Tunneling effect

The electron tunneling effect is a purely quantum-mechanical effect and first theoretical studies were published in the early 1930s [117]. Although the TMR effect was discovered early in 1975 [73], it took two more decades, and the discovery of the Giant Magneto Resistance (GMR) [5,12], until the interest in the TMR effect grew quickly.

Figure 1.13: Tunneling in metal/insulator/metal (M/I/M) structures, from [84]

[Wave function $\Psi (x)$ for electron tunneling through an insulating barrier. While the barrier would classically be forbidden, quantum-mechanically the wave function decays exponentially and, for thin barriers $\Psi (x)$, is attenuated beyond the barrier.]         [Sketch of a M/I/M model with an applied bias Voltage $eV$. The barrier height $\nu$, the thickness b and the asymmetry d$\nu$ can be obtained from a BRINKMANn fit.]

Figure 1.13(a) shows the wave function $\Psi (x)$ of two electrodes separated by an insulating barrier. Although classically forbidden, a part of the wave function continues beyond the barrier. Because the wave function has to be continuous, it decays exponentially within the insulator. If the barrier is too thick, the wave function vanishes beyond the barrier.

In a MTJ the electrons tunnel only through thin insulating barriers (only a few nanometers thick) and, therefore, a reasonable tunneling current can be measured. Those metal/insulator/metal systems are mostly analysed by measuring the current/voltage ($I/V$) characteristic. Figure 1.13(b) shows a sketch of such a system. The Fermi-levels $E_{\rm F}$ of the two metals are shifted because of the applied bias Voltage $eV$. The tunneling through an insulator mainly depends on the the density of states (DOS) in the left and right electrode. The current from the left to the right electrode can be written as:

$$I_{l\rightarrow r}(E) = \int_{-\infty}^\infty \rho_l(E)\c... ..._r(E + eV)\cdot \vert T(E)\vert^2\cdot f(E)\cdot(1-f(E+eV)) dE$$ (1.11)

where $\rho_l$ is the DOS in the left electrode at energy $E$ and $\rho_r$ is the DOS at the same energy plus the applied bias voltage. $\vert T(E)\vert^2$ is the probability of transmission through the barrier and $f(E)$ the Fermi-Dirac function. Because electrons also tunnel in the other direction, you have to subtract the current from right to left to get the total current $I_{total} = I_{l\rightarrow r} - I_{r\rightarrow l}$.

While the easiest way to get the properties of the barrier is the Simmons-fit [116], which assumes a rectangularly shaped barrier, in this thesis the more elaborate BRINKMAN-fit is used. BRINKMAN et al. [17] used the WKB-approximation to numerically calculate the transmission probability $\vert T\vert^2$ for a trapezoidally shaped barrier. The first terms of the WKB-approximation give for the conductance: $G= A\cdot V^2+B\cdot V+C$. So when the conductance is measured, the barrier parameters can be obtained by fitting the parameters A,B and C:

$$\nu = \frac{e^2C}{32A}\ln^2\left(\frac{h^3}{\sqrt{2}\pi e^3m_{\rm eff}}\sqrt{AC}\right)$$ (1.12)

$$b = - \frac{\hbar}{2\sqrt{2 m_{\rm eff}\nu}} \ln\left(\frac{h^3}{\sqrt{2}\pi e^3m_{\rm eff}}\sqrt{AC}\right)$$ (1.13)

$$d\nu = - \frac{12\hbar\nu^{\frac{3}{2}}B}{\sqrt{2m_{\rm eff}}ebC}$$ (1.14)

with the effective electron mass $m_{\rm eff}$ set to 0.4 [16]. Although BRINKMANS approach neglects any dependence of the transport characteristics on the DOS of the electrodes, equation 1.13 provides good results for the barrier thickness. In the experiments, the $I/V$ curve of a MTJ is measured, numerically differentiated and fitted with a standard code [104].