Voltage switching and domain relocation in semiconductor superlattices

A numerical study of domain wall relocation during voltage switching with different ramping times is presented for weakly coupled, doped semiconductor superlattices exhibiting multistable domain formation in the first plateau of their current-voltage characteristics. Stable self-oscillations of the current at the end of stable stationary branches of the current-voltage characteristics have been found. These oscillations are due to periodic motion of charge dipoles near the cathode that disappear inside the SL, before they can reach the receiving contact. Depending on the dc voltage step, the type of multistability between static branches and the duration of voltage switching, unusual relocation scenarios are found including changes of the current that follow adiabatically the stable $I\text{−}V$ branches, different faster episodes involving charge tripoles and dipoles, and even small amplitude oscillations of the current near the end of static $I\text{−}V$ branches followed by dipole-tripole scenarios.

Figure 1

Tunneling current density for $n_i=n_{i+1}=N_D$ as a function of field $F_i=F$, showing that $F_M\approx 3.945\mathrm{kV}∕\mathrm{cm}$ and $J_M\approx 3.1269\mathrm{A}∕{\mathrm{cm}}^2$ for $T=5\mathrm{K}$ and the SL values of Table .

Figure 2

(Color online) $I\text{−}V$ characteristics of the 40-period 9-4 SL of Ref. 8 obtained by up- and down-sweeping adiabatic processes for $V∊[0,4]\mathrm{V}$. Parameters correspond to Table at $5\mathrm{K}$ and with cathode resistivity of $25.2\Omega \mathrm{m}$ $(\tilde{\sigma }=0.5)$. The branch number increases with voltage: the $i$ th branch has a CAL separating low and high field domains which is located at the $(N-i+1)\text{th}$ well.

Figure 3

(Color online) Bistability between branches 5 and 6 of the $I\text{−}V$ characteristics in Fig. 2 $(V∊[0.55,0.6]\mathrm{V})$. The stationary solution becomes unstable to a small-amplitude oscillatory solution at the upper end of branch 5.

Figure 4

(Color online) Total current density during voltage switching from $V_i=0.83\mathrm{V}\text{to}{V}_f=1.37\mathrm{V}$ (seven branches) for ramping times: (a) ${\tau }_r=30\mu \mathrm{s}$ (dimensionless value, ${\tilde{\tau }}_r=2\times {10}^4$), and (b) ${\tau }_r=15\mu \mathrm{s}$ $({\tilde{\tau }}_r={10}^4)$. Thick black line, upper part of the $I\text{−}V$ branches; red thin line, lower part of the static $I\text{−}V$ branches; thin blue line, response curve $(V\left(t\right),J\left(t\right))$. In (a), the current follows adiabatically the $I\text{−}V$ curve, whereas in (b), the ramping time is so short that the fourth and sixth branches are skipped. The dimensionless cathode conductivity is $\tilde{\sigma }=0.5$.

Figure 5

Time trace of the total current density $J\left(t\right)$ after sudden voltage switching characterizing the tripole-dipole scenario. Here $V_i=1.0\mathrm{V}$ (for $t<0$) and $\Delta V=0.2\mathrm{V}$.

Figure 6

Field profiles corresponding to Fig. 4 during relocation of the domain wall. (a) Detail of the emission and the travel of the CAL during the second relocation stage in Fig. 4a. (b) The fourth branch in Fig. 4b has been skipped, therefore the fourth CAL is missing and there is no relocation stage.

Figure 7

(Color online) Voltage switching with $\Delta V=0.5\mathrm{V}$ and ramping time ${\tau }_r=30\mu \mathrm{s}$ for different $V_i$ on the $I\text{−}V$ characteristics: (a) $V_i=0.5\mathrm{V}$, (b) $V_i=1\mathrm{V}$, (c) $V_i=1.5\mathrm{V}$, (d) $V_i=2\mathrm{V}$, (e) $V_i=2.5\mathrm{V}$, (f) $V_i=3\mathrm{V}$, (g) $V_i=3.5\mathrm{V}$, (h) $V_i=4\mathrm{V}$, $\Delta V=1.5\mathrm{V}$. In (a), (b), (g), and (h), the current follows the $I\text{−}V$ curve along the upper part of each branch, whereas $J$ ranges from $6\text{to}16\mathrm{A}∕{\mathrm{cm}}^2$. In (c), a branch (the fifth) is skipped for the first time. In (d) and (e), branches 3 and 5 are skipped. In (f), the upper part of the fifth branch is reached but the sixth is skipped.

