by the magnetic field:

     ------------------  (3)

In the SI unit system, χ is dimensionless, while both M and H are expressed in Am⁻¹. Most materials display little magnetism, and these are classified either as paramagnets or diamagnets. The χ value for paramagnets is usually in the range of 10⁻⁶ to 10⁻¹, while the χ value for diamagnets is usually in the range −10⁻⁶ to −10⁻³. Negative χ value of diamagnets indicates that in such materials, the magnetization M and the magnetic field H are in opposite directions. However, some materials exhibit ordered magnetic states, and they are usually classified as ferromagnets, ferrimagnets and antiferromagnets. The prefixes of these names refer to the nature of the coupling interactions between the electrons within the material. Such couplings may lead to large spontaneous magnetizations, and this is the reason why ordered magnetic materials usually have much larger χ values than paramagnetic or diamagnetic materials.

It should be noted that the susceptibility in ordered materials also depends on applied magnetic field H. This magnetic field gives rise to the characteristic sigmoidal shape of the M–H curve, with M approaching a saturation value at high magnetic field. In ferromagnetic and ferrimagnetic materials, hysteresis loops can be observed, as shown in Figure 1. The shape of a hysteresis loop is partly determined by the particle size. A particle in the order of micron size or more usually has a multi-domain structure. As it is easy to make the domain walls move, the hysteresis loop of such particles is narrow. In a smaller particle, the single-domain structure leads to a broad hysteresis loop. When particle size becomes even smaller, in the order of tens of nanometers or less, superparamagnetism can be found. The magnetic moment of a superparamagnetic particle as a whole is free to fluctuate in response to thermal energy, while the individual atomic moments maintain their ordered state relative to each other. As shown in Figure 2, the M–H curve of a superparamagnetic particle is anhysteretic, but still sigmoidal.

1.2       Magnetic Anisotropy

Most materials contain some type of anisotropy affecting their magnetization behaviors. The magnetic anisotropy of a material can be modeled as uniaxial in character and represented by:

   ------------------------- (4)

where K is the effective uniaxial anisotropy energy per unit volume, θ is the angle between the moment and the easy axis, and V is the particle volume.

1.3       Single-domain Particles

A domain is a group of spins whose magnetic moments are in the same direction, and in the magnetization procedure, they act cooperatively. In a bulk material, domains are separated by domain walls, which have a characteristic width and energy associated with their formation and existence. The movement of domain walls is a primary means of reversing magnetization and a major source of energy dissipation.

Figure 3 schematically shows the relationship between the coercivity in particle systems and particle sizes. In a large particle, energetic considerations favor the formation of domain walls, forming a multi-domain structure. The magnetization of such a particle is realized through the nucleation and motion of these walls. As the particle size decreases toward a critical particle diameter, D_c, the formation of domain walls becomes energetically unfavorable. So there is no domain wall in such a particle, and this particle is called a single-domain particle. For a single-domain particle, the magnetization procedure is realized through the coherent rotation of spins. The particles with size close to D_c usually have large coercivities. As the particle size is much smaller than D_c, the spins in this particle are affected by thermal fluctuations, and such a single-domain particle is usually called a superparamagnetic particle, which will be discussed in later section.

Frenkel and Dorfman (1930) theoretically predicted the existence of single-domain particles. The Dc values for some typical magnetic materials of spherical shape are listed in Table 2. It should be noted that particles with significant shape anisotropy usually have a larger effective critical single-domain diameter than corresponding spherical particles

Table 2. Critical single-domain sizes, Dc, for spherical particles with no shape anisotropy.

Material	Dc (nm)
Co	70
Fe	14
Ni	55
Fe3O4	128
g-Fe2O3	166

1.4       Time Dependence of Magnetization

The time dependence of magnetization of a material is important for its engineering applications and for investigating the fundamental mechanisms of magnetism. The variation of magnetization with time of a magnetic material can be described by:

  --------------- (5)

where  is the magnetization at the equilibrium state, and  is a characteristic relaxation time given by:

 --------- (6)

For a uniaxial anisotropy, the energy barrier, , is equal to the product of the anisotropy constant and the volume. In most cases,  is often taken as a constant of value 10⁹ s⁻¹. As the behavior of  is dominated by the exponential argument, the accurate value of  is usually not necessary. However, the particle size greatly affects the relaxation time. If we choose typical values  = 10⁹ s⁻¹, K= 10⁶ erg/cm³, and T = 300 K, the relaxation time of a particle with diameter of 11.4 nm is 0.1 s, while the relaxation time of a particle with diameter of 14.6 nm is 10⁸ s.

If all components of a system have the same relaxation time, Equation (5) offers the simplest solution. However this assumption is not applicable to real systems because of the distribution of energy barriers in real systems. The energy barrier distribution may be related to the variations of a lot of parameters, such as particle sizes, anisotropies or compositional inhomogeneity, and the distribution of energy barriers causes a distribution of relaxation times. If the distribution of energy barriers is constant, the magnetization decays logarithmically:

  ----------- (7)

where the magnetic viscosity, S, is related to the energy barrier distribution. If the distribution of energy barriers is constant, deviations from the  behavior can be observed. To keep Equation (7) applicable, the magnetic viscosity, , should be accordingly modified.

1.5       Superparamagnetism

Néel theoretically demonstrated that H_c approaches zero when particles become very small because the thermal fluctuations of very small particles prevent the existence of a stable magnetization. This is a typical phenomenon of superparamagnetism. There are two experimental criteria for superparamagnetism. First, the magnetization curve exhibits no hysteresis, and second the magnetization curves at different temperatures must superpose in a plot of . Figure 4   shows the magnetization curves of iron amalgam on  bases. Measurements were made at 77K and 200K respectively, and the magnetization curves at 77K and 200K superpose each other. The imperfect  superposition may be due to a broad distribution of particle sizes, changes in the spontaneous magnetization of the particle as a function of temperature or anisotropy effects.

The basic mechanism of superparamagnetism is based on the relaxation time  of the net magnetization of a magnetic particle:

 -------- (8)

where  is the energy barrier to moment reversal, and  is the thermal energy. For non-interacting particles the pre-exponential factor  is in the order of 10⁻¹⁰–10⁻¹² s  and only weakly dependent on temperature. The energy barrier has several origins, including both intrinsic and extrinsic effects such as the magnetocrystalline and shape anisotropies, respectively. However, in the simplest cases, it is given by, where  is the anisotropy energy density and  is the particle volume. For small particles,  is comparable to  at room temperature, so superparamagnetism is important for small particles.

It should be noted that, for a given material, the observation of superparamagnetism is dependent not only on temperature, but also on the measurement time  of the experimental technique used. As shown in Figure 5, if , the flipping is fast relative to the experimental time window and the particles appear to be paramagnetic; while if , the flipping is slow, and such a state is called a blocked state. In a block state, the quasi-static properties of the material can be observed. The blocking temperature  can be obtained by assuming . In typical experiments, the measurement time can range from the slow timescale of 10² s for DC magnetization, and medium timescale of 10⁻¹–10⁻⁵ s for AC susceptibility, through to the fast timescale of 10⁻⁷–10⁻⁹ s for ⁵⁷Fe Mössbauer spectroscopy.