3.3 Varieties of Magnetism

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Chapter 3

Fundamental concepts

3.1 Why ultrahigh vacuum is needed?

There are two reasons why ultrahigh vacuum is needed in studying surface properties.

First is to get a suitable environment for the electrons emitted from the specimen to reach the analyzer, and the second is to avoid the unexpected gas contaminating the surface.

In order to decrease the probability of electron emitted from the specimen scattering by the ambient gas, they should meet as few gas molecule as possible on their way to the analyzer. Generally, the vacuum pressure in the range 10⁻⁵–10⁻⁶ Torr is enough to deal with the problem. But the question is, the electrons which reached the analyzer are those we really want? Are they emitted from the surface contamination? This is a very serious question that aﬀects the precise of the data.

A good clean surface must be under an condition that the contamination may play a minor role for molecular beam epitaxy (MBE) growth and surface studies. The ambient pressure P is an easy way to determine the amount of unexpected gas contaminates on the surface. Denoting n to be the average amount per unit area∆A per unit time ∆t of the ambient gas normally strike the surface, the change of total momentum of gas∆p per unit time∆t can be expressed as eq. (3.1).

∆p

∆A·∆t = 2mvn (3.1)

1

2mv² 3kT

2 (3.2)

The symbol m is the mass of the gas molecules, v is their root-mean-square thermal velocity at temperature T (in Kelvin), and k is the Boltzmann’s constant. By combining

21

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eq. (3.1) with eq. (3.2), one can get the expression of the ambient pressure as shown in eq. (3.3).

P = ^∆p

∆A·∆t 6kT n

v (3.3)

The number n can be easily obtained from eq. (3.3).

n = P· v

6kT (3.4)

Assuming the average molecule weight of gas is 28, one can get v  5.17 × 10⁴ cm/s at T = 300 K. The surface concentration of element Pt is about 10¹⁵ atoms/cm²[38]. The sticking coeﬃcient which is the probability that an impinging molecule remains adsorbed after striking the surface is denoted to be s. Substituting the above values to eq. (3.4), n can be expressed as eq. (3.5).

n 2.8 · 10²⁰· P s

cm²· sec · torr  2 · 10⁵· P · s M L

sec· torr (3.5) This means that it needs approximately only 1 to 10 sec for the ambient gas to build up a monolayer of adsorbate on the Pt surface at a pressure of 10⁻⁶ Torr. The mean free path of the observed electrons is only several monolayers. If we want to have enough time to study the surface without the inﬂuence of contamination, we need to decrease the pressure of the chamber to at least 10⁻¹⁰torr. That is an important reason why ultrahigh vacuum is needed in studing surface properties.

3.2 Growth and surface structure

3.2.1 How to study the growth mode in our lab?

There are two methods for us to study the growth mode and to calibrate the deposition rate of adsorbate. The ﬁrst is evolution of AES intensity versus deposition time; the other is the LEED specular beam oscillated with the processing deposition.

(a) AES uptake curve

The time evolution of AES intensity is so called AES uptake curve. Before Auger electron escaping from the specimen surface, it must meet the adsorbate atoms on its way. Since the AES electron may be scattered by the adsorbate atoms, the intensity of the AES signal will be aﬀected by an exponential decay term. Assuming the growth is layer-by-layer and

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3.2. Growth and surface structure 23

there is already has n ML of complete adsorbate layers building up on the surface, one can expresses the time evolution of AES intensity as shown in eq. (3.7).

I(t) = tRI₀e^−nd/λe^−d/λ+{1 − tR}I0e^−nd/λ (3.6)

I(t) = tRI0e^−nd/λ{e^−d/λ− 1} + I0 (3.7)

I(t) and I₀ are the Auger intensities at deposition time t and at the initial stage.

The symbols R, d and λ are the deposition rate of the adsorbate, interlayer distance of the adsorbate, and the mean free path of the Auger electron, respectively. Under the conditions of steady deposition rate and the same n, it is easy to see that I(t) is proportional to the deposition time t from eq. (3.7). AS the ﬁlm ﬁnishing a complete layer, the AES uptake curve starts to change its slope. Hence, the AES uptake curve has a kink between two neighboring layer growth regions. Alternatively, if the adatom starts to grow up before the previous layer building up completely, the AES uptake curve will not obey the proportional relation in eq. (3.7).

