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Written by Kirill Rivkin, John B Ketterson | ||||||||||||||||||||||||||||||||||||||||||||
Friday, 30 June 2006 10:35 | ||||||||||||||||||||||||||||||||||||||||||||
Author(s): Kirill Rivkin, Ph.D., Prof. John B Ketterson Institution: Northwestern University, Evanston, IL USA Level: High School, College
DESCRIPTION: *Material is classified according to the intended audience.
Magnetic DipolesFOR HIGH SCHOOL LEVEL:Most of the high school graduates think about magnetism in terms of currents and magnetic fields produced by such currents. However, when it comes to most of the nanomagnetics, the easiest way to describe the physics involved is by using magnetic dipoles. In electrostatics the fields produced by individual electric charges can be describe via Coulomb's Law:
where is an electric field vector, q is the charge, r is the distance from the position at which the electric field is measured to the charge, is a vector connecting the charge and the position at which the electric field is measured, k is a coefficient depending on the choice of units. The force exerted on some other, test, charge will be:
Electric fields can be graphically presented by using field lines - originating at the charges these lines are always staying parallel to the fields, with the density of lines being proportional to the field's strength. For example Figure 1 presents the field lines of a single negative charge.
The material presented above can be found, usually in a much greater detail, in most high school textbooks. Now imagine that we have two charges, with opposite signs of equal magnitude q, separated from each other by distance D (Figure 2).
The field, produced by these two charges is a superposition of the fields they produce separately from each other - each of these fields can be obtained via Coulomb's formula. In the immediate proximity to these charges one can clearly see the fields of individual charges, distorted by the presence of another charge; however it can be shown, that far away from the charges the fields are given by the formula:
where is what is called dipole moment , is the vector that starts at the negative charge and ends on the positive charge. Therefore, the bigger the charges and the greater the distance that separates them, the bigger is the dipole moment's magnitude. While the total charge of the system is 0, the field produced is finite at finite distances, but instead of falling off like 1/R^{2} (like it does for a single charge) it now falls off like 1/R^{3}.
FOR COLLEGE LEVEL: In fact, it can be shown that when one has a combination of charges q 1,2,3.N the field far away from the charges equals:
The first term is called a monopole term. It is analogous to the field given by Coulomb's formula for a single charge, only now the role of a single charge is performed by the total charge of the system - if the total charge is 0 (for example the system has equal positive and negative charges, like the one on Figure 2), the first term is absent. The second term is a dipole term we already familiar with, however this time it's due to all dipole moments present in the system. As one can see the dipole moment of the entire system is equal to a vector sum of all the dipole moments in the system. The third term can be referred as a field due to higher moments - as one can see it falls off with R even faster than dipole and monopole fields. We shall illustrate these concepts with a few figures:
FOR HIGH SCHOOL LEVEL:
On the Figure 3 you can see the fields produced by a combination of a single dipole and a single monopole. Since dipole field falls off like 1/R^{3} , and monopole field falls off like 1/R^{2} , far away from the system the monopole field due to a single excessive negative charge dominates the dipole field. Closer to the charges you can see both dipole and monopole fields.
Figures 4 and 5 show the fields produced by two dipoles - parallel to each other in the first case, and anti-parallel in the second case. As one can see in the second case the field falls off extremely rapidly with the distance from the charges - total dipole moment of the system is 0, therefore the only field that remains is the one due to the higher moments - in our case so called quadrupole moment. However if the dipoles are parallel to each other, their total dipole moment is twice their individual dipole moments - as one can see of Figure 4. One of the most important applications of dipoles is the fact that they can be used to store binary information. If the dipole is reversed (positive and negative charges are interchanged with each other), the field produced by the dipole reverses its direction. This can be used to store and read the information - one can assign "0" to the dipole pointing up, and "1" to the dipole pointing down. One can read the information by measuring the direction of the field produced by the dipole. It turns out that to store the information in most cases it is more practical to use magnetic dipoles rather than electric dipoles. While magnetic field is created by currents, one can show that, for example, a current loop creates magnetic moment :
where is a unit vector along the direction perpendicular to the plane of current loop, I is current, S is the surface area encircled by the loop. In fact, most of the things to which we refer as "magnets" are actually magnetic dipoles, with North-South denoting the direction of the dipole. In addition to macroscopic objects like current loops, certain elementary particles, such as electrons are natural magnetic dipoles (scientists are still researching if there are particles that are natural magnetic monopoles). This property of electrons is called spin : one can imagine that the electron, having negative charge, is constantly spinning around some axis - therefore there is always some current present in the electron, which is responsible for producing the magnetic field. However such picture is extremely simplified and based on classical ideas, while the correct description of electron's properties requires the use of quantum field theory. The fields produced by magnetic dipoles are constructed identically to that of electric dipoles (Eq.(3)). To us, the most famous magnetic dipole is obviously the Earth itself (Figure 5). This dipole if formed by currents circulating in the molten iron in Earth's core.
