Absolute Negative Resistance in Ballistic Variable Threshold Field Effect Transistor

Michel I. Dyakonov

Laboratoire de Physique Théorique et Astroparticules
Université Montpellier II, France

Michael S. Shur

ECSE Department and Broadband Center
Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A.

We consider a ballistic variable threshold field effect transistor (BVTFET),^1-4 where the electron transport is ballistic (no collisions with impurities or lattice vibrations), see Fig. 1(a). We will use the hydrodynamic approach,⁵ which is justified if the electron-electron collisions are much faster than collisions with impurities andlor phonons. Qualitatively, our results hold even if this condition is not met, so long as the mean free path is greater than the characteristic length scale for the variation of the electron concentration.

Within this approach, the situation we consider is similar to a water flow through a pipe with varying cross-section, see Fig. l(b) and (c). If there is no flow, obviously the pressure is the same on both sides. In the presence of a stationary water flow, the Bernoulli law (which follows from energy conservation) tells us that the pressure will be lower in the narrow part of the pipe, where the velocity is larger. Note, that this fact does not depend on the direction of the flow. This means that when water flows from the narrow part towards the wide part, the direction of the flow is locally opposite to the pressure gradient. One can say that the interface region has an absolute negative resistance. We will show below, that a similar effect should exist for the BVTFET.

To be specific, we assume, as shown in Figure 1, that the device threshold voltages for the section with coordinates 0 < x < L₁, and L₁ < x < L are V_T1 and V_T2, respectively (with V_T2 < V_T1) and that the source contact at x = 0 is grounded. The gate and drain voltages are V_G and V_D, respectively. The voltage drop in the ballistic sections of the device is zero, so the entire voltage drop occurs at the interface between the two sections at x = L₁ , see Fig. 1(a). We will treat this interface as abrupt, assuming that the Debye length and the separation between the gate and the channel are small compared to L₁.

The basic equations describing this device are the equation of motion and the continuity equation,

as well as the equations relating the concentrations n to the local gate-to-channel voltage swing in the gradual channel approximation:

Here C is the gate-to-channel capacitance, e is the electron charge, V_G is the gate voltage, V_T1 and V_T2 are the threshold voltages for the left and right sides of the transistor respectively, V_D = V₁ − V₂ is the source-drain potential difference, V₁ and V₂ are the channel potentials on the left and on the right side of the device.

Figure 1.  (a) Electron concentration distribution in a variable threshold field effect transistor; (b) a water pipe with varying cross section. In the absence of flow the pressures on both sides are obviously equal; (c) the same water pipe in the presence of flow – the pressure in the narrow part of the pipe is lower (Bernoulli law), regardless of the flow direction.

In the stationary situation that we are considering, Eqs. (la) and (1b) yield:

where v₁ and v₂ are the left and right electron drift velocities, The first of Eqs. (4) is analogous to the Bernoulli law in hydrodynamics.

We denote by n₁₀ and n₂₀ the equilibrium electron concentration on the two sides of the interface when the current is absent. The concentration. velocity, and gate voltage swing profiles for the case n₁₀ < n₂₀ are shown in Fig. 2. We will be interested also in the opposite case, when n₁₀ > n₂₀. We will consider the situation when the left side of the device with the electron concentration n₁₀ is grounded (V₁ = 0). Then the concentration on this side is always fixed: n₁= n₁₀. while the concentration on the right side, n₂, will vary with changing current.

Figure 2.  Concentration, velocity, and gate voltage swing profiles in BVTFET for the case n₁₀ < n₂₀. Obvious modifications should be made for the opposite case.

If n₁₀ < n₂₀ and v₁ > v₂ (see Figure 2), V_D is negative (meaning that the high concentration side has a more negative potential than the low concentration side with the electron concentration n₁). In the opposite case, when n₁₀ > n₂₀ and v₁ < v₂, V_D is positive.

