1.3 Introduction to mass storage media
1.3.2 Exchange bias
Hysteresis is well known in ferromagnetic (FM) materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized. The relationship between magnetic field strength H and magnetic flux density B(H) is not linear in such materials but has a sort of S−shaped loop as shown in Fig. 1.5a for grain–oriented electrical steel. For increasing levels of field strength, the magnetic induction curves up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation. If the magnetic field is now reduced linearly, the plotted relationship will follow a different curve back towards zero field strength at which point it will be offset from the original curve by an amount called the remanent flux density BR. In antiferromagnetic (AFM∗∗) materials on the other hand, the magnetic moments of atoms or molecules align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. Therefore, the total magnetization vanishes. Ferromagnetic materials are characterized by the Curie or critical temperature TC, whereas antiferromagnetic materials are characterized by the Néel temperature TN. Above the critical temperature, the collective magnetism merges into paramagnetism with the corresponding characteristic behavior of the inverse susceptibility χ(T ) as function of temperature [31]. If an antiferromagnetic and a ferromagnetic material are stacked on top of each other, magnetic exchange coupling, or exchange bias, is induced at the interface between the FM and AFM system. At the interface, an extra source of
∗∗Throughout the remainder of this thesis the abbreviations FM and AFM will stand for frequency modulation and atomic force microscopy, respectively, unless indicated differently.
BS BR
HS HC
BH()
(a) B(H)−loops
m[10emu]–4
2
–2
–4 4
–800 –400 400 800
H [Oe]
T = 10 K HE
HC 0
0
(b) AFM–FM system
Figure 1.5: (a) A family of B(H)−loops measured with a flux density that is sinusoidally modulated at 50 Hz with an amplitude varying from 0.3 T to 1.7 T. The material is conventional grain–oriented electrical steel. BR
denotes the remanent magnetic flux density, BSdenotes the saturation magnetic flux density, HS the saturation field and HCis the coercitivity. B(H)−loop is another name for hysteresis loop. (Adapted from [36]) (b) Hysteresis loop m(H), with m(H) the magnetic moment, of a FeFe2/Fe bilayer at T = 10 K after field cooling. The exchange bias HE and the coercitivity HC are indicated in the figure. (Adapted from [37]) In order to avoid any confusion about the units of magnetic quantities, the units in the SI system and the cgs system are listed in table 1.1.
anisotropy is introduced, leading to magnetic stability. This exchange coupling can be used to suppress the superparamagnetic limit [38]. Interface coupling due to exchange anisotropy is observed cooling the AFM–FM couple in the presence of a static magnetic field from a temperature above the Néel temperature, TN, but below the Curie temperature (TN < T < TC) to temperatures T < TN. The hysteresis loop of the AFM–
FM system at T < TN after the field cool procedure, is shifted along the field axis generally in the opposite (‘negative’) direction to the cooling field, see Fig. 1.5b, i.e. the absolute value of coercive field for decreasing and increasing field is different. This loop shift is generally known as exchange bias, HE. The hysteresis loops also have an increased coercitivity HC, after the field cool procedure. Both these effects disappear at, or close to, the AFM Néel temperature confirming that it is the presence of the AFM material which causes this anisotropy. In last decade, much research has been conducted on ferromagnetic nanoparticles embedded in a paramagnetic matrix or in an antiferromagnetic matrix [38]. More recently however, also ferromagnetic semiconductors, in particular GaMnAs, where interlayer exchange coupling (IEC) between the GaMnAs – either antiferromagnetic or ferromagnetic – is expected to play a crucial role in controlling the spin configuration within structures. Unfortunately, the magnetic anisotropy, which is the most important physical quantity in determining the direction of the magnetization in a ferromagnetic, is found to depend on many parameters in GaMnAs, including temperature, strain and carrier density, which largely complicates the development of a practical spintronic application. However, the possibility of electrical control of magnetic
Table 1.1: Units in the SI system and the cgs⋆ system (G: Gauss, Oe: Oerstedt, erg: energy, emu: electromagnetic unit)
Quantity Symbol SI unit cgs unit
magnetic induction B 10−4 T = 1 G
magnetic field H 103/4π A/m = 1 Oe
magnetic moment m 10−3 J/T = 1 erg/G⋆
magnetization M 103 A/m = 1 Oe
magnetic susceptibility χ 4π = 1 emu/cm3
⋆cgs = centimetre–gramme–seconde.
