Change Wordpress login method

Create a file named force-email-login.php containing the plugin script below and put it in the wp-content/plugins folder. This will force you to log in with the email (tested by me in version 4.1):

<?php
/*
Plugin Name: Force Email Login
Author: Takayuki Miyauchi
Plugin URI: https://github.com/miya0001/force-email-login
Description: Use email address for login to your WordPress.
Version: 0.5.0
Author URI: https://github.com/miya0001/
*/
$force_email_auth = new Force_Email_Auth();
$force_email_auth->register();
class Force_Email_Auth {
    function register()
    {
        add_action( 'plugins_loaded', array( $this, 'plugins_loaded' ) );
    }
    public function plugins_loaded()
    {
        remove_filter( 'authenticate', 'wp_authenticate_username_password', 20, 3 );
        add_filter( 'authenticate', array( $this, 'authenticate'), 20, 3 );
    }
    public function authenticate( $user, $username, $password )
    {
        if ( is_a( $user, 'WP_User' ) ) {
            return $user;
        }
        if ( ! empty( $username ) && is_email( $username ) ) {
            $user = get_user_by( 'email', $username );
            if ( isset( $user, $user->user_login, $user->user_status ) ) {
                if ( 0 === intval( $user->user_status ) ) {
                    $username = $user->user_login;
                    return wp_authenticate_username_password( null, $username, $password );
                }
            }
        }
        if ( ! empty( $username ) || ! empty( $password ) ) {
            return false;
        } else {
            return wp_authenticate_username_password( null, "", "" );
        }
    }
}

Source: link

With the function below you can log in with the email or the password. Just put the code in functions.php

//remove wordpress authentication
remove_filter('authenticate', 'wp_authenticate_username_password', 20);

add_filter('authenticate', function($user, $email, $password) {

    //Check for empty fields
    if (empty($email) || empty($password)) {
        //create new error object and add errors to it.
        $error = new WP_Error();

        if (empty($email)) { //No email
            $error->add('empty_username', __('<strong>ERROR</strong>: Email field is empty.'));
        } else if (!filter_var($email, FILTER_VALIDATE_EMAIL)) { //Invalid Email
            $error->add('invalid_username', __('<strong>ERROR</strong>: Email is invalid.'));
        }

        if (empty($password)) { //No password
            $error->add('empty_password', __('<strong>ERROR</strong>: Password field is empty.'));
        }

        return $error;
    }

    //Check if user exists in WordPress database
    $user = (get_user_by('email', $email))? : get_user_by('login', $email);

    //bad email
    if (!$user) {
        $error = new WP_Error();
        $error->add('invalid', __('<strong>ERROR</strong>: O email ou o password é inválido.'));
        return $error;
    } else { //check password
        if (!wp_check_password($password, $user->user_pass, $user->ID)) { //bad password
            $error = new WP_Error();
            $error->add('invalid', __('<strong>ERROR</strong>: Either the email or password you entered is invalid.'));
            return $error;
        } else {
            return $user; //passed
        }
    }
}, 20, 3);