Increase in mass of battery of my smartphone

There will be an increase of the mass of the battery when you charge it, though that increase is going to be undetectably small.

I would do the calculation in reverse i.e. start with a fully charged battery and calculate how much it decreases in mass when you run it down. The mass decreases because the battery does work $E$ on the electrons that flow through it and that work is related to the mass lost $\Delta m$ by Einstein's famous equation:

$$ E = \Delta m c^2 \tag{1}$$

We can do an approximate calculation using the information you provided:

Maximum capacity of the smartphone's battery: 2000mAh

That capacity means the battery can supply a current of $2$ amps for an hour, so if the battery voltage is $V$ it can supply a power of $2V$ for an hour and the total energy is therefore:

$$ E = 7200V \tag{2} $$

If we substitute this into equation (1) and rearrange we get:

$$ \Delta m = \frac{7200V}{c^2} $$

Smartphone batteries have $V\approx 4$V and this gives us:

$$ \Delta m \approx 3.2 \times 10^{-13}\,\text{kg} \approx 0.32\,\text{nanograms} $$

This is approximate because the battery voltage isn't constant as the battery discharges so our equation (2) isn't exact. However it gives us a good estimate of how much the mass decreases on discharging and increases again on recharging.

Your rechargeable battery certainly relies on an electrochemical reaction which gives electrical power to your device when going one way, and goes the other way when you provide the necessary amount of energy when charging.

Since you just moved electrons the other way around, there should be no change in mass.