(b) LEED specular beam oscillation

The diagram of LEED diffraction pattern is a result of superposition of the wavefunction of the diffraction electron beam. A sharp and bright diffraction spot results from a well organized surface. By this way, the time evolution of LEED specular beam intensity ((0,0) beam) has local maximum when the adsorbate form a complete layer. Alternatively, its intensity will decrease with the growth of the film without appearance of local maximum.

During the measurement of LEED (0,0) intensity in our lab, the kinetic energy of the incident electron beam must be chosen carefully to be 60 eV. At this energy level, LEED diﬀraction pattern should be very surface sensitive. Another way of expressing this requirement is that the out-of-phase condition of diﬀraction is more sensitive to surface step density on the (0,0) beam.[39] The phase in our experimental conditions is ϕ(0, 0) = h K_⊥ = 2hKcosθ ≈ 5π for 60 eV electron energy and incident angle θ = 5^◦, where K_⊥ is the momentum transferred perpendicular to the surface, and h_{N i} = 0.203 nm and h_Co = 2.04 nm are the step height of the adsorbed Ni adatom and Co adatom, respectively.

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Figure 3.1: The schematic plots of three kinds of growth mode.

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3.2. Growth and surface structure 25

3.2.2 Growth modes

In general, the growth mode of ultrathin ﬁlm can be described as three marked types [40].

The schematic plot of the three kinds of growth modes is plotted as shown in Fig. 3.1.

The AES uptake curve and LEED specular beam oscillation of each growth mode are also plotted.

(a) Frank-van der Merwe growth mode (FM mode)

When the interaction between substrate atom and adatom is greater than that between the neighboring adatoms, the subsequent film favors growing after the previous deposited layer formed a complete film. This is also called layer-by-layer growth. The corresponding AES uptake curve has a kink when a complete layer is formed. At this place, we can also find a local maximum of LEED (0,0) beam intensity as shown in Fig 3.1.

(b) Stranski-Krastanov growth mode (SK mode)

The film grows one or several complete layers and following the 3-dimensional island growth instead. In Fig. 3.1(b), the AES intensity increases linearly with deposition time before turning into 3-dimensioinal island growth. There LEED (0,0) beam intensity has only one local maximum. This means that the film builds up only one complete layer at first, after which it turns to growing 3-dimensional island.

(c) Volmer-Weber growth mode (VW mode)

If the interaction between neighboring adatoms is stronger than that between adatom and substrate atom, the ﬁlm like to grow up initially in 3-dimensional island growth mode.

There shows no linear relation between AES signal and deposition time, and no local maximum can be observed in the time evolution plot of LEED (0,0) beam intensity.

3.2.3 Lattice mismatch and surface free energy

Lattice match and surface free energy are usually used to interpret the growth mode of ultrathin ﬁlm.

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Table 3.1: The lattice mismatch parameter η in percentage (%).

Adsorbate atoms

Co Ni Pt

Substrate a_B a_A= 3.550 ˚A 3.524 ˚A 3.924 ˚A

Co 3.550 ˚A 0 −0.73 10.54

Ni 3.524 ˚A 0.74 0 11.35

Pt 3.924 ˚A −9.53 −10.19 0

(a) Lattice match

The lattice mismatch parameter η of A atom epitaxial growing on B single crystal is deﬁned as eq. (3.8), where a_Aand a_B are the lattice constant of bulk A and bulk B.

η = a_A− aB

a_B (3.8)

If the lattice mismatch between the lattice parameters is not too large, minimizing the total energy leads to a situation whereby, below a critical thickness, the misﬁt can be accommodated by introducing a tensile strain in one layer and a compressive strain in the other such that ultimately the two materials A and B adopt the same in-plane lattice parameter. This regime is called the coherent regime; the lateral planes are in full lattice- registry. The values of η are listed in Table 3.1. The deposited ﬁlm faces a tensile strain if η < 0, on the other hand, it faces a compressive strain if η > 0.

(b) Surface free energy

The growth mode of a film on a particular substrate must follow a universal law which is to minimize the total energy of the system. The energy difference of surface free energy and interface energy can be use to discuss this problem. Surface free energy is defined as the work required increasing the area of a substance by one unit area. The excess of free energy per unit area of film/substrate interface is defined as interface energy. Thus, the energy difference∆γ can be written as eq. (3.9).