One of the most important roles of this field is that it deflects the massive flow of charged particles from the Sun - therefore protecting the life of Earth from being exposed to deadly radiation. Earth field is not constant - the dipole is constantly changing its orientation and occasionally (sometimes once in a few thousand, sometimes once in a few million years) - undergoes a complete reversal - North becomes South and South becomes North. From Macro to NanoWhat makes a body magnetic? As we discussed above, electrons have an intrinsic dipole moment. If the electrons inside of the body's atoms are somehow capable of realigning themselves under the influence of the magnetic fields, the body can possess a total dipole moment: more electrons align their magnetic moments along certain direction than along other directions. In particular, the energy E of a magnetic dipole in a magnetic field equals:
where is the magnetic field. As one can see the energy is minimal when magnetic moments are aligned along the magnetic fields (both external magnetic fields due and internal magnetic fields - the latter ones being due to their neighboring electrons). It is impossible to describe the process of magnetization (creating a total magnetic moment in a body) without saying a few words about temperature effects. Using a very imprecise definition, temperature is a measure of the thermal energy present in the system. Thermal energy is the energy due to random interactions between particles forming the system. The higher the temperature, the higher thermal energy is present in the system, the more random the system is. For example, if we have a magnetic body at a high temperature, individual dipoles are constantly experiencing random "punches" from all sides. As a result they move in a random fashion. If the temperature is higher than certain value (so called Curie temperature), the system can not have a total magnetic moment, because individual moments are simply randomly rotating with respect to one another. When we lower the temperature, the system becomes more and more ordered - the dipoles choose the configuration that corresponds to the energy minimum . If there is some external magnetic field applied to the system, the direction of this magnetic field will be preferred by magnetic moments, the body becomes magnetized : the stronger applied field is, the bigger portion of dipoles is going to be aligned in parallel to the field's direction. The more dipoles are aligned in the same direction, the bigger the total dipole moment the system possesses and so is the magnetic field produced by the system as a whole. One should also mention that the density of electrons that can reorient themselves varies between different magnetic materials - samples with higher number of such electrons per unit volume are obviously "more magnetic". This can be characterized by the value called "saturation magnetization" - maximum magnetic moment per unit value. A good example of magnetization: when molten lava is ejected by volcanoes, despite the fact that it contains certain iron-based chemicals that are in general magnetic, the high temperature prevents the lava from possessing any total magnetic moment. When the lava cools down, a significant portion of its magnetic dipoles becomes oriented parallel to the Earth magnetic field (Figure 1).
Therefore by studying the magnetization of cooled lava (paleomagnetism) we can study the time evolution of Earth's magnetic field - rock formations "record" the information about the magnetic field in the time of their cooling. As we said before, the polarity of Earth magnetic field have been reversed many times in the past, therefore what we are going to see are stripes of magnetized rocks with their magnetization anti-parallel between stripes formed in between of reversals of Earth magnetic field. Similar ideas can be applied when trying to date certain historical objects (archaeomagnetism). By knowing the strength of Earth magnetic field in the past we can assign an approximate date to materials such as ceramics - small iron particles inside of such objects orient their magnetic moment along the Earth magnetic field at the time of their last cooling. This method does not usually work with metallic objects made from magnetic metals due to the fact that the strong interaction between neighboring dipoles negates the effects due to the Earth magnetic field. For example Figure 2 shows a magnetized dagger that exhibits an unusually complex magnetization.
