Contents

List of Figures

    2.1  Phonon \mathaccent "017E\relax k-vectors with \voidb@x w_{\mathaccent "017E\relax k} < 5
    2.2  Distribution function over w for w < 0.1 ·10¹⁵   \voidb@x \voidb@x s^-1
    2.3  Distribution function over w for w < 10¹² \voidb@x \voidb@x s^-1
    2.4  Distribution function over w for w < 10⁹   \voidb@x \voidb@x s^-1
    2.5  Distribution function over q for w < 10¹² \voidb@x \voidb@x s^-1
    2.6  Kapitza resistance between gold heater and glass substrate as a function of T

    3.1  Pumped ⁴He cryostat
    3.2  Experimental cell
    3.3  Twin bars setup
    3.4  Setup for electropolishing the copper bar
    3.5  Electrical circuit for electropolishing
    3.6  Typical electropolishing I,V-diagram
    3.7  Heater on copper
    3.8  Excitation electronics
    3.9  Tunnel diode and filtering electronics
    3.10  Detection system
    3.11  Gas handling system

    4.1  Third sound signal on glass. \voidb@x n _{\voidb@x in} = 16.83 mmol
    4.2  Third sound signal on glass. \voidb@x n _{\voidb@x in} = 35.9 mmol
    4.3  Third sound signal on copper. \voidb@x n _{\voidb@x in} = 35.9 mmol
    4.4  Third sound speed c₃ on copper and on glass as a function of the amount of ⁴He in the cell
    4.5  Third sound speed c_3,Cu on copper as a function of ⁴He film thickness d_Cu
    4.6  Third sound attenuation a on glass as a function of ⁴He film thickness d
    4.7  Third sound attenuation a on copper as a function of ⁴He film thickness \voidb@x d_Cu
    4.8  Brisson's heater

    A.1  Bottom plate in experimental cell
    A.2  Left frame piece
    A.3  Right frame piece
    A.4  Copper resonator bar
    A.5  Glass resonator bar
    A.6  Mounting platform

Chapter 1
Third Sound

1.1  Introduction

Liquid ⁴He becomes a superfluid, i.e., it can flow without friction, below the l-temperature (2.18 K). Superfluid ⁴He can be described by the two-fluids model[1]. This model states that the liquid consists of two parts:

• A normal part that is viscous and carries entropy.
• A superfluid part that is not viscous and does not carry entropy.

In a superfluid ⁴He film, a surface wave called third sound can be excited. Only the superfluid part takes part in the surface wave, because the normal part is stuck to the wall as long as the viscous penetration depth is smaller than the film thickness. Third sound consists of both temperature and film thickness oscillations and can be excited by either a heat pulse or a strong electric field.

Third sound has been studied extensively, mostly on a glass substrate[7]. Subjects of study have been the propagation speed and the damping coefficient, as a function of film thickness and temperature.

To extend these experiments to ³He, the film has to be cooled below 0.93 mK, the ³He superfluid transition temperature. Because it is hard to cool glass to these temperatures, a copper substrate has to be used. Experiments with third sound on a copper substrate, using ⁴He, have been described previously [14]. In these experiments a copper cylinder was used which enabled resonant measuments; in previous experiments of the same group a flat glass surface was used so that only time-of-flight measurements could be performed.

The present work was undertaken to link previous experiments with a copper cylinder to those with glass. For this, an experimental cell that contains both a glass and a copper cylinder was built in order to measure properties, especially third sound speed and attenuation, of third sound on both substrates under the same circumstances.

The experiments described in this report involve the use of a thin gold heater to excite the third sound. Theoretical work on phonon modes and Kapitza resistances at low temperatures and with thin materials was performed as well (Chapter 2).

All symbols used in this paper are given in Appendix B.

1.2  Third sound speed

From the film equations of the two-fluids model the following equation for the third sound speed can be derived[1]:

1.3  Third sound attenuation

The results from this theory, for 4.17 kHz third sound and the infinite geometry, are presented in Figure 1.2 for a glass substrate and in Figure 1.3 for a copper substrate. These calculations were performed using a program written by Brouwer[5]. The minimums in these plots indicate the transition from the thin-film to the thick-film region, where the dominant damping mechanism shifts from thermal to acoustic waves.

1.4  Film thickness

There are several ways to determine the helium film thickness on the walls of a container. The determination can be performed with the use of one of the following parameters:

• The vapour pressure p_v
• The number of ⁴He atoms in the cell N
• The third sound speed c₃.

1.4.1  The vapour pressure p_v

1.4.2  The number of ⁴He atoms in the cell N

1.4.3  The third sound speed c₃

Chapter 2
Phonon modes in thin films recalculated

2.1  Introduction

Therefore an alternative for the standard (Debye) phonon distribution function was developed and tested by Monte Carlo simulation. Furthermore the Kapitza resistance between the film and another material was recalculated using the new phonon distribution function.

2.2  A new phonon distribution function

2.2.1  Crystal dimensions

All calculations will be performed assuming the film is a solid of dimensions V = L_x×L_y×L_z with a cubic lattice with lattice parameter a. There are N = N_x×N_y×N_z atoms in the crystal. The speed of sound in the solid is c. Only longitudinal phonons will be take into account. This treatment can easily be generalized to other phonon polarizations.

The phonon (reciprocal lattice) vectors take the form [k\vec] = (k_x,k_y,k_z) with

Assuming L_z << L_y << L_x it follows that the values of k_x and k_y are much more closely spaced than the values of k_z. Therefore k_x and k_y will be assumed to be continuous, while k_z will be considered discrete.