∆γ = (γ_{f ilm}+ γ_{interf ace})− γsubstrate (3.9)

,where γ_{f ilm}and γ_substrateare the surface free of the adsorbate and the substrate. γ_{interf ace} is the interface energy of the adsorbate/substrate interface. If ∆γ < 0, this means that

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3.3. Varieties of Magnetism 27

Table 3.2: Surface free energy

Ni Co Pt

γ (J/m²) 2.364 2.709 2.691 Ref. [41]

the system can decrease the total energy by decreasing the area of substrate and increasing the areas of ﬁlm and interface. Hence, the ﬁlm favors layer-by-layer growth mode.

Alternatively, it favors 3-D island growth if∆γ > 0.

3.3 Varieties of Magnetism

When a magnetic field H is applied, a magnetic flux will flow though the space. The magnetic flux density is called magnetic induction B. An easy way to express the relation between B and H is the B− H curve. The ratio of these two quantities, B and H, is the permeability µ .

µ = B

H (3.10)

Magnetization is the amount of magnetic moment per unit volume of a material. Mag- netization curve of a material is the evolution of magnetic induction inside the material versus external applied ﬁeld, it is also called the B−H curve of a material. The magnetic properties of a material are characterized by magnetization M and the way in which M varies with H. Therefore, the relation between B, M, and H can be expressed as follow

B = µ₀(H + M ) (3.11)

, where µ₀ = 4π× 10⁻⁷ N/A² is the permeability of free space. The MKS units of B, H, and M are webers/meter² (N/A· m; Tesla), amperes/meter (A/m), and amperes/meter (A/m), respectively. A relation between M and H exists in many materials.

χ_m = M

H (3.12)

, where χ_m is called the magnetic susceptibility (dimensionless). Thus, the varieties of magnetism can be roughly characterized by their corresponding values of χ_m and µ.

1. Empty space: χ_m= 0; µ = µ₀ = 4π× 10⁻⁷ N/A².

2. Diamagnetism: χ_m is negative and at a order of 10⁻⁶; µ is slight less than µ0. 3. Paramagnetism and antiferromagnetism: χ_m is positive and at a order of 10⁻³ −

10⁻⁵; µ is slight less than µ₀.

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4. Ferromagnetism and ferrimagnetism: χ_m and µ are large and positive; Both are function of H.

3.3.1 Diamagnetism and Paramagnetism

Diamagnetism is the most common magnetic behavior, it occures for filled and partially filled orbitals. The diamagnetic magnetization is proportional and opposing to H. This can be explained by the Lenz’s law. When a orbiting electron is affected by H, it will meet a magnetic force on it in the form of −→F = q−→v × −→B . This force causes it to either speed up or slow down in its orbital motion. This modifies the magnetic moment of the orbital in a direction against the external field. The magnetic susceptibility is at an order of 10⁻⁶ and is independent of temperature. For example, the most strongly diamagnetic material is Bismuth (Bi, Z = 83), its χ =−166 × 10⁻⁶. All materials show a diamagnetic response in an applied magnetic field. But, when the material exhibits another magnetic property such as ferromagnetism and paramagnetism, the diamagnetism is completely swamped.

Paramagnetic materials present a magnetization that is proportional to the applied field. It requires that the atoms individually have dipole moments even without an applied field, which typically implies a partially filled electron shell. These atomic dipoles have no interaction between them and are randomly oriented in the absence of an external field. Thus, paramagnetism has zero net magnetization with the absence of applied magnetic field. Paramagnetic materials in magnetic fields will act like magnets, but thermal motion will quickly disrupt the magnetic alignment when the field is removed. Param- agnetism behavior can also be observed in ferromagnetic materials that are above their Curie temperature, and in antiferromagnets above their Neel temperature. In general, paramagnetic effects are very small (magnetic susceptibility of the order of 10⁻³ to 10⁻⁵) and is dependent of temperature. The magnetic susceptibility χ_m varies inversely with temperature was first reported by Pierre Curie in 1895 as shown in eq. 3.13. It is called the Curie’s law.

χ_m = C

T (3.13)

C is the Curie constant per gram and T is the absolute temperature in Kelvin.