So what are the interactions between individual dipoles? In the previous Chapters we spoke about the fields, produced by dipoles:
Here we chose to use symbols more characteristic to magnetic dipoles (M instead of D and H instead of E - as a reminder M denotes magnetic dipoles and H denotes magnetic fields), however the configuration of dipole fields does not depend on magnetic or electric nature of the dipoles.
Figure 3 presents magnetic fields due to a single magnetic dipole (in order to show the similarity with electric dipoles we also show the charges that would correspond to the electric dipole of similar properties) . As one can see the field is parallel to the dipole's orientation above and below the dipole and anti-parallel on each of the sides of a dipole. This means that if dipole-dipole interaction would be the prevalent force in magnetic bodies one would expect the individual dipoles to form strips of dipoles aligned parallel to each other inside each strip and anti-parallel with respect to dipoles of neighboring strips (again, the lowest energy configuration corresponds to dipoles aligned parallel to the magnetic field). The only way to change the configuration would be to apply external magnetic field that would overcome the dipole-dipole interaction and force the alignment of magnetic dipoles parallel to each other. However it is known that it is usually not the case in real life - when external magnetic field is weak there is a variety of configurations that can appear in the system, depending on its magnetic properties, shape and size. From now on we will mostly discuss only one particular type of magnetic materials - so called ferromagnets. While there are many other types, this one is nearly the only one used in the information storage. The reason behind the fact that even in the absence of external magnetic field magnetic bodies can have a significant magnetization, i.e. total magnetic moment, it is that we ignored the atomic scale interactions between dipoles. Due to the fact that the properties of individual atoms are quantum in nature and can not be completely described by using classical theories we have been using so far, there is another force we need to account for: so called exchange interaction . It can be introduced into the classical picture by assuming that each of the dipoles produces additional field, equal to:
where J is exchange constant. Moreover this field acts only on the nearest neighbors of the dipole - it can be shown that beyond them this field drops off with distance as 1/R^{6}. It is clear that depending on the sign of J the field created by a dipole is either parallel (J>0) or anti-parallel (J<0) to the dipole. If J>0 and is big enough to counteract the magnetic field due to "classical" dipoles given by Eq. 1 (hereandafter referred to as "dipole-dipole" interaction), the lowest energy of the magnetic body corresponds to magnetic moments aligned parallel to each other - such magnetic materials are called to be ferromagnetic , otherwise the "natural" orientation is the one with dipoles anti-parallel to each other, and corresponding magnetic materials are called to be antiferromagnetic . In most cases the exchange interaction, external fields and dipole-dipole interactions are "competing" with each other while forming the stable magnetization configuration. As a result some areas, so called domains, can be formed by dipoles parallel to each other, while other domains, while also formed by dipoles parallel to each other, have each of their dipoles oriented anti-parallel or at some angle with respect to the dipoles of neighboring domains. The total magnetization of the body in this case can be close to 0, but if the external field is applied a large number of individual domains become oriented in parallel with the applied field, inducing a total magnetization in the sample. Another force one can encounter is called anisotropy . The presence of such force means that due to some structural properties of a particular magnetic body, its magnetic dipoles prefer to be oriented along certain directions, called "easy axis". One of the reasons behind the existence of anisotropy can be deformations appearing in magnetic materials on the scale of their atomic lattice.