⋆⋆1 erg = 10−7J.
anisotropy of a GaMnAs layer along with controlling IEC in GaMnAs multilayers by doping, suggests that such multilayers can potentially provide significant device advantages over metallic ferromagnetic multilayers [39].
As already suggested, a lot of additional research needs to be performed on GaMnAs layers, of which on of the most challenging will be to extend its Curie temperature to room temperature. Therefore, characterizing its magnetic properties by means of low temperature magnetic force microscopy could indeed proof to be a very usefool tool.
Fundamentals
2.1 Non–contact force microscopy
In this section, we will outline the most fundamental theories involved with atomic force microscopy as studied by Uwe Hartmann [40]. We will start with the simplest mode of operation of a non contact scanning force microscope, which consists in lifting the cantilever probe up to a certain distance from the sample surface to measure a long–range interaction in terms of a static force exerted on the probe. This mode of operation is in general referred to as the static or dc mode operation. When we monitor the deflection of the cantilever end resulting from the applied force, we define the deflection in one dimension as
∆z = Fz/k, (2.1)
z
y x
Figure 2.1: Three dimensional representation of the microcantilever–tip system.
where Fz is the force applied in the z direction. The static mode operation however, is not preferred in reality. Far more sensitive detection can be realized by utilizing the dynamic properties of the probe, e.g. by involving sinu-soidal excitation of the probe, also called the dynamic or ac mode operation. Excitation of the probe can be achieved by attaching the cantilever a bimorph piezoelec-tric plate, the dither piezo. In contrast to the detection of quasistatic forces, the response of the cantilever in the dy-namic mode is more complex. A thorough understanding of dynamic SPM operation requires to solve the equation of motion of the cantilever–tip ensemble under the influ-ence of tip–surface forces. A complete descriptive analysis is a formidable task that involves the solution of the equa-tion of moequa-tion of a three–dimensional object, a vibrating cantilever (Fig. 2.1). Some symmetry considerations allow to approximate the microcantilever by a one–dimensional object [41], then
EΥ∂4w(x, t)
∂x4 + µ∂2w(x, t)
∂t2 = F (x, t), (2.2)
here w(x, t) is the transverse displacement of the cantilever, E is Young’s modulus, Υ is the moment of inertia, µ is the mass per unit of length of the cantilever, and F (x, t) is the force per unit of length exerted on the cantilever. Combined with the point–mass approximation∗, the cantilever sensor may be represented by a simple harmonic oscillator consisting of a spring with a spring constant k and an effective mass m, yielding a resonance frequency of
ω0= rk
m. (2.3)
∗In the point–mass approximation the cantilever–tip ensemble is considered as a point–mass spring, see for example Rugar et al.[42]
If the cantilever is excited sinusoidally at its clamped end with a frequency ω and an amplitude A0= F0/m, the probe tip likewise oscillates sinusoidally with a certain amplitude Aδ, exhibiting a phase shift ∆ϕ with respect to the drive signal applied to the piezoelectric actuator. The deflection sensor of the force microscope monitors the motion of the probe tip, provided that its bandwidth is large enough. In the absence of tip–sample forces, the equation of motion in one dimension of the cantilever is given by
∂2z(t)
∂t2 +ω0
Q
∂z(t)
∂t + ω20(z(t) − z0) = A0cos (ωt), (2.4) where z0 is the probe–sample distance at zero oscillation amplitude and z(t) is the momentum probe–sample separation. The quality factor Q of the spring itself stems from intrinsic properties of the cantilever, which are the lumped effective mass and the resonant frequency, and is determined by the damping factor γ in the following manner,
Q = mω0
γ . (2.5)