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3.3.2 Ferromagnetism

Ferromagnetism presents a magnetization much larger than other materials. It arises from the strong coupling between the magnetic dipoles in the material. According to classical electromagnetism, two nearby magnetic dipoles will tend to align in opposite directions. But, in ferromagnetism, they like to align in the same directions. This is a purely quantum effect. The property of ferromagnetism is due to the spin and the Pauli exclusion principle. Because of the Pauli principle, two electrons with the same spin state cannot lie at the same position, and thus feel an effective additional repulsion that lowers their electrostatic energy. This difference in energy is called the exchange energy and induces nearby electrons to align in the same direction. In addition, the dipoles of ferromagnetism tend to align spontaneously without any applied magnetic field. At this place, the magnetization is called spontaneous magnetization. Only atoms with partially filled shells can experience a net magnetic moment in the absence of an external field.

Ferromagnetism favors to form many magnetic domains (as shown in Fig. 3.2) to decrease the magnetostatic energy raised from the spontaneous magnetization. A magnetic domain is a region that all the magnetic dipoles inside it align to a same direction.

However, the directions of two neighboring magnetic domains are not the same. The transition between two domains, where the magnetic dipoles rotate, is called a Domain wall (e.g., for a Bloch/Neel wall, the magnetic dipoles rotate parallel/perpendicular to the domain interface). The thickness of a domain wall is on the atomic scale (covering a distance of about 300 ions for iron). Those domains whose magnetic dipoles inside them have the same direction with applied magnetic ﬁeld will expand their area by domain wall motion. Hysteresis phenomenon is a characteristic of this motion. A typical hysteresis

Figure 3.2: Ferromagnetic domains. (R.W. DeBlois) [42]

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Figure 3.3: Hysteresis loop of ferromagnetic material and its domain wall motion (unaxial easy axis).

Figure 3.4: M_S− T curve of ferromagnetism.

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Table 3.3: The value of critical exponent β for several theoretical models.

Model β Reference

2-D Ising 0.125 [44][45][46]

2-D XY 0.23 [47]

3-D Ising 0.325 [48][49][46]

3-D Heisenberg 0.365 [48]

mean ﬁeld theory 0.5 [50][46]

loop (M− H curve) of ferromagnetism is shown in Fig. 3.3. MS, M_R, and H_C are the saturated magnetization, remnant magnetization (magnetization at zero external field), and coercive force (field strength needed to demagnetize the material), respectively. The saturated magnetization is temperature dependence. As the temperature increases, thermal fluctuation decreases the tendency for dipoles to align. When the temperature rises beyond a certain temperature, thermal fluctuation will overcome the exchange energy and the spontaneous magnetization drops to zero. But the paramagnetic property still persists beyond that temperature. Thus, the M_S − T curve (as shown in Fig. 3.4) has a small tail of weak spontaneous magnetization extending into the paramagnetic region.

The temperature T_C deﬁned by the extrapolation (dotted line) of the main part of the curve is called Curie temperature [43]. As T → TC, the magnetization M_S becomes proportional to (T_C − T )^β [43]. β is called critical point exponent and is dependent on theoretical models. Some of the β values corresponding to several theoretical models are shown in Table 3.3.

The Curie temperature is dependent on the strength of the magnetic interaction, namely, the total exchange energy per spin [51, 52]. Increasing the ﬁlm’s thickness will results in an increase of the average number of nearest neighbors, therefore, the total exchange energy per spin is thickness dependent. This causes T_C to increase with the ﬁlm thickness. In contrast, the value of β does not vary with the total exchange energy per spin, it is sensitive to the actual type of the lattice or the form of the interactions.

This phenomenon is called university. In magnetic phase transition, a decreased critical exponent is often interpreted to result from a decrease in the dimensionality of a magnetic system [24]. The dimensionality of a magnetic system can be characterized by two lengths:

the thickness d of the ferromagnetic ﬁlms and the size∆of the largest lateral ﬂuctuations.

Since d∆, all ﬁlms fall into 2-D universality class [51].

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Figure 3.5: Sublattices of (a) Antiferromagnet and (b) Ferrimagnet.