FOR COLLEGE LEVEL: Before we proceed to study the real applications, we need to introduce the most important equation used by scientists working on designing magnetic memory devices, the equation that describes the dynamics of magnetic dipoles in the applied field - Landau-Lifshitz equation:
where γ is gyromagnetic coefficient that depends entirely on the choice of units (for example whether SI or Gauss systems of units are chosen), β is a dimensionless damping coefficient. First we will consider the case when β is equal to 0. Direction of the torque is such that magnetic moment precesses around the direction of the magnetic field. This torque is always perpendicular to both and - in fact its direction can be described like this: suppose we draw a circular orbit in plane perpendicular to the direction of magnetic field in such a way that this orbit touches the end of vector . In this case is always going to be tangential to such an orbit - if one would imagine someone rotating a ball on this circular orbit, the velocity of the ball is always going to be parallel to . For sufficiently small times the change in will be parallel to :
and therefore when a single dipole is subject to the external magnetic field H 0 it precesses around it on a circular orbit. The circular frequency of such precession will be .
As one can see, this term forces the dipole, instead of just precessing on a circular orbit, to slowly converge towards the direction of the applied field. As a result of this, the dipole converges to the state of minimal energy - with the direction of the dipole being parallel to the direction of the applied field. The extra energy that the dipole initially had is usually transferred to the heat. If more than one dipole is present in the system, the magnetic field H in Eq. 3 includes above mentioned dipole-dipole and exchange fields. The effects these fields have on precession of the dipoles can be observed in Figures 5 and 6 (small damping is present, external magnetic field is equal to 0).
FOR HIGH SCHOOL LEVEL: Therefore when describing dynamic behavior of magnetic systems we can say that:
Let's illustrate these concepts with a few simulations: * You will need VRML Viewer to open the simulations below. » Install Free VRML Viewer » Simulation 1 - This movie demonstrates a "frictionless" dipole precessing (rotating) around the external magnetic field. » Simulation 2 - we turn the energy dissipation. As it was said before now the system moves towards the equilibrium, when magnetic dipoles are parallel to the magnetic field. » Simulation 3 - the system of a few dipoles in the presence of external field. Competition between dipole-dipole fields and the external field is responsible for the equilibrium. » Simulation 4 - same with exchange interaction present. Exchange interaction "wants" to align the dipoles along each other, making much more uniform magnetization. » Simulation 5 - Cooling of a large magnetic system. Imagine that initially we have a system at a high temperature, and as it was said before all of the dipoles are basically oriented randomly. If we rapidly cool the system, damping is going to suck the magnetic energy out of the system, leading it towards the equilibrium. Notice the multi-domain configuration. » Simulation 6 - Same, but now in the presence of a strong external field. Java version of all the simulations listed above: » Java Version What makes a body magnetic? As we discussed above, electrons have an intrinsic dipole moment. If the electrons inside of the body's atoms are somehow capable of realigning themselves under the influence of the magnetic fields, the body can possess a total dipole moment: more electrons align their magnetic moments along certain direction than along other directions. In particular, the energy E of a magnetic dipole in a magnetic field equals:
where is the magnetic field. As one can see the energy is minimal when magnetic moments are aligned along the magnetic fields (both external magnetic fields due and internal magnetic fields - the latter ones being due to their neighboring electrons). It is impossible to describe the process of magnetization (creating a total magnetic moment in a body) without saying a few words about temperature effects. Using a very imprecise definition, temperature is a measure of the thermal energy present in the system. Thermal energy is the energy due to random interactions between particles forming the system. The higher the temperature, the higher thermal energy is present in the system, the more random the system is. For example, if we have a magnetic body at a high temperature, individual dipoles are constantly experiencing random "punches" from all sides. As a result they move in a random fashion. If the temperature is higher than certain value (so called Curie temperature), the system can not have a total magnetic moment, because individual moments are simply randomly rotating with respect to one another. When we lower the temperature, the system becomes more and more ordered - the dipoles choose the configuration that corresponds to the energy minimum . If there is some external magnetic field applied to the system, the direction of this magnetic field will be preferred by magnetic moments, the body becomes magnetized : the stronger applied field is, the bigger portion of dipoles is going to be aligned in parallel to the field's direction. The more dipoles are aligned in the same direction, the bigger the total dipole moment the system possesses and so is the magnetic field produced by the system as a whole. One should also mention that the density of electrons that can reorient themselves varies between different magnetic materials - samples with higher number of such electrons per unit volume are obviously "more magnetic". This can be characterized by the value called "saturation magnetization" - maximum magnetic moment per unit value. A good example of magnetization: when molten lava is ejected by volcanoes, despite the fact that it contains certain iron-based chemicals that are in general magnetic, the high temperature prevents the lava from possessing any total magnetic moment. When the lava cools down, a significant portion of its magnetic dipoles becomes oriented parallel to the Earth magnetic field (Figure 1).