Figure 8

Current response to voltage switching from $V_{\mathrm{i}}=3.68\mathrm{V}$ to five different values of $V_f$ near $V_{\text{th}}$ [(1) 3.74, (2) 3.747, (3) 3.747 2732, (4) 3.7474, and (5) $3.76\mathrm{V}$]. The ramping time is ${\tau }_r=100\mathrm{ns}$ and we have set $t=0$ at the end of voltage switching. Note the oscillatory behavior for $V_f=3.747\mathrm{V}$.

Figure 9

Current response to voltage switching for $V_i=3.5\mathrm{V}$, two different values of $V_f$ and three different ramping times in each case: (a) $V_f=3.555\mathrm{V}>V_{\text{th}}$, for ${\tau }_r=1\mathrm{ns}$, $1\mu \mathrm{s}$, and $100\mu \mathrm{s}$. (b) $V_f=3.5545\mathrm{V}<V_{\text{th}}$, for ${\tau }_r=0.1\mu \mathrm{s}$, $1\mu \mathrm{s}$, and $100\mu \mathrm{s}$. In all cases, we set $t=0$ at the end of voltage switching. For ${\tau }_r=100\mu \mathrm{s}$, the current undergoes a small-amplitude oscillation in the case of Fig. 9b.

Figure 10

Evolution of the 41 eigenvalues determining linear stability of the static branch $B_{35}$ for 20 values of the applied voltage from $V=3.54\text{to}3.56\mathrm{V}$.

Figure 11

(Color online) Current response to voltage switching between branches 5 and 6 of the $I\text{−}V$ characteristics in Fig. 2 for $V_i=0.56\mathrm{V}$, ramping time ${\tau }_r=10\mu \mathrm{s}$, and four different values of $V_f$ near $V_{\text{th}}$: (1) 0.59, (2) 0.591, (3) 0.59203, and (4) $0.593\mathrm{V}$. Note that curve (3) is shifted down $0.2\mathrm{A}∕{\mathrm{cm}}^2$ for the sake of clarity.

Figure 12

(Color online) Current response to voltage switching between branches 5 and 6 of the $I\text{−}V$ characteristics in Fig. 2 for $V_i=0.56\mathrm{V}$. (a) $V_f=0.593\mathrm{V}>V_{\text{th}}$ and seven different ramping times from $1\text{to}25\mu \mathrm{s}$. (b) Details of current response for $V_f=0.59203\mathrm{V}<V_{\text{th}}$ and three short ramping times: 0.1, 2.1994, and $2.1995\mu \mathrm{s}$.

Figure 13

Evolution of the electric field profile corresponding to Fig. 12b, where $V_i=0.56\mathrm{V}$ and $V_f=0.59203\mathrm{V}<V_{\text{th}}$, for the two (very similar) ramping times (a) ${\tau }_r=2.1994\mu \mathrm{s}$ and (b) ${\tau }_r=2.1995\mu \mathrm{s}$. In (a), the CAL is emitted after a short oscillatory transient. In (b), a CAL is emitted from the cathode, it disappears in the interior of the sample and this behavior is repeated periodically. This is similar to Gunn effect oscillations confined to one part of the SL.

Figure 14

Evolution of the electric field profile corresponding to curve number (6) in Fig. 12a, which has longest oscillatory interval before a CAL is emitted from the cathode (ramping time ${\tau }_r=20\mu \mathrm{s}$). (a) Detail of the oscillatory transient regime. (b) Detail of the profile near the cathode (closest 15 wells) when the CAL is finally emitted.

Figure 15

(a) Evolution of the 41 eigenvalues determining linear stability of the static branch $B_5$ for 20 values of the applied voltage from $V=0.56\text{to}0.6\mathrm{V}$. The insets show the overall picture with all the eigenvalues and a zoom of the region near the imaginary axis. (b) Real part of the eigenvalues as a function of voltage. The gap in the figure corresponds to the oscillatory instability.