2.2.2  The density of states over w

Because the circular frequency of a phonon mode [k\vec] equals

As an example, in figure 2.1, the sphere depicts

The number of modes with circular frequency between w and w+dw at the fixed value of k_z now equals

This equation is tested using Monte Carlo techniques (section 2.3). Note that for large values of w equation 2.4 becomes equal to equation 2.2.

2.2.3  Angular dependence of phonon density

The values of [k\vec] do not have a uniform angular distribution because of the discreteness of the values of k_z. Therefore the phonon density of states is not only a function of w (depending on the absolute value of [k\vec] only), but also a function of the phonon directions.

The density of states in every k_x,k_y-plane is [(L_x L_y)/((2 p)²)] (there is one phonon state in every area [(2 p)/(L_x)] ·[(2 p)/(L_y)]). The Dirac d-function must be used because the k_x,k_y-planes lie a distance [(2 p)/(L_z)] apart. The phonon density of states as a function of [k\vec] is

This differential function is transformed into the new coordinates w,q,f with

If the discreteness of the values of k_z is neglected, the density of states as a function of [k\vec] becomes

2.2.4  The density of states over w recalculated as a check

The density of states over w is recalculated to check equation 2.7 by integrating it with respect to q and f.

2.2.5  Taking into account discreteness of k_y

If it is assumed that L_z << L_y << L_x the discreteness of k_y can be taken into account, just like the discreteness of k_z in the previous sections. The phonon density of states as a function of [k\vec] now becomes

In the rest of this chapter, the discreteness of k_ywill not be taken into account. Only the discreteness of k_zwill be taken into account because in the experiments described in Chapters 3 and 4 the effects of the discreteness of k_y are too small to play a role of importance. (With L_y = 0.1 mm, the discrete steps in k_y are sized [(2 p)/(L_y)] = 63  m^-1, which corresponds to (with c = 3390  m s^-1) an energy of E = (^h/_2p) c k_y = 2.3·10^-29 J. This is small compared to the thermal energy at T = 1 K, E = k_B T = 1.4 ·10^-23  J).

2.2.6  The density of states over q

The density of states over q (i.e. the number of phonon states with q between q and q+ dq) is calculated by integrating g(w,q,f) with respect to w and f. Only values of [k\vec] with w_[k\vec] smaller than a given W will be taken into account, because the modes will be occupied with a Bose-Einstein distribution, which doesn't allow a noticable number of phonons above a certain energy. The Bose-Einstein distribution will be used in section 2.4 to calculate the equilibrium occupation of the phonon state at a given temperature and hence the heat flux. In the following treatment the case q = p/2, phonons with k_z = 0 will be excluded. This case will be examined at the end of this section.

Also note that the preceding treatment is only valid for q ¹ [(p)/2], because then cosq = 0. Of course there are a lot of possible modes with q exactly equal to [(p)/2], since this corresponds with the k_z = 0-plane. The number of modes with q = [(p)/2], N_{w < W; q = [(p)/2]; 0 £ f £ 2 p}, can be calculated as follows:

Equations 2.14 and 2.15 will also be checked with a Monte Carlo-simulation in section 2.3.

If the classical phonon distribution function of Equation 2.9 is used, calculation of the density of states over q gives

2.3  Monte Carlo simulations

The idea of using Monte Carlo techniques to check our calculations was first conceived after finding Equation 2.4 and its unexpected discontinuities. It was decided to generate a large number of phonon [k\vec]-vectors of the form [k\vec] = (k_x,k_y,k_z) with k_i = [(2 p)/(L_i)] n_i and to choose a random integer number in the range -[(N_i)/2] ... [(N_i)/2] for each n_i (see section 2.2.1). With a large number of such random [k\vec]-vectors, a good approximation of the phonon distribution function g(w,q,f) will be generated. Next, all these [k\vec]-vectors can be classified with respect to their values of w, q and f to generate, for instance, the density of states over w.

In figures 2.2 to 2.4, g(w) from these Monte Carlo simulations is plotted against w. The material properties have been chosen to resemble a gold strip of dimensions 30 mm × 0.1 mm × 100 nm with c = 3390  m s^-1 and a = 0.257  nm. The three figures plot the same function but on a different w-scale: Figure 2.2 for w < 10¹⁴  s^-1, Figure 2.3 for w < 10¹²  s^-1 and Figure 2.4 for w < 10⁹  s^-1. In Figure 2.2, it is demonstrated that, for w >> c[(2 p)/(L_z)] but w < c [(p)/a] Monte Carlo simulation gives the same results for g(w) as the classical result of equation 2.2. For w > c [(p)/a] the finite crystal dimensions that the classical approximation does not take into account causes a breakdown of g(w). In Figure 2.3 the discrete steps of Equation 2.4 can be seen both in the analytical results and in the Monte Carlo simulation. In Figure 2.4, on an even smaller scale, the discontinuities of Equation 2.12 are shown to comply with the Monte Carlo simulation.

In figure 2.5, the distribution function over theta g_W(q) is plotted. An upper limit W = 1 ·10¹²  s^-1 was used. The figure shows the correspondence between Equation 2.14 and the Monte Carlo simulation. The number of modes with q = p/2 also agrees with the analytical result from Equation 2.15.

2.4  Kapitza resistance

2.4.1  Heat flux

The expression for the heat flux from a thin heater of dimensions V = L_x ×L_y ×L_z, with the interface to the substrate in the z-direction, is (from [12])