Therefore by studying the magnetization of cooled lava (paleomagnetism) we can study the time evolution of Earth's magnetic field - rock formations "record" the information about the magnetic field in the time of their cooling. As we said before, the polarity of Earth magnetic field have been reversed many times in the past, therefore what we are going to see are stripes of magnetized rocks with their magnetization anti-parallel between stripes formed in between of reversals of Earth magnetic field. Similar ideas can be applied when trying to date certain historical objects (archaeomagnetism). By knowing the strength of Earth magnetic field in the past we can assign an approximate date to materials such as ceramics - small iron particles inside of such objects orient their magnetic moment along the Earth magnetic field at the time of their last cooling. This method does not usually work with metallic objects made from magnetic metals due to the fact that the strong interaction between neighboring dipoles negates the effects due to the Earth magnetic field. For example Figure 2 shows a magnetized dagger that exhibits an unusually complex magnetization.
So what are the interactions between individual dipoles? In the previous Chapters we spoke about the fields, produced by dipoles:
Here we chose to use symbols more characteristic to magnetic dipoles (M instead of D and H instead of E - as a reminder M denotes magnetic dipoles and H denotes magnetic fields), however the configuration of dipole fields does not depend on magnetic or electric nature of the dipoles.
Figure 3 presents magnetic fields due to a single magnetic dipole (in order to show the similarity with electric dipoles we also show the charges that would correspond to the electric dipole of similar properties) . As one can see the field is parallel to the dipole's orientation above and below the dipole and anti-parallel on each of the sides of a dipole. This means that if dipole-dipole interaction would be the prevalent force in magnetic bodies one would expect the individual dipoles to form strips of dipoles aligned parallel to each other inside each strip and anti-parallel with respect to dipoles of neighboring strips (again, the lowest energy configuration corresponds to dipoles aligned parallel to the magnetic field). The only way to change the configuration would be to apply external magnetic field that would overcome the dipole-dipole interaction and force the alignment of magnetic dipoles parallel to each other. However it is known that it is usually not the case in real life - when external magnetic field is weak there is a variety of configurations that can appear in the system, depending on its magnetic properties, shape and size. From now on we will mostly discuss only one particular type of magnetic materials - so called ferromagnets. While there are many other types, this one is nearly the only one used in the information storage. The reason behind the fact that even in the absence of external magnetic field magnetic bodies can have a significant magnetization, i.e. total magnetic moment, it is that we ignored the atomic scale interactions between dipoles. Due to the fact that the properties of individual atoms are quantum in nature and can not be completely described by using classical theories we have been using so far, there is another force we need to account for: so called exchange interaction . It can be introduced into the classical picture by assuming that each of the dipoles produces additional field, equal to:
where J is exchange constant. Moreover this field acts only on the nearest neighbors of the dipole - it can be shown that beyond them this field drops off with distance as 1/R^{6}. It is clear that depending on the sign of J the field created by a dipole is either parallel (J>0) or anti-parallel (J<0) to the dipole. If J>0 and is big enough to counteract the magnetic field due to "classical" dipoles given by Eq. 1 (hereandafter referred to as "dipole-dipole" interaction), the lowest energy of the magnetic body corresponds to magnetic moments aligned parallel to each other - such magnetic materials are called to be ferromagnetic , otherwise the "natural" orientation is the one with dipoles anti-parallel to each other, and corresponding magnetic materials are called to be antiferromagnetic . In most cases the exchange interaction, external fields and dipole-dipole interactions are "competing" with each other while forming the stable magnetization configuration. As a result some areas, so called domains, can be formed by dipoles parallel to each other, while other domains, while also formed by dipoles parallel to each other, have each of their dipoles oriented anti-parallel or at some angle with respect to the dipoles of neighboring domains. The total magnetization of the body in this case can be close to 0, but if the external field is applied a large number of individual