3.3.3 Antiferromagnetism and Ferrimagnetism

Below a particular temperature, called N´eel temperature, the lattice of magnetic ions in the crystal of a material break up into two sublattices having magnetic moments antiparallel to each other. Such material is called antiferromagnet (eg. MnO). Owing to the magnetic moment of the magnetic ions have the same strength; the net magnetization of antiferromagnet is zero. The schematic plot of this kind of lattice is shown in Fig. 3.5(a).

Generally, antiferromagnetic materials become disordered above Néel temperature. Above the Néel temperature, the material is typically paramagnetic. Antiferromagnets can also couple to ferromagnetic materials through exchange anisotropy. This provides the ability to ”pin” the orientation of a ferromagnetic film, which provides one of the main uses in so-called spin valves, which are the basis of magnetic sensors including modern hard drive read heads.

When the two sublattices of antiferromagnet have unequal moments as shown in Fig. 3.5(b), the net magnetization is not zero. Such material is called ferrimagnet, for example, Fe₃O₄. The iron ions at diﬀerent sites have diﬀerent strength of magnetic moment.

3.3.4 Superparamagnetism

When the size of the magnetic particle decreases to a certain dimension (1-10 nm), the thermal energy is suﬃcient to change the direction of magnetization of the entire particle.

Such a material is called superparamagnetism. It may exhibit a behavior similar to paramagnetism at temperatures below the Curie or the N´eel temperature.

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0 100 200 300 400 500 600

0 100 200 300 400 500

Ni: fcc

<110>

<111>

<100>

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<111>

Medium

<110>

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<100>

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Hard

<111>

Medium

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<100>

0 200 400 600 800 1000

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H (Oe) 1200

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Fe: bcc

0 2000 4000 6000

0 200 400 600 800 1000

[0001]

M(emu/cm)3

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1200 1400

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Co: hcp

[1010]

Figure 3.6: Crystal structures of Fe [53] , Ni [54] , and Co [55].

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3.4 Magnetic anisotropy

The preference for a magnetization to lie in a particular direction is called magnetic anisotropy. The direction for a sample like most to be saturate magnetized is called easy axis, it was usually found in ferromagnets, such as single crystals of Fe, Co, and Ni (see Fig. 3.6). The energy required to change the direction of magnetization is called the magnetic anisotropy energy E_A. If we ignore the higher order terms, it can be written as eq. 3.14

E_A= K_{ef f}sin²θ (3.14)

, where K_{ef f} is the eﬀect magnetic anisotropy and θ is the angle of the magnetization with respect to surface normal. A system prefers PMA if its K_{ef f} > 0, while it favors a in-plane easy axis if K_{ef f} < 0. (Another expression of E_Ais K_{ef f}cos²θ, and thus, the sign of K_{ef f} is opposite to that we discuss in this dissertation.) The eﬀective magnetic anisotropy K_{ef f} can be expressed by a volume contribution K_V and a interfaces contribution K_S. The two contributions approximately obeyed the relation:

K_{ef f} = K_V +2K_S

t (3.15)

, where t is the thickness of the magnetic thin ﬁlm. The factor 2 in the interface con- tribution is corresponding to the two same identical magnetic/nonmagnetic interfaces of multilayers. For example, each period of (Co/Pt)_n multilayers has two Co/Pt interfaces and one t thickness of Co. The interface term shows a ¹_t response. This is represents the diﬀerence between the anisotropy of the interface atoms with respect to the inner or

Figure 3.7: K_{ef f} · t vs t plot of Co/Pd multilayers [56].

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3.4. Magnetic anisotropy 35

Table 3.4: Magnetic anisotropy value Reference

K_v^Co −0.8 MJ/m³ [57]

K_v^{N i} −0.14 MJ/m³ [58]

K_s^Co−vacuum −0.28 mJ/m² [59]

K_s^{N i−vacuum} −0.48 mJ/m² [60]

K_s^{Co−P t} 0.6 mJ/m² [16]

K_s^{Co−P t} 0.97 mJ/m² [58]

K_s^{Co−P t} 1.15 mJ/m² [61]

K_s^{Co−N i} 0.42 mJ/m² [57]

K_s^{N i−P t} 0.17 mJ/m² [16]

bulk atoms. den Broeder et al [56] is the ﬁrst group who used eq. (3.15) in the determi- nation of K_V and K_S. The two contributions can be obtained by a plot of the product K_{ef f}·t versus t as shown in Fig. 3.7. The vertical axis intercept equals twice the interface anisotropy, whereas the slope gives the volume contribution.

