Stability of metallic thin films studied with a free electron model

The stability of metallic thin films is studied with a free electron model, which is popularly known as the model of “particle in a box.” A detailed theoretical framework is presented, along with discussion on typical metals, such as Pb, Al, Ag, Na, and Be. This simple model is found to be able to describe well the oscillation pattern of stability for continuous metallic films. In particular, it yields even-odd oscillations in the stability of Pb(111) film, consistent with both experimental observation and ab initio results. However, the free electron model is too crude to predict at what thickness a film is stable. Film stability is further examined with a model of “particle in a corrugated box,” where a lattice potential is added along the vertical direction of a film. The effect of lattice potential is found not substantial.

Figure 1

Energy subbands of a metallic thin film.

Figure 2

Energy barriers at interfaces. The barrier at the vacuum side has a height of $E_F+W_m$; the barrier height $V_0$ at the substrate side depends on the specifics of the substrate material.

Figure 3

Continuous thin film. The top is a rough film before annealing; after annealing, the film may become atomically flat (bottom left) or a film of two different heights (bottom right). In this case, the contact area between the film and the substrate is conserved during morphology evolution.

Figure 4

Well-separated islands in a thin film. After annealing, the islands may evolve to have the same height as indicated by dashed-lined objects. As the total volume of the film is conserved, the total contact area of the islands with the substrate is changed.

Figure 5

(a) Number of subbands below the Fermi surface, (b) Fermi energy $E_f$, and (c) energy per electron $E_a$ as a function of $\kappa$ for a metallic thin film. Interfaces are modeled as infinite energy barriers. Insets show the enlarged portions of $E_f$ and $E_a$ for $\kappa ⪢1$. The dashed lines indicate the correspondence between the emergence of a new subband and a cusp.

Figure 6

Film interface energy $E_s$ as a function of $\kappa$. Interfaces are modeled as infinite energy barriers. The inset shows the enlarged portion of $E_s$ for large $\kappa$. The diamonds are the results for Pb(111) film. $E_s$ is in units of $E_F{k}_F^2∕4\pi$.

Figure 7

Film interface energy $E_s$ as a function of layer number $L$ for metallic films: Be(0001), Al(111), Pb(111), Ag(111), and Na(110). $E_s$ is in units of $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 8

The second difference $d^{2}E$ as a function of layer number for metallic films: Be(0001), Al(111), Pb(111), Ag(111), and Na(110). $d^{2}E$ is in units of $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 9

Symmetric well of finite energy barriers.

Figure 10

Fermi energy $E_f$ of thin films. The solid lines are for $V_0=2.2E_F$, $V_0=1.8E_F$, and $V_0=1.4E_F$ from top to bottom, respectively. The dashed curve is for the case of infinite barriers. The inset shows an enlarged portion of $E_f$ curves.

Figure 11

Interface energy $E_s$ of thin films. The solid lines are for $V_0=2.2E_F$, $V_0=2.0E_F$, $V_0=1.8E_F$, $V_0=1.6E_F$, and $V_0=1.4E_F$. The dashed curve is for the case of infinite barriers and is shifted downward by a trivial constant for easy comparison. The unit of $E_s$ is $E_F{k}_F^2∕4\pi$.

Figure 12

Energy per electron $E_a$ of thin films. The solid lines are for $V_0=2.2E_F$, $V_0=1.8E_F$, and $V_0=1.4E_F$ from top to bottom, respectively. The dashed curve is for the case of infinite barriers.

Figure 13

Interface energy $E_s$ and its second difference for a freestanding Be(0001) thin film. For Be, $W_m=4.98\mathrm{eV}$ and $V_0=1.35E_F$. The open triangles with dashed lines are for infinite barriers. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 14

Interface energy $E_s$ and its second difference for a freestanding Na(110) thin film. For Na, $W_m=2.75\mathrm{eV}$ and $V_0=1.85E_F$. The open squares with dashed lines are for infinite barriers. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 15

Interface energy $E_s$ and its second difference for a freestanding Ag(111) thin film. For Ag, $W_m=4.3\mathrm{eV}$ and $V_0=1.78E_F$. The open triangles with dashed lines are for infinite barriers. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 16

Interface energy $E_s$ and its second difference for a freestanding Pb(111) thin film. For Pb, $W_m=4.25\mathrm{eV}$ and $V_0=1.45E_F$. The open diamonds with dashed lines are for infinite barriers. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 17

Interface energy and its second difference of a freestanding Al(111) thin film. For Al, $W_m=4.28\mathrm{eV}$ and $V_0=1.37E_F$. The open circles with dashed lines are for infinite barriers. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 18

Interface energy $E_s$ and its second difference of Pb(111) thin films on semiconductor substrates. From top to bottom in (a), the results are for $V_2=1.45E_F=V_1$ (open diamonds), $V_2=1.34E_F$ (solid diamonds), $V_2=1.22E_F$ (open triangles), and $V_2=1.11E_F$ (solid triangles). The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 19

Corrugated potential well.

Figure 20

Subband energy in Pb(111) film with a corrugated potential. (a) 5 monolayers, (b) 12 monolayers, and (c) 20 monolayers. The dashed line represents the Bloch bands without any confinement. $v=0.3E_F$ in Eq. (29). The Bloch wave vector $k$ is in units of $\pi ∕{\kappa }_0$ with ${\kappa }_0=k_F{d}_0∕\pi$.

Figure 21

Interface energy of Pb(111) film. The discrete set of diamonds are for real Pb(111) film, whose thickness changes only monolayer by monolayer. The dashed line is for the case where the film thickness is allowed to change continuously. $v=0.3E_F$. The unit of $E_s$ is $E_F{k}_F^2∕4\pi$.

Figure 22

Interface energy and its second difference of Pb(111) film. (a) Surface energy $E_s$ for $v=0.0$, $v=0.2E_F$, and $v=0.4E_F$ from top to bottom. (b) The second difference $d^{2}E$. The unit is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 23

Interface energy and its second difference of Al(111) film. (a) Surface energy $E_s$ for $v=0.0$, $v=0.2E_F$, and $v=0.4E_F$ from top to bottom. (b) The second difference $d^{2}E$. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 24

Interface energy and its second difference of Be(0001) film. (a) Surface energy $E_s$ for $v=0.0$, $v=0.2E_F$, and $v=0.4E_F$ from top to bottom. (b) The second difference $d^{2}E$. The unit of $E_s$ is $\mathrm{eV}∕{\mathrm{\AA }}^2$.

Figure 25

Interface energy and its second difference of Pb(111) film. (a) Surface energy and (b) its second difference $d^{2}E$. Open squares are for infinite barriers; solid circles are for the free standing case with $V_0=1.45E_F$. $v=0.1E_F$. The unit is $\mathrm{eV}∕{\mathrm{\AA }}^2$.