Now my question is : is the applied voltage same as the voltage across
the diode?
In the equation, the term \$V_D\$ is the voltage that's across the diode's terminals. Note that you can either
- use a voltage source to apply a constant voltage of \$V_D\$ Volts across the diode's terminals and then measure the current \$I_D\$ that's flowing through the diode, or
- use a current source to inject a constant current of \$I_D\$ Amps through the diode and then measure the voltage \$V_D\$ across the diode's terminals.
For what it's worth, the commonly-used form of the Shockley diode equation (as shown in your question) does not make evident that
- the value of the reverse saturation current term \$I_S\$ depends upon the diode's junction temperature T—i.e., \$I_S\$'s value changes as the diode's junction temperature T changes, and
- \$I_S\$ plays a critical role in determining how the diode's current changes \$\partial I_D\$ as the diode's junction temperature changes \$\partial T\$—i.e., the rate \$\partial I_D / \partial T\$ is heavily dependent upon \$\partial I_S / \partial T\$.
For the sake of argument, let's assume \$I_S\$'s value is constant (more-or-less), and the values for \$q\$, \$V_D\$, \$\eta\$, and \$k\$ are all constants. In fact, for simplicity, let's define a new constant \$x\$ whose value is \$x=(q\,V_D/\eta\,k)\$, so that
$$
I_D = I_S(e^{x/T}-1)\;\;\;\;\;\;\;\;(1)
$$
For the conditions mentioned above, Eqn. (1) predicts that as the diode's junction temperature T increases, the current through the diode \$I_D\$ tends to zero:
$$
\lim_{T \to \infty}I_S(e^{x/T}-1)\\
\Rightarrow I_S \cdot (e^{x/\infty}-1)\\
\Rightarrow I_S \cdot (e^{0}-1)\\
\Rightarrow I_S \cdot (1-1)\\
\Rightarrow I_S \cdot (0)\\
\Rightarrow 0
$$
Of course in a real diode, if \$q\$, \$V_D\$, \$\eta\$, and \$k\$ are more-or-less constant, we know (from making measurements with test equipment) that the diode's current \$I_D\$ increases as its junction temperature T increases; this contradicts the result shown above which says that \$I_D\$ should decrease toward zero as the junction temperature T increases.
Therefore, we can conclude that the \$I_S\$ term a) cannot be constant-valued, and b) must be the term in the Shockley diode equation that causes the diode current \$I_D\$ to increase as the junction temperature T increases. In other words, the term \$(e^{(q\,V_D/\eta\,kT)}-1)\$ tries to decrease \$I_D\$ as T rises, and the term \$I_S\$ tries to increase \$I_D\$ as T rises, and for a given change in junction temperature \$\partial T\$ the following relation must hold if \$I_D\$ is to increase for increasing T:
$$
\left | \frac{\partial I_S}{\partial T} \right |
>
\left | \frac{\partial}{\partial T} \left (e^{(q\,V_D/\eta\,k\,T)}-1 \right ) \right |
$$
i.e., for a given change in junction temperature \$\partial T\$, the \$I_S\$ term has greater influence on the change in diode current \$\partial I_D\$ compared to the \$(e^{(q\,V_D/\eta\,k\,T)}-1)\$ term.
For what it's worth, Eqn. (2) is a commonly used formula (model) for calculating the reverse saturation current term \$I_S\$ as a function of junction temperature T:
$$
I_S = I_K \cdot e^{(-q\,E_g/\eta\,k\,T)}\;\;\;\;\;\;\;\;(2)
$$
And so an improved model for the diode current \$I_D\$ would be Eqn. (3):
$$
I_D = (I_K \cdot e^{(-q\,E_g/\eta\,k\,T)}) \cdot (e^{(q\,V_D/\eta\,k\,T)}-1)\;\;\;\;\;\;\;(3)
$$
For more information here's a useful reference that provides descriptions for the terms \$I_K\$ and \$E_g\$ in Eqn. (2).