The volume and interface contributions listed in Table 3.4 are useful for us to discuss the direction of magnetization easy axis.

3.4.1 Magnetic dipolar anisotropy, Kmd

Magnetic dipolar anisotropy is raised from magnetostatic energy. Generally, it depends not only on the magnetic properties but also on the shape of the sample. The magnetostatic energy results from the magnetic poles on the surface. As shown in Fig. 3.8, when a magnetized material has magnetic flux lines flow through its surface with normal component, there exist free standing poles on this surface. This results in an internal magnetic field, demagnetizing field, exists inside the material. Thus, the magnetic field really passes through the material is the sum of external applied field H_appl and demagnetizing field H_d. The strength of demagnetizing field is related to the shape of the sample. For an arbitrarily shaped sample with a magnetization−M , it can be expressed as eq. (3.16)→

−

→H_d=−N−M→ (3.16)

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Figure 3.8: Schematic plot of microscopic dipoles of a magnetized material.

,where N is called the demagnetization factor, and in general, it is a shape-dependent tensor function. If the ends of the sample are flat, N = 1. This is approximately persisted in a thin film. For a thin film, the magnetostatic energy density can be expressed as eq. (3.17).

E_d=−µ₀ 2V = 1

2µ₀M_S²cos²θ (3.17)

Here, θ is the angle between −M and surface normal. Because the thickness of magnetic→ ﬁlm does not aﬀected E_d, thus, the magnetostatic energy contributes only to volume anisotropy K_V. To minimize the magnetostatic energy, the magnetic dipolar anisotropy favors an in-plane easy axis (θ = 0^◦).

3.4.2 Magnetocrystalline anisotropy, Kmc

Magnetocrystalline anisotropy is the energy cost per volume to align its magnetization from one crystallographic direction to another. The spin-orbit interaction is the primary source of the magnetocrystalline anisotropy. If the magnitude of the dipolar interface contribution is of minor importance, the spin-orbit coupling appears to be dominant.

Consequently, the spin-orbit interaction is more important for the magnetocrystalline anisotropy in Fe, Ni and Co.

How is the spin-orbital interaction coupled to the lattice? To solve this problem, the crystalline electric field seen by an atom must be taking into consideration. If the crystalline electric field seen by an atom is of high symmetry, the z component of the atomic orbital angular moment (Lz) can register in several orientations. Hence, the spin couple with the orbit has less anisotropy. Alternatively, if the crystalline electric field is of less symmetry, the orientation ofLz) will prefer a special direction. Thus, the

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3.4. Magnetic anisotropy 37

spin will prefer a orientation relative to −→L . This results in a strong magnetocrystalline anisotropy.

3.4.3 Magneto-elastic anisotropy, Kme

If the lattice is changed by strain the distances between the magnetic atoms is altered and hence the spin-orbital interaction energies are changed. This produces magneto-elastic anisotropy. Strain in ﬁlms can be induced by various sources. Some of them are dependent of −→H_appl (eg. magnetostriction), some are not (eg. thermal expansion, dislocation, lattice mismatch etc.).

A substance changes its dimension when exposed to a magnetic ﬁeld, this is called magnetostriction. For an elastically isotropic medium with isotropic magnetostriction, the magneto-elastic energy per unit volume E_me can be written as

E_me= 3

2λσsin²θ = K_mesin²θ (3.18) , where λ and σ are the magnetostriction constant and stress, respectively. θ is the angle between the magnetization and stress directions. The magnetostriction result in magnetic anisotropy can be expressed as anisotropy constant K_me. For positive, the easy magnetic direction will be along a direction of tensile stress, or perpendicular to a compressive stress.

When the parameters are constant (not depending on the magnetic layer thickness, t ) this contribution can be identiﬁed with a volume contribution K_V.

3.4.4 Magnetic Surface Anisotropy

The magnetic surface anisotropy is resulted from symmetry-breaking around surface atoms, it is a straightforward consequence of the fact that surface atoms are located in a diﬀerent environment than the bulk ones. The low symmetry behavior raises a strong magnetic anisotropy which contributes to the surface term K_S.

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