Using transistors to linearise biochemistry C. Toumazou and L. Shepherd A fundamental relationship exists between diffusion characteristics within semiconductors and Nernstian equilibrium in biological systems. In a transistor operating in weak inversion the potential difference between terminals governs electron concentration in an exponential way according to the Boltzmann distribution of charged particles while in a biochemical cell the potential difference across a membrane is governed by ionic concentration in a logarithmic way according to the Nernst equation. These two nonlinear physical phenomena form an interaction that potentially leads to linearisation and subsequent modelling of or interaction with biological systems by integrated semiconductor devices. To demonstrate the authors’ hypothesis a silicon transistor-based biosensor is considered. This natural bridge between biochemistry and semiconductor silicon chips will enable the potential mass production of portable biochemical devices for the consumer market. Diffusion is ubiquitous in Life. Nature’s fundamental processes are governed by the diffusion of charged species and the potentials they generate at equilibrium. From the distribution of matter in the atmo- sphere to the steady state of ions in biological cells, the opposing phenomena of diffusion down a concentration gradient and drift due to a field, be it gravitational or electrical, govern the distribution of particles in thermal equilibrium. For example, in biological cells, passive ion transport under the influence of concentration gradients and electrical forces requires negative work to be done by cells, providing them with a ‘power supply’ or ‘battery’. The Nernst equation quantifies the relation between the electrical potential across an interface and the activity (see Note) of ions on either side. It is derived from the change in free energy (DG c ) resulting from both diffusive and electrical work for transporting one mole of positive ions across the interface: DG c ¼ RT ln ½a þ  i ½a þ  o þ zFE m ð1Þ where R is the gas constant, T is the absolute temperature, [a þ ] i and [a þ ] o are the activities either side of the interface, z is the charge, F is the Faraday constant and E m is the membrane potential. At thermal equilibrium, the change in free energy (DG c ) from the movement of particles across the membrane is zero. Thus, the terms in (1) associated with diffusive work and electrical work are equated, and by simple rearrangement, we derive the Nernst equation [1] E m ¼ RT zF ln ½a þ  i ½a þ  o ð2Þ We show herein how the Nernstian behaviour observed in biochemistry can be linearised using diffusion current in semiconductor devices, leading to increased capability in sensing, modelling and bio-inspired feedback control of biological systems. Potentiometric sensor characteristics: Nernst: Potentiometric sensors exploit Nernstian electrodiffusion to measure the potential generated across a membrane separating two compartments of differing ion concentration (activity). This electrical potential is a log-compression of the ratio of ionic concentrations in each compartment. By keeping the concentration on one side of the membrane constant (using a reference solution), the unknown concentration of ions in an electro- lyte is found to be proportional to the exponential of the potential across the membrane. For the example of a pH-electrode with a membrane sensitive to hydrogen ions (charge z ¼ 1), (2) is commonly rearranged into the form E m ¼ 2:3U T pH þ E const ð3Þ where U T is the ‘thermal voltage’ RT=F, pH ¼ log 10 [H þ ] o and E const is a fixed potential relating to the standard electrode potential of the internal electrolyte solution of constant concentration [H þ ] i . Real-world dynamics depends on ionic concentration though, not on logarithms. For linear systems processing of real-world biochemical signals therefore, we require a linear relation between our system output and concentration. Thus we must reconvert to the ‘concentration domain’ by an exponential expansion of the form: ½H þ ¼½H þ  i exp E m U T   ð4Þ The question of how we can transduce chemical inputs in such a way as to preserve the linearity of concentration information thus arises. The answer, as we shall see, lies in the exploitation of diffusion processes in silicon semiconductor devices, with the prospect of bringing the low- cost and mass-fabrication benefits of the microelectronics industry to the world of biochemical sensing. Silicon device characteristics: Boltzmann: Governed by the Boltzmann distribution, the number of free electrons in the channel of a subthreshold or ‘weak inversion’ field effect transistor (FET), and thus the current which flows, varies with the exponential of the potential applied. V B n+ p+ p-Si n+ V S V G V D depletion layer inversion layer (B) (S) (G) (D) polysilicon SiO 2 Fig. 1 n-channel field effect transistor Potentials applied to the four terminals (bulk, gate, source, drain) control current in the channel below the SiO 2 insulator In a FET, as gate voltage is increased, positive charge is initially repelled from the channel forming a depletion layer with no mobile charge carriers and net negative charge. As the gate voltage is further increased, this depletion layer widens until electrons begin to be drawn from source and drain into the channel, forming an ‘inversion’ layer: the region having been inverted from hole-rich to electron-rich (Fig. 1). The transistor is most commonly operated in the region known as ‘strong inversion’, where the gate voltage is higher than the threshold voltage and the mobile electrons in the inversion layer drift across the electric field in the channel when a potential difference is applied between drain and source. For this mode of operation, drain current is related to the gate voltage by a square law or linear relation. The less common ‘weak inversion’ mode of operation involves maintaining the gate voltage lower than the threshold voltage such that the channel is depleted and only a thin inversion layer of mobile electrons exists. Drain current in weak inversion is due to the diffusion of electrons across the concentration gradient between source and drain. Since the electron concentrations at source and drain and along the channel are related to the barrier potentials at those points by the Boltzmann distribution, it follows that drain current is exponentially related to V S , V D and V G relative to V B , scaled by the thermal voltage U T ¼ kT=q which is equivalent to the term RT=F preferred by chemists [2]: I ¼ I d0 e V G =nU T e V S =U T  e V D =U T   ð5Þ where I d0 is the pre-exponential multiplier and n is the subthreshold slope factor which accounts for capacitive division of potential across inversion and depletion layers. This can be rearranged into the form I ¼ I 0 e V GS =nU T 1  e V DS =U T   ð6Þ where V GS and V DS denote the potential difference between gate and source terminals and drain and source terminals, respectively, and I 0 includes a factor to take into account the potential difference between bulk and source terminals. Clearly, when V DS exceeds four to five times the thermal voltage U T , we have a saturation effect and the equation can be simplified to the form shown in (7): I ¼ I 0 e V GS =nU T ð7Þ Preservation of linearity with complementary nonlinear phenomena: Equation (7) performs the required exponential expansion function proposed in (4). It is clear that if the potential across the channel of the transistor V GS tracks the membrane potential E m of a potentiometric sensor, then the current through the transistor is a suitable output to code ion ELECTRONICS LETTERS 18th January 2007 Vol. 43 No. 2