CONCLUSIONES Y RECOMENDACIONES 1 CONCLUSIONES

A field dependent Fermi surface leads to a field dependent frequency, however the observed frequency is not simply related to the changing Fermi surface size by F = ~

2πeA. Under these conditions, it is not possible to make a direct measurement of the extremal area causing the oscillation. To understand why this is the case, it is enough to consider a Taylor expansion of the frequencyF(B) =a0+a1B+a2B2+. . .. The oscillatory part of the Lifshitz-Kosevich formula

then becomes R_∝cos _2π(a 0+a1B+a2B2+. . .) B +γ (1.32) and it is obvious thata1 is indistinguishable from the phaseγ. A microscopic view of this phe-

nomena can be provided by recalling that the origin of the quantum oscillations is the expansion of Landau tubes through the Fermi surface. The frequency of the oscillations is given by the spacing between the tubes. If the Fermi surface is moving in or out, the oscillating DOS will be

Doppler shifted to a different frequency[7].

The argument of the of the oscillatory part of the Lifshitz-Kosevich formula is still given by: 2πF(B)

B +γ (1.33)

Where we are continuing to write the ‘true frequency’ F as a shorthand for ~

2πeA, which is different the observed frequency Fobs. When considering the oscillations as a function of _B1, we need to know the rate of change of the above with _B1, so:

Fobs= d d(1 B) _F(B) B + γ 2π =F(B)₋BdF(B) dB (1.34)

Which, if the form ofFobsis known, yields a linear ordinary differential equation forF. Note that it is also an equation for a straight line through the pointF(B =B0_{) with gradient} dF(B)

dB

B=B0

and interceptFobs. This leads to a common description ofFobs as the frequency back-projected to zero field. It is immediately obvious that this differential equation admits a complementary function of the formF =a1B, for anya1, in agreement with the simpler arguments above.

We can proceed no further without knowing the form ofFobs. For all measurements presented here,Fobsis smooth and slowly varying, so it can be represented as a Maclaurin series of relatively few terms. In order to continue our calculation then we assume a form

Fobs= ∞ X

n=0

αnBn

To solve equation1.34shall begin with the ansatz

F(B) =a0+alogBlog(B) +

∞ X

m=2

amBm

Differentiating the latter and substituting both expressions into equation1.34yields ∞ X n=0 αnBn = a0+alogBlog(B) + ∞ X m=2 amBm ! (1.35) −B alog(1 + log(B)) + ∞ X m=2 ammBm−1 ! =a0+a1B+ ∞ X n=2 am(1−m)Bm (1.36)

Finally we can equate coefficients in the two Maclaurin series, to obtain the result that:

a0=α0

alog=₋α1

an= αn

1₋n n >1 (1.37)

Which, combined with the complementary function noted above, is used in later chapters to convert observed frequencies into a family of possible real frequencies and thus extremal areas. In reality, we cannot use an infinite series, so must truncate it at some value ofn. If we choose nmax so that Fobs is well-represented, we can have confidence thatF is well represented as the final relation in equation1.37ensures that large-nterms contribute less to the series forF than Fobs.

It is useful to consider how this simplifies in some of the more familiar cases. In the case of a single, constant Fermi surface, the only term in either series is the zeroth term, andFobs=F. The other well known case is the paramagnetic case with spin. In this case, the field dependence of the extremal area is very small, and results from Zeeman splitting. It is linear, and depends on the Land´egfactor and the effective mass. The linear term is lost into the phase, andFobs=F(0), so no special consideration is required when convertingF toA. There is still a difference between the two orbit’s areas, but it changes with field at the same rate as the spacing between landau tubes changes. This means the phase difference between the two oscillations remains the same, and there is no change in properties with field. The phase difference is non zero though, so the total amplitude is reduced. This is usually represented by a spin reduction factor RS when analysing quantum oscillations, but in a ferromagnet, where different spin bands are often well separated, it is usually better to treat the bands separately. That is why the spin reduction factor was omitted in the discussions in section1.3.2.

The ferromagnetic case is different for two reasons. Firstly, in any ferromagnet the magnetiz- ationM is no longer small, and may be a strong function of applied field. As the field driving the oscillations isB, notµ0H, the changing M makes the splitting, though linear inB, nonlinear in

µ0H. This is the same effect which leads to magnetic interaction, though that term is normally

reserved for things arising from the oscillatory magnetization of the dHvA effect. Two examples of such materials are ZrZn2[7] and UPt3[8]. In the latter case, it is not strictly a ferromagnet,

rather the magnetization increases withB as one would expect from a paramagnet,∗ _{then jumps}

to a higher value. But the same logic applies.

The second effect is in itinerant ferromagnets, and is, where present, much stronger than the first. In this case, the magnetization and some of the Fermi surfaces are strongly coupled. The energy gap between the_↑ and _↓ bands can be much more than one would expect from Zeeman

∗_UPt

3 is discussed in the cited reference as a Pauli paramagnet, but more recent work suggests it is a weak

antiferomagnet[9]. In either case, the discussion about field dependent Fermi surfaces remains relevant.

splitting. Also, the bands in question are often heavy, flat bands with substantialf-weight. This means that as the exchange splitting changes the filling of the band, the size and shape can change drastically, leading to strongly field dependent orbits. These bands are also much less likely to be undergoing a rigid shift, rather they will be changing shape and nature too as the magnetization evolves.

In extreme cases, a band may move totally above (or below) the Fermi energy, resulting in an electronic topological transition, as has been suggested to be the case in YbRh2Si2[10]. In other

cases, phase transitions can occur, and it may be possible to follow the field dependent Fermi surface up to and even through the transitions, as in the metamagnet Sr3Ru2O7[11].