For the resonance drive frequency ωr, which takes into account the damping of the cantilever, but neglects the tip–sample interaction, we write
ωr= ω0
r 1 − 1
2Q2. (2.6)
The influence of operation conditions, e.g. ambient air, liquid, ultrahigh vacuum (UHV) or low–temperature (LT), is reflected by the damping factor γ. The Q factor reflects the coupling to dissipative forces and can range from values below 100 for liquid or ambient gas conditions at an appropriate pressure, to more than 100, 000 which is sometimes obtained in UHV. Eq. 2.4 is the equation of a forced oscillator with damping. The general solution to this equation has a transient term and a steady state solution. Initially, both terms are prominent, after a time 2Q/ω0 however, the transient term is reduced by a factor 1/e [41], and the motion is dominated by the steady state solution
z(t) = z0+ A cos(ωt + α). (2.7)
The amplitude A of the probe oscillation is given by
A(ω) = A0ω20 q
(ω2− ω02)2+ 4γ2ω2
. (2.8)
The phase shift between the probe oscillation and the excitation signal amounts to
∆ϕ = arctan
2γω ω2− ω20
. (2.9)
The above simplified formalism is based on the assumption that the oscillation amplitude A is sufficiently small in comparison with the length of the cantilever. The results derived so far however, describe only free cantilever oscillations, meaning that z0 is large enough, to prevent any influence of the sample on the probe oscillation. If z0 is now decreased such that a force Fts affects the motion of the cantilever then a term Fts/m has to be added to the right–hand side of Eq. 2.4, yielding
m∂2z
∂t2 + γ∂z
∂t + k (z(t) − z0) = F0+ Fts. (2.10) In order to consider almost all interactions that could be relevant in SPM, one has to assume
F = F
which takes into account both the static and dynamic interactions. Since F covers probe–sample interactions of various types, in particular spatially nonlinear ones, the z(t) curves monitored by the deflection sensor and defined by Eq. 2.4 represent anharmonic oscillations, rather than the commonly know harmonic oscillations of the spring–type model. If however F (z(t)) can be substituted by a first–order Taylor series approximation, which is valid for Aδ ≪ |z0|, then the force microscope detects the compliance or vertical component of the
force gradient ∂F/∂z. On the basis of this approximation, the cantilever behaves under the influence of the probe–sample interactions as if it had a modified spring constant, given by
kef f= k −∂F
∂z, (2.12)
where k is the intrinsic spring constant, which has to be substituted into Eq. 2.3. As a result, an attractive probe–sample interaction, with ∂F/∂z > 0 will effectively soften the cantilever spring, while a repulsive interaction with ∂F/∂z < 0 will make it effectively stiffer. Converting Eq. 2.3 with the modified spring constant yields
The minus sign comes from the fact that F is increasingly attractive as the tip approaches the surface; thus, a decrease of the resonance frequency is observed. Provided that ∂F/∂z ≪ k, or ∆ω ≪ ω0 – which is always the case in MFM and AFM applications – the shift in resonant frequency is given by
∆ω ≈ −1 2k
∂F
∂z. (2.14)
Summarizing, if we drive the lever at ω0, we can measure the amplitude eA0 due to this excitation and use Eq. 2.8 to calculate the new resonance fω0(t) =p
where a = A0/ eA0 [43]. According to equations 2.8 and 2.9, a shift in the resonant frequency will result in a change of the probe oscillation amplitude Aδ and of the phase shift ∆ϕ between probe oscillation and driving signal. ∆ω, ∆A, and ∆ϕ are experimentally measurable quantities that can be used to map the spatial variation of ∂F/∂z. Additionally, phase and amplitude contain information about the damping coefficient γ.
The simple harmonic solution in Eq. 2.7 evidently shows that the dynamic mode of operation can be based on the employment of lock–in signal detection methods. The additional use of suitable feedback mechanisms opens up different variants of operation. These modes of operation will be discussed in the next sections.