We now introduce dimensionless variables U for the source-drain voltage and J for the current, together with a dimensionless parameter N defined as the ratio of the carrier concentrations in the right section of the device to that in the left section in equilibrium (i.e. at zero current):

U ≡ C|V_D|/en_l0, J ≡ V₁/V₀, N ≡ n₂₀/n₁₀ = 1 + C(V_T1 – V_T2)/en₁₀,

where the characteristic electron velocity v₀ is given by:

We find from Eqs. (2)–(4) for negative V_D:

and for positive V_D, respectively:

For typical values of an AIGaAs/GaAs ballistic HEMT, we can assume V_G = 0, V_T2 = −1 V, V_T1, = −0.5 V, m = 0.067m₀ (where m₀ is the free electron mass), and C = 5×10^-7 F/cm² (corresponding to approximately 20 nm gate-to-channel separation). These values yield a characteristic current density of approximately 4 μA/μm.

In the dimensionless units introduced above, Eqs. (5) and (6) can be rewritten as:

For N > 1 (i.e. n₁₀ > n₂₀), V_D is negative and increases the carrier concentration in the high-concentration section of the device. As seen from Eq. (7a), for N > 1 at large U, J² is proportional to U. At small U << N, the current-voltage characteristic for both cases becomes:

Since typical current densities achieved in AIGaAs/GaAs HEMTs correspond to the range of J between 0.01 and 0.2, in many cases, Eq. (8) should apply.

Figure 3.  Dimensionless current-voltage characteristics of BVTFET for N = 1.01, 1.1, 1.6, and 10 plotted on linear (top) and double log (bottom) scales.

The resulting current-voltage characteristics for are shown in Fig. 3. For n₁₀ < n₂₀, the carrier concentration in the right section of the device increases proportionally to the absolute value of V_D, as seen from Eq. 6(a). The dimensionless electron velocity in this section is given by

For n₁₀ < n₂₀, velocity v₂ reaches a maximuin value at:

Figure 4.  Dependence of v₂/v₀ on U for N = 1.01, 1.l, 1.6 and 10 plotted on linear (top) and double log (bottom) scales.

Figures 4(a) and 4b show the dependence of v₂/v₀ on U for n₁₀ < n₂₀. In the opposite bias polarity, when n₁₀ > n₂₀, we have N < 1 and V_D is positive. The resulting current-voltage characteristics for n₁₀ > n₂₀ obtained from Eq. (7b) are shown in Fig. 5, where we also superimpose the dependence of the choking current J_CH on the choking voltage U_CH (found from the requirement that the electron velocity in region 2 for n₁₀ > n₂₀, N < 1 is equal to the plasma wave velocity in this region)⁶

As can be seen, the maxima of the current-voltage characteristics at N > 0.75 correspond to the choking current.

Figure 5.  Dimensionless current-voltage characteristics of BVTFET for n₁₀ > n₂₀ and N = 0.5, 0.75 and 0.99 plotted on linear (a) and double log (b) scales. The choking current dependence on the choking voltage is also shown.

Figure 6 shows the dimensionless electron velocity in region 2 as a function of U for N < 1, where we also superimpose the dependence of the choking velocity on the choking voltage (found from the requirement that the electron velocity in region 2 for n_l0 > n₂₀, N < 1 is equal to the plasma wave velocity in this region):

The physical meaning of the obtained results is as follows. The electrons crossing the boundary at x = L₁ must keep their energy. which can be only achieved by creating a dipole layer at the boundary leading to the voltage drop that changes the electron concentration in one section of the device and adjusts the electron velocities in both device sections accordingly. In the device section with the fixed potential, the carrier concentration is independent of the current. The other section accommodates any change in current by adjusting the carrier concentration. If the concentration in this section is higher, it increases further with an increase in the device current. If the concentration in this section is smaller, it gets depleted with an increase in the device current. The striking result is that this voltage drop is independent of the direction of the current, so that, depending on the direction of the current, the device should exhibit equal positive or absolute negative resistance. The consequences of this absolute negative resistance will be discussed elsewhere.

Figure 6.  Dimensionless velocity v₂/v₀ in section 2 for n₁₀ > n₂₀, for N = 0.5, 0.75 and 0.99 vs. U. The choking velocity dependence on the choking voltage is also shown.

Acknowledgments

This work was supported in part by ONR under EMMA MURI contract (project manager Dr. Colin Wood).

References

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