domains become oriented in parallel with the applied field, inducing a total magnetization in the sample. Another force one can encounter is called anisotropy . The presence of such force means that due to some structural properties of a particular magnetic body, its magnetic dipoles prefer to be oriented along certain directions, called "easy axis". One of the reasons behind the existence of anisotropy can be deformations appearing in magnetic materials on the scale of their atomic lattice. FOR COLLEGE LEVEL: Before we proceed to study the real applications, we need to introduce the most important equation used by scientists working on designing magnetic memory devices, the equation that describes the dynamics of magnetic dipoles in the applied field - Landau-Lifshitz equation:
where γ is gyromagnetic coefficient that depends entirely on the choice of units (for example whether SI or Gauss systems of units are chosen), β is a dimensionless damping coefficient. First we will consider the case when β is equal to 0. Direction of the torque is such that magnetic moment precesses around the direction of the magnetic field. This torque is always perpendicular to both and - in fact its direction can be described like this: suppose we draw a circular orbit in plane perpendicular to the direction of magnetic field in such a way that this orbit touches the end of vector . In this case is always going to be tangential to such an orbit - if one would imagine someone rotating a ball on this circular orbit, the velocity of the ball is always going to be parallel to . For sufficiently small times the change in will be parallel to :
and therefore when a single dipole is subject to the external magnetic field H 0 it precesses around it on a circular orbit. The circular frequency of such precession will be .
As one can see, this term forces the dipole, instead of just precessing on a circular orbit, to slowly converge towards the direction of the applied field. As a result of this, the dipole converges to the state of minimal energy - with the direction of the dipole being parallel to the direction of the applied field. The extra energy that the dipole initially had is usually transferred to the heat. If more than one dipole is present in the system, the magnetic field H in Eq. 3 includes above mentioned dipole-dipole and exchange fields. The effects these fields have on precession of the dipoles can be observed in Figures 5 and 6 (small damping is present, external magnetic field is equal to 0). FOR HIGH SCHOOL LEVEL: Therefore when describing dynamic behavior of magnetic systems we can say that:
Let's illustrate these concepts with a few simulations: * You will need VRML Viewer to open the simulations below. » Install Free VRML Viewer » Simulation 1 - This movie demonstrates a "frictionless" dipole precessing (rotating) around the external magnetic field. » Simulation 2 - we turn the energy dissipation. As it was said before now the system moves towards the equilibrium, when magnetic dipoles are parallel to the magnetic field. » Simulation 3 - the system of a few dipoles in the presence of external field. Competition between dipole-dipole fields and the external field is responsible for the equilibrium. » Simulation 4 - same with exchange interaction present. Exchange interaction "wants" to align the dipoles along each other, making much more uniform magnetization. » Simulation 5 - Cooling of a large magnetic system. Imagine that initially we have a system at a high temperature, and as it was said before all of the dipoles are basically oriented randomly. If we rapidly cool the system, damping is going to suck the magnetic energy out of the system, leading it towards the equilibrium. Notice the multi-domain configuration. » Simulation 6 - Same, but now in the presence of a strong external field. Java version of all the simulations listed above: » Java Version Information StorageHow one designs an effective memory element? In general such element should satisfy a few conditions:
Knowing what we already know about magnetic dipoles, we can imagine how one can use a magnetic dipole-like system as a memory element:
The only thing that is missing is how we are going to switch the magnetization between "dipole up" and "dipole down" states. The most simple way to do is to apply strong magnetic field parallel to the desirable configuration, i.e. if we want the dipoles to point down we need to apply external magnetic field pointing in the same direction, "down". As shown in the following movie, the system will change its magnetization so to be in this new equilibrium state: » Open Movie (You will need VRML Viewer to open the movie. » Install Free VRML Viewer) Afterwards we switch off the external field, however the anisotropy is still present and therefore is responsible for "holding" the magnetization in this new state. |
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